Basic Operations on Supertropical Quadratic Forms

Zur Izhakian and Manfred Knebusch^{0}

Received: November 19, 2016 Revised: September 6, 2017

Communicated by Ulf Rehmann

Abstract. In the case that a module V over a (commutative) supertropical semiring R is free, the R-module Quad(V) of all quadratic forms on V is almost never a free module. Nevertheless, Quad(V) has two free submodules, the module QL(V) of quasilinear forms with base 𝔇_{0} and the module Rig(V) of rigid forms with base ℌ_{0}, such that Quad(V) = QL(V) + Rig(V) and QL(V) ∩ Rig(V) = {0}.

In this paper we study endomorphisms of Quad(V) for which each submodule Rq with q ∈ 𝔇_{0} ∪ ℌ_{0} is invariant; these basic endomorphisms are determined by coefficients in R and do not depend on the base of V. We aim for a description of all basic endomorphisms of Quad(V), or more generally of its submodules spanned by subsets of 𝔇_{0} ∪ ℌ_{0}. But, due to complexity issues, this naive goal is highly nontrivial for an arbitrary supertropical semiring R. Our main stress is therefore on results valid under only mild conditions on R, while a complete solution is provided for the case that R is a tangible supersemifield.

2010 Mathematics Subject Classification: Primary 15A03, 15A09, 15A15, 16Y60; Secondary 14T05, 15A33, 20M18, 51M20

Keywords and Phrases: Tropical algebra, supertropical modules, bilinear forms, quadratic forms, quadratic pairs, minimal ordering, unique base property.

0The research of the first author has been supported by the Research Councils UK (EP-SRC), grant no EP/N02995X/1.

The authors thank the referee for many helpful suggestions.

1. Introduction

We continue a study of quadratic forms and modules over semirings, begun in [8] and [10], where now we face a general problem over the so called supertropical semrings, as explained in §1.3 and §1.7 below. Exhibiting the contribution of the present paper, our approach is indicated in §1.8. For the reader’s convenience we first recall basic terminology and results, mainly from [8] and [10], but also from other sources.

1.1. Modules over a semiring. A (commutative) semiring R is a set R equipped with addition and multiplication such that (R, +, 0) and (R, · , 1) are abelian monoids with natural elements 0 := 0_{R} and 1 := 1_{R} respectively, and multiplication distributes over addition in the standard way. In other words, R satisfies all the properties of a commutative ring except the existence of negation under addition. R is called a semifield if every nonzero element of R is invertible, i.e., R \ {0} is an abelian group. A module V over a semiring R (called also a semimodule) is an abelian monoid (V, +, 0_{V}) equipped with a scalar multiplication R × V → V, (a, v) ↦ av, such that all the customary axioms of modules over a ring are satisfied: a_{1}(bv) = (a_{1}b)v, (a_{1} + a_{2})v = a_{1}v + a_{2}v, a_{1}(u + v) = a_{1}u + a_{1}v for all a_{1}, a_{2}, b ∈ R, u, v ∈ V. We usually write 0 for both 0_{R} and 0_{V}, and 1 for 1_{R}, and often speak about elements of V as “vectors” and elements of R as “scalars”.

1.2. Quadratic forms on a free module. For any module V over a semiring R, a quadratic form on V is a function q : V → R with

q(ax) = a^{2}q(x)

for any a ∈ R, x ∈ V, together with a symmetric bilinear form b : V × V → R such that

(1.1)  q(x + y) = q(x) + q(y) + b(x, y)

for any x, y ∈ V. Here “symmetric bilinear” has the obvious meaning, but – in contrast to the case where R is a ring – b is often not uniquely determined by q. We call such b a companion of q, or say that b accompanies q. In this paper we assume throughout that V is a free R-module with base (ε_{i} | i ∈ I), i.e., every vector x ∈ V is a linear combination

(1.2)  x = ∑_{i∈I} x_{i}ε_{i}

with unique family of scalars (x_{i} | i ∈ I) ⊂ R, only finitely many x_{i} ≠ 0, called the coordinates of x. Then, after choosing a companion b of q, a quadratic form q : V → R can be written as (for notational convenience we choose a total order on I):

(1.3)  q(x) = ∑_{i∈I} α_{i}x_{i}^{2} + ∑_{i<j} α_{ij}x_{i}x_{j},

where α_{i} = q(ε_{i}) and α_{ij} = b(ε_{i}, ε_{j}), cf. [8, §1].

Basic Operations on Supertropical Quadratic Forms 1663

Although the case that I is infinite is relevant for applications, we assume in this and the next introductory subsection that I = {1, …, n} is finite, for simplicity. Then, as customary, the presentation (1.3) of a quadratic from q is written as a triangular scheme

(1.4)  q = [\begin{matrix}α_{1} & α_{12} & ⋯ & α_{1n} \\ & α_{2} & ⋯ & α_{2n} \\ & & ... & ⋮ \\ & & & α_{n} \end{matrix}]

using square brackets. A quadratic form may have presentations by different triangular schemes (cf. [8, §1]). To cope with this difficulty, we use the sign ≅ (“equivalent”) to indicate such a case. Note that the entries α_{i} in (1.4) are uniquely determined by q, since α_{i} = q(ε_{i}). If R is embeddable as subsemiring in a ring R′, then also the α_{ij} are uniquely determined by q, since by identifying R ⊂ R′ we have

α_{ij} = q(ε_{i} + ε_{j}) − q(ε_{i}) − q(ε_{j}).

However, this situation is far apart from the semirings in this paper, the so called “supertropical semirings”, to be described bellow.

1.3. The problem. Assume that (ε_{i} | i ∈ I) is a fixed base of a module V. We search for families of scalars

(1.5)  (μ_{i} | 1 ≤ i ≤ n) ∪ (μ_{ij} | 1 ≤ i < j ≤ n)

with the following property: For any two equivalent triangular schemes

[\begin{matrix}α_{1} & α_{12} & ⋯ & α_{1n} \\ & α_{2} & ⋯ & α_{2n} \\ & & ... & ⋮ \\ & & & α_{n} \end{matrix}] ≅ [\begin{matrix}α_{1} & β_{12} & ⋯ & β_{1n} \\ & α_{2} & ⋯ & β_{2n} \\ & & ... & ⋮ \\ & & & α_{n} \end{matrix}]

also

[\begin{matrix}μ_{1}α_{1} & μ_{12}α_{12} & ⋯ & μ_{1n}α_{1n} \\ & μ_{2}α_{2} & ⋯ & μ_{2n}α_{2n} \\ & & ... & ⋮ \\ & & & μ_{n}α_{n} \end{matrix}] ≅ [\begin{matrix}μ_{1}α_{1} & μ_{12}β_{12} & ⋯ & μ_{1n}β_{1n} \\ & μ_{2}α_{2} & ⋯ & μ_{2n}β_{2n} \\ & & ... & ⋮ \\ & & & μ_{n}α_{n} \end{matrix}] .

Then the multiplication of the entries of a triangular scheme by the scalars μ_{i} and μ_{ij} yields a well defined map of the set Quad(V) of all quadratic forms on V into itself. These maps are the “basic operations” on quadratic forms appearing in the title of the paper. Two quadratic forms q_{1}, q_{2} on V can be added by the rule

(q_{1} + q_{2})(x) = q_{1}(x) + q_{2}(x),

and a quadratic form q can be multiplied by a scalar a ∈ R by the rule

(aq)(x) = a · q(x).

In this way, the set Quad(V) becomes an R-module. The above presentation (1.3) of a quadratic form q translates to

(1.6)  q = ∑_{i∈I} α_{i}d_{i} + ∑_{i<j} α_{ij}h_{ij},

with d_{i}, h_{ij} defined by

(1.7)  d_{i}(x) = x_{i}^{2},  h_{ij}(x) = x_{i}x_{j},

where as before the x_{i} are the coordinates of x, cf (1.2). We read off from (1.6) that the d_{i} and h_{ij} generate the R-module Quad(V), which gives us a linear algebraic interpretation of the basic operations as certain endomorphisms of the R-module Quad(V), as follows. An endomorphism φ of Quad(V) is called basic (w.r. to a given base (ε_{i} | i ∈ I) of V) if it maps the submodules Rd_{i} (1 ≤ i ≤ n) and Rh_{ij} (1 ≤ i < j ≤ n) to itself, and so

φ(d_{i}) = μ_{i}d_{i}, φ(h_{ij}) = μ_{ij}h_{ij},

with scalars μ_{i}, μ_{ij}, called the coefficients of the basic endomorphism φ. It is now immediate that these systems of coefficients are the same families of scalars as occurring for the basic operations (cf. (1.5)), and so basic operations are the same objects as basic endomorphisms in a different disguise. In general the set of basic endomorphisms of Quad(V) depends on the choice of the base (ε_{i} | i ∈ I) of V. But, when R is a supertropical semiring (to be discussed below), the framework of the present paper, it happily turns our that any free module V has only one base up to scalar multiplication by units [8, Theorem 0.9], a phenomenon for which we use the catch-phrase “V has unique base”. Actually, this property holds over a much broader class of semirings than the supertropical ones [11, §1]. If V has unique base, then the set of endomorphisms of Quad(V) is independent of the choice of the base (ε_{i} | i ∈ I) of V (also for infinite I). In fact, if (ε′_{i} | i ∈ I) is another base, ε′_{i} = u_{i}ε_{i}, with units u_{i} of R, then the generators of Quad(V) associated to this base are

(1.8)  d′_{i} = μ_{i}^{−2}d_{i},  h_{ij}; = μ_{i}^{−1}μ_{j}^{−1}h_{ij},

as easily verified. Thus in the presence of the unique base property, the problem of finding basic endomorphisms of Quad(V) gains extra momentum.

1.4. Supertropical semirings. A semiring R is called supertropical ([8, Definition 0.3] and [3, §3]) if e := 1 + 1 is an idempotent (i.e., e = 1 + 1 = 1 + 1 + 1 + 1 = e + e), and the following axioms hold for all a, b ∈ R :

(1.9)  If ea ≠ eb, then a + b ∈ {a, b},
(1.10)  If ea = eb, then a + b = eb.

Then the ideal eR of R is a semiring with unit element e, which is bipotent, i.e., for any u, v ∈ eR the sum u + v is either u or v. It follows that eR carries a total ordering, compatible with addition and multiplication, which is given by

u ≤ v ⇔ u + v = v.

The addition in a supertropical semiring is determined by the map a ↦ ea and the total ordering on eR as follows: If a, b ∈ R, then

(1.11)  a + b = \begin{cases}b & if ea < eb, \\ a & if ea > eb, \\ eb & if ea = eb. \end{cases}

In particular (taking b = 0 in (1.11) or in (1.10)), for any a ∈ R

(1.12)  ea = 0 ⇒ a = 0,

in other terms

(1.12’)  a + a = 0 ⇒ a = 0.

Note also that

(1.13)  e + 1 = e,

as follows from (1.11) for a = e and b = 1. For later use we quote another fact, true in any supertropical semiring R:

(1.14)  (a + b)^{2} = a^{2} + b^{2}

for all a, b ∈ R, cf. [8, p.65]. When R is a supertropical semiring, the elements of T (R) := R\(eR) are called tangible elements, and those of G(R) := (eR) \ {0} are called ghost elements. The zero of R is regarded both as tangible and ghost. The semiring R itself is called tangible if R is generated by T (R) as a semiring. Clearly, this happens iff eT (R) = G(R). If T (R) ≠ ∅, then the set

R′ := T (R) ∪ eT (R) ∪ {0}

is the largest subsemiring of R which is tangible supertropical. (We have discarded the “superfluous” ghost elements.) The map

ν_{R} : R → eR

is a homomorphisms of semirings, which we call the ghost map of R. When there is no ambiguity, we write T , G, ν for T (R), G(R), ν_{R}. Sometimes we adhere to the very convenient “ν-notation” for a, b ∈ R: a ≤_{ν} b means that ea ≤ eb, a ≅_{ν} b (“ν-equivalent”) means that ea = eb, while a <_{ν} b means that ea < eb. We call a supertropical semiring a supersemifield if all nonzero tangible elements are invertible in R and all nonzero ghost elements are invertible in the bipotent subsemiring eR, whence both T and G are abelian groups under multiplication. Supertropical semirings have been previously introduced as a tool to refine certain aspects of tropical geometry (e.g. [13]), linear algebra [6, 7], starting with [2], and tropical valuation theory [3]. Up to now supertropical semifields have been prevalent in applications, but more general supertropical semirings are definitely needed for any coherent theory (cf. e.g. [3, 4, 5]). The relevance of quadratic forms over supertropical semirings to classical quadratic forms over rings is explained in [8, §9].

1.5. Partial orderings on R, V, and Quad(V). Assume that V is any module over a supertropical semiring R. Then it is known from more general facts (e.g. [10]), that the binary relation defined by

(1.15)  x ≤ y ⇔ ∃z ∈ V : x + z = y

for any x, y ∈ V is a partial ordering on V. For the reader’s convenience, we provide a direct argument giving this important fact. Reflexivity (x ≤ x) and transitivity (x ≤ y, y ≤ z ⇒ x ≤ z) are evident, but antisymmetry is subtler. Given x, y, z, w such that x + z = y, y + w = x, we need to verify that x = y. First we get x + (z + w) = x, then x + e(z + w) = x. From (1.13) we infer that ez + z = ez, and so y = x + z = x + ez + ew + z = x + ez + ew = x, as desired. The ordering (1.15) is called the minimal ordering of V, since it is the coarsest (partial) ordering on V compatible with addition, such that 0 ≤ x for all x ∈ V. In particular we have a minimal ordering on R itself. It is immediate that scalar multiplication is compatible with both minimal orderings, i.e., for a, b ∈ R, x, y ∈ V,

a ≤ b, x ≤ y ⇒ ax ≤ by.

Note also that, if V is free with base (ε_{i} | i ∈ I), then

∑_{i∈I} x_{i}ε_{i} ≤ ∑_{i∈I} y_{i}ε_{i} ⇔ ∀i ∈ I : x_{i} ≤ y_{i}.

In this paper, the sign “≤” is used for both orderings on R and V. These orderings lead to a “functional ordering” on Quad(V), again denoted by “≤”, defined for q_{1}, q_{2} ∈ Quad(V) as

q_{1} ≤ q_{2} ⇔ ∀x ∈ V : q_{1}(x) ≤ q_{2}(x).

On the other hand, since Quad(V) is an R-module, it carries a minimal ordering, denoted here by “⪯”. Definition (1.15) now reads as: If q_{1}, q_{2} are quadratic forms on V, then

q_{1} ⪯ q_{2} ⇔ ∃χ ∈ Quad(V) : q_{1} + χ = q_{2}.

The functional ordering refines the minimal ordering, q_{1} ⪯ q_{2} ⇒ q_{1} ≤ q_{2}. The interplay between these orderings is the major theme in the second half of [10], whose results will be very useful below.

1.6. The submodules QL(V) and Rig(V) of Quad(V). A quadratic form q on a module V over a semring R is called quasilinear if the zero bilinear form b = 0 is a companion of q, i.e., ( cf. (1.1))

q(x + y) = q(x) + q(y)

for all x, y ∈ V, and q is called rigid if q has only one companion. It is obvious that the set QL(V) of all quasilinear forms on V is an R-submodule of Quad(V). Assuming that the R-module V is free with base (ε_{i} | i ∈ I) and R is supertropical, we have the forms d_{i} and h_{ij} with i, j ∈ I, i < j, cf. (1.7). In consequence of property (1.14) of R, every d_{i} is quasilinear. On the other hand a quadratic form q is rigid iff q(ε_{i}) = 0 for all i ∈ I [8, Theorem 3.5] ^{1}. This implies that the set Rig(V) of all rigid forms on V is a submodule of Quad(V) and that all forms h_{ij} (i < j) are rigid. Having this starting point, it is an easy matter to verify that both QL(V) and Rig(V) are free modules with bases

𝔇_{0} := {d_{i} | i ∈ I}, ℌ_{0} := {h_{ij} | i > j}

respectively [10, Proposition 7.2]. For any κ ∈ QL(V) and ρ ∈ Rig(V) we have (as a special cases of (1.3)) the presentations

(1.16)  κ = ∑_{i∈I} κ(ε_{i})d_{i}.

ρ = ∑_{i<j} b(ε_{i}, ε_{j})h_{ij} = ∑_{i<j} ρ(ε_{i} + ε_{j})h_{ij},

where b is the unique companion of ρ [8, §4]. From these presentations it follows that the functional ordering of Quad(V) restricts on both QL(V) and Rig(V) to the minimal ordering on these free modules [10, Proposition 7.3]. Every q ∈ Quad(V) has a decomposition

(1.17)  q = q_{QL} + ρ

with q_{QL} ∈ QL(V) and ρ ∈ Rig(V), as it is now evident from (1.3). Moreover, for any decomposition (1.17) clearly q(ε_{i}) = q_{QL}(ε_{i}), and so we infer from (1.16) that

q_{QL} = ∑_{i∈I} q(ε_{i})d_{i},

which proves that q_{QL} is uniquely determined by q. We call q_{QL} the quasilinear part of q and ρ in (1.17) a rigid complement of q_{QL} in q. Most often ρ is not unique [8, §6 and §7]. If I is finite and a triangular scheme for q is given (cf. (1.4)), then q_{QL} is represented by the diagonal part of the scheme, while the upper triangular part gives a rigid complement of q_{QL} in q. We have

Quad(V) = QL(V) + Rig(V),

QL(V) ∩ Rig(V) = {0},

but nevertheless Quad(V) is not a direct sum of the submodules QL(V) and Rig(V), as soon as Rig(V) ≠ {0}, i.e., |I| > 1. Indeed, then different indices i, j give us a relation

d_{i} + d_{j} = d_{i} + d_{j} + h_{ij},

since, in consequence of (1.14), a^{2} + b^{2} = a^{2} + b^{2} + ab for any a, b ∈ R.

1More generally, this remains true if the semiring R has the properties (1.12’) and (1.14)

[loc. cit].

1.7. A refinement of the problem in §1.3 for R supertropical. As before we assume that the semiring R is supertropical and V is free with base (ε_{i} | i ∈ I). We then have the set of generators 𝔅_{0} = 𝔇_{0} ∪ ℌ_{0} of Quad(V) with

𝔇_{0} := {d_{i} | i ∈ I} and ℌ_{0} := {h_{ij} | i < j},

which up to multiplication by scalars does not depend on the choice of the base (ε_{i} | i ∈ I), cf. (1.8). We call a submodule Z of Quad(V) basic, if Z is generated by a subset 𝔅′_{0} of 𝔅_{0}, which then is a union 𝔇′_{0} ∪ ℌ′_{0} with 𝔇′_{0} ⊂ 𝔇_{0}, ℌ′_{0} ⊂ ℌ_{0}. In this case necessarily 𝔇′_{0} = 𝔇_{0} ∩ Z, ℌ′_{0} = ℌ_{0} ∩ Z and so 𝔅′_{0} = 𝔅_{0} ∩ Z. For these intersections we write 𝔇_{0}(Z), ℌ_{0}(Z), 𝔅_{0}(Z) respectively. Instead of the basic endomorphisms of Quad(V) addressed in §1.3, in the present paper we search, more generally, for endomorphisms of a fixed basic submodule Z of Quad(V) that map each submodule Rq, q ∈ 𝔅_{0}(Z) into itself. Such map φ is called a basic endomorphism of Z. Given a fixed base (ε_{i} | i ∈ I) of V, we denote by I^{[2]} the set of all 2-element subsets of I, and write

(1.18)  𝔇_{0}(Z) = {d_{i} | i ∈ K},  ℌ_{0}(Z) = {h_{ij} | {i, j} ∈ M},

where K ⊂ I, M ⊂ I^{[2]}. Then a basic endomorphisms φ of Z is determined by a family of scalars

(1.19)  (μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M),

which we again call the coefficients of φ, via the formulas (i ∈ K, {i, j} ∈ M)

φ(d_{i}) = μ_{i}d_{i}, φ(h_{ij}) = μ_{ij}h_{ij}.

In the case that I = {1, …, n} is finite, we may use triangular schemes to present quadratic forms. Now the refined problem means that we focus on quadratic forms which are represented by a scheme as in (1.4), with zero entries at fixed places (i, j), i ≤ j, namely at (i, i), with i ∉ K and {i, j} ∉ M. The task is to find all systems of scalars (μ_{i})∪(μ_{ij}) such that any two equivalent schemes of this type remain equivalent after multiplication of entries by the scalars μ_{i} and μ_{ij} respectively. So this is indeed a very natural expansion of the problem described at the beginning of §1.3. In what follows we call a basic submodule of Quad(V) simply a basic module. A basic endomorphisms φ of a given basic module is called a basic projector on Z, if its coefficients are all 1 or 0, and thus φ(z) = z or 0 for any z ∈ 𝔅_{0}(Z). Then X = φ(Z) is a basic module with X ⊂ Z and φ is uniquely determined by X. We call these submodules X of Z the basic projections of Z. For example, QL(V) is a basic projection of Quad(V) whose associated basic projector is the endomorphism π_{QL} of Quad(V) which maps any q ∈ Quad(V) to its quasilinear part q_{QL}. But Rig(V) is not a basic projection of Quad(V) whenever Rig(V) ≠ {0}, i.e., |I| > 1. Indeed, the existence of an endomorphism of Quad(V) with φ(d_{i}) = 0 for all i ∈ I and φ(h_{ij}) = h_{ij} for i ≠ j is prevented by the relations d_{i} + d_{j} = d_{i} + d_{j} + h_{ij}.

1.8. Paper outline and main results. §3 and §4 are devoted to a study of basic projectors to obtain a classification of all basic projections of a basic module Z in combinatorial terms under the mild assumption that eR is “multiplicatively unbounded”, i.e., for any x, y ∈ G there exists some z ∈ G such that y < xz (cf. Corollary 3.6 and Theorem 3.12). In particular it turns out (without the assumption of multiplicatively unboundedness) that any basic module X ⊂ Z with 𝔇_{0}(X) = 𝔇_{0}(Z) is a basic projection of Z. The associated projectors are constructed in §3 for Z = Quad(V) under the name of partial quasilinearizations. For any subset Λ of the set I^{[2]} of 2-element subsets of I we have a basic projector

π_{Λ,QL} : q ⟼ q_{Λ,QL}

on Quad(V) with π_{Λ,QL}(d_{i}) = d_{i} for i ∈ I, π_{Λ,QL}(h_{ij}) = h_{ij} for {i, j} ∈ Λ, and π_{Λ,QL}(h_{ij}) = 0 otherwise. This projector then restricts to a basic projector on Z for any basic module Z ⊂ Quad(V). Its image is a basic module X ⊂ Z with

𝔇_{0}(X) = 𝔇_{0}(Z), ℌ_{0}(X) = {h_{ij} ∈ ℌ_{0}(Z) | {i, j} ∈ Λ ∩ M}

in Notation (1.18). A basic module Z is a direct sum X ⊕Y of basic modules X and Y iff 𝔅_{0}(Z) is a disjoint union of 𝔅_{0}(X) and 𝔅_{0}(Y) and both X and Y are basic projections of Z (Proposition 4.2). Thus it is not surprising that the classification of the basic projections of Z in §3 in combinatorial terms leads to a description of all (possibly infinite) direct decompositions of Z, again in a combinatorial way. In particular we learn in §4 that Z has (up to permutation of summands) only one decomposition Z = ⨁_{α∈A} Z_{α}, such that all Z_{α} are indecomposable basic modules, and these components Z_{α} of Z can be described combinatorially. In the important special case that for every h_{ij} ∈ ℌ_{0}(Z) both d_{i} and h_{ij} are in Z, and so are elements of 𝔇_{0}(Z), this description can be given in terms of graphs. We associate to Z a graph Γ(Z) whose sets of vertices and edges are 𝔇_{0}(Z) and ℌ_{0}(Z) respectively, an edge h_{ij} connecting the vertices d_{i} and d_{j}, and we call the module Z graphic. It turns out that the components Z_{α} of Z are again graphic and the graphs Γ(Z_{α}) are precisely all path components of Γ(Z) (Theorem 4.18). Starting from this, we also obtain, under a mild restriction of the supertropical semring R, a description of all components of Z when Z is not graphic (Corollary 4.19). In the last three sections §5–§8 we work on more general basic endomorphisms than basic projectors. The main result in §5 is that, under still mild conditions on R (in particular if the semiring eR is cancellative), every basic endomorphism φ of a basic module Z yields a basic projector p_{φ} on Z by the rule p_{φ}(z) = z if φ(z) ≠ 0 and p_{φ}(z) = 0 otherwise, for z ∈ 𝔅_{0}(Z). Conversely, given a basic projector π on Z we can describe all basic endomorphisms φ of Z with p_{φ} = π, called the satellites of π (Theorem 5.17 and Corollary 5.18). In §6 we develop other ways to obtain new basic endomorphisms from old ones. Given scalars μ, υ ∈ R, we say that υ is obedient to μ, if υ ≤_{ν} μ and υ is faithful to μ in the following sense: for all x, y ∈ R, if μx = μy, then υx = υy. Assume, for simplicity, that Z is graphic and φ is a basic endomorphism of Z with system of coefficients (μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M), cf. (1.19). Then it turns out that every tuple of scalars (μ_{i} | i ∈ K) ∪ (υ_{ij} | {i, j} ∈ M), with υ_{ij} obedient to μ_{ij} for all i, j, is again the coeffient system of some basic endomorphism ψ of Z (Theorem 6.9). We call such a basic endomorphism ψ an H-modification of φ. To give the flavor we point out what this theorem means in the case that Z = Quad(V), φ = id_{Z}, I = {1, …, n}. It says that for any family of scalars (λ_{ij} | 1 ≤ i < j ≤ n) with λ_{ij} ≤_{ν} 1 for all i, j the assignment

[\begin{matrix}α_{1} & α_{12} & ⋯ & α_{1n} \\ & α_{2} & ⋯ & α_{2n} \\ & & ... & ⋮ \\ & & & α_{n} \end{matrix}] ⟼ [\begin{matrix}α_{1} & λ_{12}α_{12} & ⋯ & λ_{1n}α_{1n} \\ & α_{2} & ⋯ & λ_{2n}α_{2n} \\ & & ... & ⋮ \\ & & & α_{n} \end{matrix}]

is a well defined basic operation on Quad(V). The reason is that every λ_{ij} is clearly obedient to 1. Starting again with the system of coefficients (μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M) of φ we call a basic endomorphism ψ of Z a D-modification of φ if for the coefficients (υ_{i} | i ∈ K) ∪ (υ_{ij} | {i, j} ∈ M) of ψ we have υ_{ij} = μ_{ij} for all {i, j} ∈ M and μ_{i} ≤_{ν} υ_{i} for all i ∈ K. While for H-modifications we obtained a best possible result, here our knowledge is less complete. We only know that a tuple (υ_{i}) ∪ (μ_{ij}) is the coefficient system of a basic endomorphism ψ if μ_{i} ≤ υ_{i} for all i ∈ K (minimal ordering ≤ instead of ν-dominance ≤_{ν}), cf. Theorem 6.9. In the last section §8 we determine, for a tangible supersemifield, R, all basic endomorphisms of any basic module Z. If Z is “linked”, i.e., Z is graphic and Γ(Z) has no isolated vertices (the main case to be studied), it turns out that the possible coefficient systems (μ_{i} | i ∈ K)∪(μ_{ij} | {i, j} ∈ M), cf. (1.19) are given by the condition

μ_{ij}^{2} ≤_{ν} μ_{i}μ_{j},

provided that the ghost map ν_{R} : T (R) ↠ G(R) is not bijective. Otherwise there may exist more basic endomorphisms, cf. Theorem 8.5.

2. Partial quasilinearisation

Henceforth, R is a supertropical semiring and V is a free R-module with base (ε_{i} | i ∈ I). Let I^{[2]} denote the set of 2-element subsets of I. We choose a total ordering of I and often identify I^{[2]} with the set of pairs (i, j) ∈ I × I such that i < j. If Λ is a subset of I^{[2]}, let Λ^{c} denote the complement I^{[2]} \ Λ. We define a quasilinear quadratic form d_{i} on V for every i ∈ I by

(2.1)  d_{i}(x) = x_{i}^{2}

and a rigid quadratic form h_{ij} for every i, j ∈ I with i ≠ j by

(2.2)  h_{ij}(x) = x_{i}x_{j}.

Here, as always, the x_{i} are the coordinates of the vector x ∈ V, x = ∑_{i∈I} x_{i}ε_{i}. We work with the bases (d_{i} | i ∈ I) and (h_{ij} | (i, j) ∈ I^{[2]}) of the free R-modules QL(V) and Rig(V), respectively. For any set Λ ⊂ I^{[2]} we introduce the free submodule

Rig(Λ, V) := ∑_{{i,j}∈Λ} Rh_{ij}.

Rig(Λ, V) is a lower set in the R-module Quad(V) both in the minimal and the functional ordering of Quad(V). Clearly

(2.3)  Rig(V) = Rig(Λ, V) ⊕ Rig(Λ^{c}, V).

In other words, for any rigid form ρ on V we have a unique decomposition ρ = ρ_{1} + ρ_{2} with ρ_{1} ∈ Rig(Λ, V) and ρ_{2} ∈ Rig(Λ^{c}, V). We call ρ_{1} and ρ_{2} the Λ-component and Λ^{c}-component of ρ, respectively.

Definition 2.1. We call a form q ∈ Quad(V) Λ-quasilinear, if q is quasilinear on the submodule Rε_{i} + Rε_{j} for every {i, j} ∈ Λ.

Λ-quasilinearity of q means that for every {i, j} ∈ Λ the set C_{ij}(q) in the companion table of q (cf. [8, §6]) contains zero. Notice that a priori every set C_{ii}(q) contains zero, cf. [8, Example 2.4].

Proposition 2.2. q is Λ-quasilinear iff the set Rig(q) of rigid complements of q_{QL} in q contains some ρ ∈ Rig(Λ^{c}, V).

Proof. This is a consequence of the 1-1-correspondence between the off-diagonal companions of q and the rigid complements of q_{QL} in q described in [8, Proposition 4.6]. □

Given Λ ⊂ I^{[2]} we intend to associate to any q ∈ Quad(V) a Λ-quasilinear form q_{Λ,QL} ∈ Quad(V) in a somewhat canonical way, generalizing the map q ↦ q_{QL} from Quad(V) to QL(V). The key to do this is provided by the following lemma.

Lemma 2.3. Let Λ ⊂ I^{[2]} and q_{0} ∈ QL(V). Further let ρ, ρ′ ∈ Rig(V) be given, and let ρ_{1}, ρ′_{1} denote the Λ-components of ρ, ρ′, respectively.

(a) If q_{0} + ρ ≤ q_{0} + ρ′, then q_{0} + ρ_{1} ≤ q_{0} + ρ′_{1}.

(b) If q_{0} + ρ ⪯ q_{0} + ρ′, then q_{0} + ρ_{1} ⪯ q_{0} + ρ′_{1}.

To prove the lemma we use part of the following notation, that shall also be helpful later.

Notation 2.4. Let J be a subset of the index set I.

(a) V_{J} := ∑_{i∈J} Rε_{i} is a free submodule of V. It comes with a natural R-linear

projection π_{J} : V → V_{J}, given by π_{J}(ε_{i}) = ε_{i} for i ∈ J, π_{J}(ε_{i}) = 0 for i ∈ I \ J. We also have the inclusion mapping i_{J} : V_{J} ↪ V, with i_{J}(ε_{i}) = ε_{i} for every i ∈ J.

(b) Any form ϑ ∈ Quad(V_{J}) gives us a form

ϑ^{I} := ϑ ∘ π_{J} ∈ Quad(V).

(c) Given q ∈ Quad(V), we define

q_{J} := (q|V_{J})^{I} = q ∘ i_{J} ∘ π_{J} ∈ Quad(V).

Proof of Lemma 2.3. We write q_{0} = ∑_{i∈I} α_{i}d_{i}, ρ = ∑_{i<j} α_{ij}h_{ij}, ρ′ = ∑_{i<j} β_{ij}h_{ij}

with α_{i}, α_{ij}, β_{ij} ∈ R.

(a): We have

∑_{i∈I} α_{i}d_{i} + ∑_{i<j} α_{ij}h_{ij} ≤ ∑_{i∈I} α_{i}d_{i} + ∑_{i<j} β_{ij}h_{ij}

From this we get for any i < j in I, (q_{0} + ρ)_{{i,j}} ≤ (q_{0} + ρ′)_{{i,j}}, which reads

(∗)  α_{i}d_{i} + α_{j}d_{j} + α_{ij}h_{ij} ≤ α_{i}d_{i} + α_{j}d_{j} + β_{ij}h_{ij}.

Using (∗) for every {i, j} ∈ Λ, we obtain

∑_{i∈I} α_{i}d_{i} + ∑_{{i,j}∈Λ} α_{ij}h_{ij} ≤ ∑_{i∈I} α_{i}d_{i} + ∑_{{i,j}∈Λ} β_{ij}h_{ij}.

This means that q_{0} + ρ_{1} ≤ q_{0} + ρ′_{1}.

(b): Same argument, employing ⪯ instead of ≤ . □

Corollary 2.5. Assume that q ∈ Quad(V) and ρ, ρ′ ∈ Rig(q). Given Λ ⊂ I^{[2]}, let now ρ_{2}, ρ′_{2} denote the Λ^{c}-components of ρ, ρ′. Then q_{QL} + ρ_{2} = q_{QL} + ρ′_{2}.

Proof. This follows by applying part a) of the lemma to the inequalities q_{QL} + ρ ≤ q_{QL} + ρ′ and q_{QL} + ρ′ ≤ q_{QL} + ρ, using Λ^{c} instead of Λ. (We could, equally well, use Lemma 2.3.(b).) □

Definition 2.6. Given q ∈ Quad(V) and Λ ⊂ I^{[2]}, we choose a rigid complement ρ of q_{QL} in q and define

(2.4)  q_{Λ,QL} := q_{QL} + ρ_{2},

with ρ_{2} the Λ^{c}-component of ρ. Evidently, this form is Λ-quasilinear. We call q_{Λ,QL} the Λ-quasilinearisation of q.

Corollary 2.5 tells us that q_{Λ,QL} does not depend on the choice of the rigid complement ρ in q. If Λ = I^{[2]}, then q_{Λ,QL} = q_{QL}, while if Λ = ∅, then q_{Λ,QL} = q.

Scholium 2.7. Let I = {1, 2, …, n}. We describe a given quadratic form q : V → R by a triangular scheme

q = [\begin{matrix}α_{1}α_{12} & … & α_{1n} \\ & ... & ⋮ \\ & & α_{n} \end{matrix}] ,

cf. §1.2. The quadratic form q_{Λ,QL} is then given by the triangular scheme, where every entry α_{ij} with {i, j} ∈ Λ is replaced by zero.

Remark 2.8.

(i) If q_{1}, q_{2} ∈ Quad(V), then

(q_{1} + q_{2})_{Λ,QL} = (q_{1})_{Λ,QL} + (q_{2})_{Λ,QL}.

(ii) If q ∈ Quad(V) and λ ∈ R, then

(λq)_{Λ,QL} = λ · q_{Λ,QL}.

Let QL(Λ, V) denote the R-submodule of Quad(V) consisting of all Λquasilinear forms on V; in other terms

(2.5)  QL(Λ, V) = QL(V) + Rig(Λ^{c}, V).

We have a natural map

(2.6)  π_{Λ,QL} : Quad(V) → QL(Λ, V),

sending q ∈ Quad(V) to its Λ-quasilinearization q_{Λ,QL}. It is R-linear by Remark 2.8. Since π_{Λ,QL} is additive, it is also plain that π_{Λ,QL} respects the minimal ordering on Quad(V), i.e.,

(2.7)  q ⪯ q′ ⇒ q_{Λ,QL} ⪯ q′_{Λ,QL}.

Viewing every map π_{Λ,QL} as an endomorphism of the R-module Quad(V), we may state that

π_{M,QL} ∘ π_{Λ,QL} = π_{M,QL},

if M ⊂ Λ ⊂ I^{(2)}. In particular (Λ = M), π_{Λ,QL} ∈ End_{R}(Quad(V)) is a projector. It can be characterized in terms of the minimal ordering of Quad(V) as follows.

Proposition 2.9. For any q ∈ Quad(V) the form q_{Λ,QL} is the unique maximal form κ ⪯ q, which is Λ-quasilinear.

Proof. If κ is Λ-quasilinear and κ ⪯ q, we conclude by (2.7) that κ = κ_{Λ,QL} ⪯ q_{Λ,QL}. □

Problem 2.10. For which supertropical semirings R, sets Λ ⊂ I^{[2]}, and quadratic forms q′ on R^{(I)}, is it true that q ≤ q′ implies q_{Λ,QL} ≤ q′_{Λ,QL}?

In addition to the cases where we know that q ≤ q′ means the same as q ⪯ q′ (cf. [10, Corollaries 9.11 and 9.12 ]), there is one case where we can give an answer now, for any supertropical semiring R.

Proposition 2.11. Let Λ = {{k, ℓ} | k ∈ J, ℓ ∈ I \ J} for some subset J of I.

(a) If ϑ ∈ Quad(V), then ϑ is Λ-quasilinear iff ϑ is quasilinear on V_{J} × V_{I\J}.

(b) If ϑ, q ∈ Quad(V) and ϑ ≤ q, then ϑ_{Λ,QL} ≤ q_{Λ,QL}. Moreover, q_{Λ,QL} is the unique maximal Λ-quasilinear form κ on V (in the functional ordering of Quad(V)) with κ ≤ q.

Proof. (a): If ϑ is Λ-quasilinear, ϑ has a companion b with b(ε_{k}, ε_{ℓ}) = 0 for k ∈ J, ℓ ∈ I \ J. It follows that b(V_{J} × V_{I\J}) = 0, and so ϑ is quasilinear on V_{J} × V_{I\J}. The converse is trivial.

(b): Let

q = ∑_{i∈I} α_{i}d_{i} + ∑_{i<j} α_{ij}h_{ij}.

Then, using Notation 2.4, we may write

q_{Λ,QL} = ( ∑_{i∈J} α_{i}d_{i} + ∑_{\begin{matrix}i<j \\ i,j∈J \end{matrix}} α_{ij}h_{ij}) + ( ∑_{i∉J} α_{i}d_{i} + ∑_{\begin{matrix}i<j \\ i,j∉J \end{matrix}} α_{ij}h_{ij})

= (q | V_{J})^{I} + (q | V_{I\J})^{I}
= q_{J} + q_{I\J}.

We have V = V_{J} ⊕ V_{I\J} and obtain for any x ∈ V_{J}, y ∈ V_{I\J} the formula

q_{Λ,QL}(x + y) = q(x) + q(y).

(N.B. This proves again that q is quasilinear on V_{J} × V_{I\J}.) If κ is Λ-quasilinear and κ ≤ q, then κ is quasilinear on V_{J} × V_{I\J} (as just proven again), and κ(x) ≤ q(x), κ(y) ≤ q(y) for x ∈ V_{J}, y ∈ V_{I\J}, and hence

κ(x + y) = κ(x) + κ(y) ≤ q(x) + q(y) = q_{Λ,QL}(x + y).
□

3. Basic modules and basic projections

We repeat that in the whole paper V is a free module over a supertropical semiring R and {ε_{i} | i ∈ I} is a base (mostly fixed) of V. Let I^{[2]} denote the set of all 2-element subsets of I. The R-module Quad(V) has the set of generators 𝔅_{0} := 𝔇_{0} ˙∪ℌ_{0} with

𝔇_{0} := {d_{i} | i ∈ I} and ℌ_{0} := {h_{ij} | {i, j} ∈ I^{[2]}},

where d_{(}x) = x_{i}^{2} and h_{ij} = x_{i}x_{j} for x = ∑_{i∈I} x_{i}ε_{i}, as said above. Assume now that eR is multiplicatively unbounded, i.e., that for any x, y ∈ G there exists z ∈ G such that y < xz (cf. [10, Definition 6.4]).^{2} Then, as proved in [10, §7], the set of generators 𝔅_{0} of Quad(V) is uniquely determined by the R-module Quad(V) up to multiplication by units of R. More precisely, the set 𝔅 := R^{∗}𝔅_{0}, consisting of all products λq with λ ∈ R^{∗}, q ∈ 𝔅_{0}, coincides with the set of all “basic elements” of Quad(V) (cf. [10, Definition 6.1]) and also with the set of all “primitive” (loc. cit.) indecomposable elements of Quad(V) [10, Theorem 7.8, Corollary 7.9 ]. Moreover, as has been shown in [10, §8], each of the sets

𝔇 := R^{∗}𝔇_{0}, ℌ := R^{∗}ℌ_{0},

2This property implies that the set R\{0} is closed under multiplication [10, Remark 6.5], i.e., R has no zero divisors.

is uniquely determined by the R-module Quad(V), up to multiplication by units. It now makes sense to extend the notation 𝔅_{0}, 𝔇_{0}, ℌ_{0} as follows, and to define “basic submodules” of Quad(V) without referring to a base of R.

Definition 3.1. Choosing sets of representatives 𝔇_{0} and ℌ_{0} of the orbit sets 𝔇/R^{∗} and ℌ/R^{∗}, we obtain a set 𝔅_{0} := 𝔇_{0} ˙∪ℌ_{0} which obviously generates the R-module Quad(V). We call 𝔅_{0} a basic set of generators of Quad(V). ^{3} Usually we choose the sets

(3.1)  𝔇_{0} := {d_{i} | i ∈ I} and ℌ_{0} := {h_{ij} | {i, j} ∈ I^{[2]}}

derived from a base {ε_{i} | i ∈ I} of the free R-module V, cf. (2.1), (2.2). Then we call 𝔅_{0} := 𝔇_{0} ∪ ℌ_{0} a geometric basic set of generators of Quad(V). ^{4}

Definition 3.2.

(a) We call an R-submodue Z of Quad(V) basic if Z is spanned by a subset S of 𝔅_{0}, i.e., every q ∈ Z has a presentation q = ∑_{s∈S} α_{s}s with s ∈ S, α_{s} ∈ R, almost all α_{s} = 0.

(b) If this holds, then S is uniquely determined by Z, namely S = Z ∩ 𝔅_{0}, as follows immediately from the fact that every s ∈ S is primitive and indecomposable. We call S a basic set of generators of Z, and also write Z = ∑ RS (while RS just means the set of all products RS with λ ∈ R, s ∈ S).

If Z_{1} and Z_{2} are basic submodules of Quad(V), then the modules Z_{1} + Z_{2} and Z_{1} ∩ Z_{2} are again basic in Quad(V). We have

(Z_{1} + Z_{2}) ∩ 𝔅_{0} = (Z_{1} ∩ 𝔅_{0}) ∪ (Z_{2} ∩ 𝔅_{0})

and, of course,

(Z_{1} ∩ Z_{2}) ∩ 𝔅_{0} = (Z_{1} ∩ 𝔅_{0}) ∩ (Z_{2} ∩ 𝔅_{0}),

whence

Z_{1} ⊂ Z_{2} ⇔ Z_{1} ∩ 𝔅_{0} ⊂ Z_{2} ∩ 𝔅_{0}.

Caution: If S_{1} ∩ S_{2} = ∅, Z_{1} = ∑ RS_{1}, Z_{2} = ∑ RS_{2}, then Z_{1} ∩ Z_{2} = {0}, but Z_{1} + Z_{2} is not necessarily the direct sum of the modules Z_{1} and Z_{2}.

We already met preeminent basic submodules of Quad(V). Both QL(V) and Rig(V) are basic in Quad(V). If Λ is any subset of the set I^{[2]} of two element subsets of I, then the submodule QL(Λ, V) consisting of the Λ-quasilinear forms on V (cf. Definition 2.1) is basic. Also the submodule Rig(Λ, V) of Rig(V) (cf. (2.3)) is basic in Quad(V). We are ready for the key definitions of this section.

Definition 3.3. Assume that Z is a basic submodule of Quad(V).

3Recall that the R-module Quad(V) is not free [10, Proposition 7.10].

4Usually not every basic set of generators of Quad(V) is geometric. In the present paper we do not exploit this phenomenon thoroughly.

(a) We call an endomorphism π of the R-module Z a basic projector on Z if π maps every q ∈ 𝔅_{0} ∩ Z either to itself or to zero. Clearly, then π = π^{2} and both X := π(Z), Y = π^{−1}(0) are basic submodules of Quad(V) with Z = X + Y, X ∩ Y = {0}. Indeed X is generated by the set {q ∈ 𝔅_{0} ∩ Z | π(q) = q}, while Y is generated by {q ∈ 𝔅_{0} ∩ Z | π(q) = 0}.

(b) We call X a basic projection of the R-module Z and Y a basic projection kernel in Z, and we also call (Z, X) a basic projection pair.

(c) Whenever it is convenient, we identify the basic projector π : Z → Z with the associated projection map Z ↠ X.

(d) Notice that the projector π : Z → Z is uniquely determined both by the pair (Z, X) and the pair (Z, Y). We write π = π_{Z,X}. We call Y = π^{−1}(0) the kernel of the projector π, and usually write Y = ker(π). If q ∈ X then obviously

π^{−1}(q) = q + ker(π).

(e) For the sake of brevity we often call a basic submodule Z of Quad(V) simply a “ basic module”, suppressing the reference to the free Rmodule V, as long as V is kept fixed.

The primordial example of a basic projection is provided by Z = Quad(V), X = QL(V), Y = Rig(V). It gives us the quasilinear-rigid decompositions of any quadratic form q on V treated in [8]. More generally the Λ-quasilinearizations of q, defined in §2, are provided by Z = Quad(V), X = QL(Λ, V), Y = Rig(Λ, V), cf. (2.4), (2.5). ^{5}

Remark 3.4. Assume that Z is a basic R-module.

(a) If X is a basic projection of Z, then for any basic module W the intersection X ∩ W is a basic projection of Z ∩ W, and π_{Z∩W,X∩W} is the restriction of π_{Z,X} to W. In particular, if W ⊂ Z, then

π_{W,X∩W} = π_{X,Z}|W.

(b) If X_{1}, X_{2} are basic projections of Z, then X_{1} ∩ X_{2} is also a basic projection of Z and

π_{Z,X_{1}∩X_{2}} = π_{Z,X_{1}} · π_{Z,X_{2}} = π_{Z,X_{2}} · π_{Z,X_{1}}.

(The products are taken in End_{R}(Z).) If Y_{i} is the kernel of π_{Z,X_{i}} (i = 1, 2), then Y_{1} + Y_{2} is the kernel of π_{Z,X_{1}∩X_{2}}.

(c) More generally, if (X_{i} | i ∈ K) is a family of basic projections of Z, then X = ⋂_{i∈K} X_{i} is a basic projection of Z. Given q ∈ 𝔅_{0}(Z), we have

π_{Z,X}(q) = q if q ∈ X_{i} for every i ∈ K and π_{Z,X}(q) = 0 otherwise. We

write π_{Z,X} = ∏_{i∈K} π_{Z,X_{i}}.

5A major reason for our interest in basic projectors is the desire to obtain similar decompositions when q is confined to a fixed proper basic submodule Z of Quad(V).

(d) If X_{1}, X_{2} are basic projections of Z with X_{1} ∩ X_{2} = {0}, then X_{1} + X_{2} is also a basic projection of Z and

π_{Z,X_{1}+X_{2}} = π_{Z,X_{1}} + π_{Z,X_{2}}.

(e) If X is a basic projection of Z and U is a basic projection of X, then U is a basic projection of Z and

π_{Z,U} = π_{X,U} ∘ π_{Z,X}

(where now π_{Z,X} is identified with the associated projection map Z ↠ X).

We strive for a combinatorial description of all basic projection pairs within the fixed R-module Quad(V).

Notation 3.5. Given a basic module Z, we set

𝔇_{0}(Z) = 𝔇_{0} ∩ Z, ℌ_{0}(Z) = ℌ_{0} ∩ Z, 𝔅_{0}(Z) = 𝔅_{0} ∩ Z = 𝔇_{0}(Z) ∪ ℌ_{0}(Z),

furthermore

Δ(Z) := {i ∈ I | d_{i} ∉ 𝔇_{0}(Z)},

and

Λ(Z) := {{i, j} ∈ I^{[2]} | h_{ij} ∉ ℌ_{0}(Z)}.

Thus

𝔅_{0}(Z) = {d_{i} | i ∉ Δ(Z)} ∪ {h_{ij} | {i, j} ∉ Λ(Z)}.

Notice that if X and Z are basic R-modules then

X ⊂ Z ⇔ Δ(X) ⊃ Δ(Z), Λ(X) ⊃ Λ(Z).

Corollary 3.6. Assume that Z is a basic module and N is a subset of I^{[2]} containing Λ(Z). Then the basic submodule X of Z with Δ(X) = Δ(Z) and Λ(X) = N is a basic projection of Z. The associated basic projector π_{Z,X} is the restriction to Z of the N-partial quasi-linearization π_{N,QL} (cf. (2.6)).

Proof. This follows from Remark 3.4.a, since X = Z ∩QL(N, V) and QL(N, V) is a basic projection of Quad(V) with associated projector π_{N,QL}. □

By this corollary we know all basic projections X of Z with Δ(X) = Δ(Z), i.e., 𝔇_{0}(X) = 𝔇_{0}(Z).

Lemma 3.7. Assume that X is a basic projection of a basic module Z, and that Λ(X) = Λ(Z) =: Λ. Let i ≠ j be indices such that {i, j} ∉ Λ, i ∉ Δ(Z), j ∉ Δ(Z). Then i ∉ Δ(X), j ∉ Δ(X).

Proof. Let π = π_{Z,X}. In Z we have the relation (cf. [10, Eq. (7.10)])

d_{i} + d_{j} + h_{ij} = d_{i} + d_{j}.

Now {i, j} ∉ Λ, whence π(h_{ij}) = h_{ij}. Applying π to the relation we obtain

π(d_{i}) + π(d_{j}) + h_{ij} = π(d_{i}) + π(d_{j}).

Suppose that d_{i} ∈ Δ(X). This means that π(d_{i}) = 0 and so

π(d_{j}) + h_{ij} = π(d_{j}).

But this is impossible in both cases π(d_{j}) = 0, π(d_{j}) = d_{j}. Thus π(d_{i}) ≠ 0, i.e., i ∉ Δ(X). For the same reason j ∉ Δ(X). □

Convention 3.8. Up to the end of this section, we assume that the supertropical semiring, R, besides multiplicative unboundedness, satisfies the following condition.

(†) If a, b ∈ R and there exists some c_{0} ∈ R such
that ac ≤ bc for all c ≥ c_{0}, then a ≤ b.

This condition is a rather mild hypothesis, as the following proposition reveals. We introduce the set

canc(G) := {c ∈ G | ∀a, b ∈ G : ac = bc ⇒ a = b},

consisting of the “cancellative” elements of G.

Proposition 3.9. Assume that the set canc(G) is unbounded in G, and furthermore that T is closed under multiplication and eT is unbounded in G. Then R has Property (†).

Proof. Let a, b ∈ R and assume that ac ≤ bc for all c ≥ c_{0} in R, for some c_{0} ∈ R. We want to verify that a ≤ b. If a <_{ν} b then a < b, and we are done. Henceforth assume that a ≥_{ν} b. Then ac ≥_{ν} bc for all c ∈ R, and we conclude that ac ≅_{ν} bc for all c ≥ c_{0}. Since canc(G) is unbounded, this implies a ≅_{ν} b. If a, b ∈ eR this means that a = b; and if a ∈ T and b ∈ G, then a ≅_{ν} b implies a < b. Assume finally that b ∈ T and pick u ∈ T with u ≥ c_{0}, which is possible since eT is unbounded in G. Now au ≤ bu ∈ T and au ≅_{ν} bu. This forces au = bu. Thus au ∈ T . which implies a ∈ T . From a ≅_{ν} b and a, b ∈ T we conclude again that a = b. Thus a ≤ b in all cases. □

Theorem 3.10. Let Z be a basic module, Λ := Λ(Z), Δ := Δ(Z). Assume that i ∈ I is an index with d_{i} ∈ Z.

(a) There exists a basic projector π on Z with ker(π) = Rd_{i} iff the following condition holds:

C_{Λ,Δ}(i): If k ∈ I \ {i} and {i, k} ∉ Λ, then k ∈ Δ.

(b) This projector π, if it exists, is compatible with the functional ordering ≤ (restricted to Z), i.e., if q, q′ ∈ Z, q′ ≤ q, then π(q) ≤ π(q′).

Proof. By Lemma 3.7 condition C_{Λ,Δ}(i) is necessary for the existence of π. Assuming now that C_{Λ,Δ}(i) holds, we want to show that π exists and respects the functional ordering on Z. Without loss of generality we may assume that I is finite, I = {1, …, n}, and i = 1, furthermore that h_{1j} ∈ Z for 2 ≤ j ≤ r, but h_{1j} ∉ Z for r < j ≤ n, with some r ∈ {1, …, n}.

We claim that, given presentations

(∗) q = α_{1}d_{1} + ∑_{1<j≤r} α_{1j}h_{1j} + ∑_{\begin{matrix}1 < k ≤ r \\ r < ℓ ≤ n \end{matrix}}α_{kℓ}h_{kℓ} + ∑_{i>r} α_{i}d_{i} + ∑_{i,j>r} α_{ij}h_{ij},

(∗∗) q′ = β_{1}d_{1} + ∑_{1<j≤r} β_{1j}h_{1j} + ∑_{\begin{matrix}1 < k ≤ r \\ r < ℓ ≤ n \end{matrix}}β_{kℓ}h_{kℓ} + ∑_{i>r} β_{i}d_{i} + ∑_{i,j>r} β_{ij}h_{ij},

of two forms q, q′ ∈ Z with q ≤ q′, then (1) α_{1j} ≤ β_{1j} for 1 < j ≤ r;

(2) α_{kℓ} ≤ β_{kℓ} for 1 < k ≤ r, r < ℓ ≤ n;

(3) ∑_{i>r} α_{i}d_{i} + ∑_{i,j>r} α_{ij}h_{ij} ≤ ∑_{i>r} β_{i}d_{i} + ∑_{i,j>r} β_{ij}h_{ij}.

If this is proven, then first assuming that q = q′, we learn that, if we omit in the presentation (∗) the summand α_{1}d_{1}, we obtain a quadratic form q̃ ∈ Z which is independent of the presentation (∗), and thus we have a well defined basic projection π : q ↦ q̃ with kernel Rd_{1}. (Use the claim for q ≤ q′ and q′ ≤ q.) Then using the claim in general, we learn that if q ≤ q′, then π(q) ≤ π(q′), establishing (a) and (b).

Proof of (3): The projector

π_{J} : q ↦ (q|V_{J})^{I} = q_{J}

(cf. Notation 2.4) with J = {i | r < i ≤ n} obviously respects the functional ordering on Quad(V). Applying π_{J} to q ≤ q′ we obtain (3).

Proof of (1): Given i with 1 < i ≤ r we insert in (∗) and (∗∗) the vector ε_{1} + cε_{i} with c running through R, and obtain from q ≤ q′ that

(∗ ∗ ∗)  α_{1} + cα_{1i} ≤ β_{1} + cβ_{1i}

for all c ∈ R. Choosing here c = 0 gives α_{1} ≤ β_{1}. If β_{1i} = 0 then cα_{1i} ≤ β_{1} for all c ∈ R. By multiplicative unboundedness of eR, we have some c_{0} ∈ eR such that cβ_{1i} >_{ν} β_{1} for all c ≥ c_{0} and then obtain from (∗ ∗ ∗) that cα_{1i} ≤ cβ_{1i} for all c ≥ c_{0}, which by property (†) implies α_{1i} ≤ β_{1i}. Thus α_{1i} ≤ b_{1i} in both cases.

Proof of (2): Let 1 < k ≤ r, r < ℓ ≤ n. we insert in (∗) and (∗∗) the vector cε_{k} + ε_{ℓ} with c running through R, and obtain from q ≤ q′ that

α_{ℓ} + cα_{kℓ} ≤ β_{ℓ} + cβ_{kℓ}

for all c ∈ R. This implies α_{ℓ} ≤ β_{ℓ} and α_{kℓ} ≤ β_{kℓ} by the same arguments as before.

□

We are ready to determine all basic projections X of a basic module Z. In view of Corollary 3.6 it suffices to look for those submodules X of Z where Λ(X) = Λ(Z), equivalently ℌ_{0}(X) = ℌ_{0}(Z).

Definition 3.11. We say that two elements d_{i} and d_{j}, i ≠ j, of 𝔇_{0}(Z) are linked in Z if h_{ij} ∈ Z.

Note that the condition C_{Λ,Δ}(i) from Theorem 3.10 means that d_{i} is not linked in Z to any d_{j} ∈ Z.

Theorem 3.12. Let X and Z be basic modules with X ⊂ Z and ℌ_{0}(X) = ℌ_{0}(Z). Then X is a basic projection of Z iff any two elements d_{i}, d_{j} ∈ 𝔇_{0}(Z) which are linked in Z are both elements of 𝔇_{0}(X). In more imaginative terms: we obtain all basic projections X of Z with ℌ_{0}(X) = ℌ_{0}(Z) by discarding from 𝔇_{0}(Z) some elements which are not linked in Z to other elements of 𝔇_{0}(Z).

Proof. (⇐): Lemma 3.7 means that, if X is a basic projection of Z, then this condition holds.

(⇒): Let Λ = Λ(Z), Δ = Δ(Z), and

E := E(Z) = {i ∈ I | C_{Λ,Δ}(i) holds }
= {i ∈ I | d_{i} ∈ Z, d_{i} is not linked to any d_{i} ∈ Z}.

Theorem 3.10 assures that for every i ∈ E there exists a basic projector ϑ_{i} on Z with kernel Rd_{i}. It now follows by Remark 3.4.c that for any subset K of E the basic projection

(3.2)  ϑ_{K} = ∏_{i∈K} ϑ_{i} : Z → Z

has kernel ∑_{i∈K} Rd_{i}. Thus

𝔇_{0}(ϑ_{K}(Z)) = {d_{i} ∈ 𝔇_{0}(Z) | i ∉ K}

and, of course, ℌ_{0}(ϑ_{K}(Z)) = ℌ_{0}(Z). □

Corollary 3.13. Let Z be any basic module. Among the basic projections X of Z with ℌ_{0}(X) = ℌ_{0}(Z) there is a unique minimal one, X_{min}. All basic submodules X of Z with X_{min} ⊂ X are basic projections of Z, of course with ℌ_{0}(X) = ℌ_{0}(Z).

Theorem 3.14. With the hypotheses of Convention 3.8, let Z be any basic submodule of Quad(V). Then any basic projector π of Z with π(q) = q for all ℌ_{0}(Z) respects the functional ordering on Z:

q ≤ q′ ⇒ π(q) ≤ π(q′).

Proof. Clear from Theorem 3.10.b, since such a projector has the shape ϑ_{K} given in (3.2). □

4. Direct decompositions; linked versus free modules

In this section, as well as in §5 and §6, it will be good to keep the following simple fact in mind.

Lemma 4.1. Assume that φ : X → Y is a linear map between R-modules X and Y (with R supertropical as always). Assume that S is a subset of Y which is convex w.r. to the minimal ordering ⪯ (i.e., if s, t ∈ S, y ∈ Y and s ⪯ y ⪯ t, then y ∈ S). Then the preimage φ^{−1}(S) is convex in X. In particular (take S = {0}) the kernel φ^{−1}(0) of φ is a convex submodule of X.

Proof. This is evident, since φ is additive and so x ⪯ x′ implies φ(x) ⪯ φ(x′). □

Note that a submodule S of an R-modules X is convex w.r. to the minimal ordering on X iff S is a lower set in X, since we always have 0 ∈ S.

In the sequel we assume that eR is mutiplicatively unbounded and Z is a “basic module”, i.e., Z is a basic submodule of Quad(V) for V a fixed free R-module with base {ε_{i} | i ∈ I}, cf. Definitions 3.2 and 3.3. We want to get insight into the presentations of Z as a direct sum of submodules (which then are again basic modules).

We start with two easy facts.

Proposition 4.2. Assume that X and Y are two basic submodules of a basic module Z with X ∩ Y = {0} and X + Y = Z.

(a) The following are equivalent (1) Z = X ⊕ Y;

(2) Both X and Y are basic projections of Z;

(3) Both X and Y are basic projection kernels of Z.

(b) If (1)-(3) hold the both X and Y are convex submodules of Z.

Proof. a): Since the elements of 𝔅_{0}(Z) are indecomposable, it follows from X ∩ Y = {0} and X + Y = Z that 𝔅_{0}(Z) is the disjoint union of 𝔅_{0}(X) and 𝔅_{0}(Y). Thus the implication (1) ⇒ (2) is obvious. Furthermore, if X is a basic projection of Z, then Y is the associated projection kernel, i.e., Y = π_{Z,X}^{−1}(0). The same holds for X, Y interchanged. This makes the implications (2) ⇔ (3) evident.

(2) ⇒ (1): Let z ∈ Z and z = x + y with x ∈ X, y ∈ Y. Then π_{Z,X}(x) = x, π_{Z,X}(y) = 0, π_{Z,Y} (x) = 0, π_{Z,Y} (y) = y, whence π_{Z,X}(z) = x, π_{Z,Y} (z) = y. Thus x and y are uniquely determined by z, which proves that Z = X ⊕ Y.

b): Obvious from Lemma 4.1, since X = π_{Z,Y}^{−1} (0) and Y = π_{Z,X}^{−1}(0). □

Proposition 4.3. Assume that {X_{α} | α ∈ A} is a family of convex submodules of Z with X = ∑_{α∈A} X_{α} and X_{α} ∩ X_{β} = 0 for α ≠ β. Then each X_{α} is a basic submodule of Z, and the basic set of generators 𝔅_{0}(Z) is the disjoint union of the sets 𝔅_{0}(X_{α}).

Proof. Since the elements of 𝔅_{0}(Z) are indecomposable, it follows from X = ∑_{α∈A} X_{α} and X_{α} ∩ X_{β} = {0} for α ≠ β that every element of 𝔅_{0}(Z) is contained in some X_{α}, α ∈ A. Let X′_{α} denote the R-submodule spanned by X_{α}∩𝔅_{0}(Z). This is the maximal basic module of Z containing X_{α}. Since 𝔅_{0}(Z) is the union of the sets X_{α} ∩ 𝔅_{0}(Z) it is clear that Z = ∑_{α∈A} X′_{α}. Picking any α ∈ A, we will be done by verifying that X_{α} = X′_{α}. Let x ∈ X_{α} and write x = ∑_{β∈A} x′_{β} with x′_{β} ∈ X′_{β} (almost all x′_{β} = 0). Clearly x′_{β} ⪯ x for all β and thus x′_{β} ∈ X_{α}. If β ≠ α, then x′_{β} ∈ X_{α} ∩ X_{β} = {0}, and so x = x′_{α}. □

But if X and Y are convex submodules of the basic module X + Y = Z, where X ∩ Y = {0}, then it is not necessarily true that Z is the direct sum of X and Y, as the following key example shows. As usual, we call an R-module U decomposable if there exist submodules U_{1} ≠ U, U_{2} ≠ U such that U = U_{1} ⊕ U_{2}; otherwise, we call U indecomposable.

Example 4.4. Let I = {1, 2} and let Z = Quad(V) = Rd_{1} +Rd_{2} +Rh_{12}. Then

Rd_{1} ∩ (Rd_{2} + Rh_{12}) = {0},

and Rd_{2} + Rh_{12} is convex in Z since it is the kernel of a basic projector π_{1} : Z → Rd_{1}, namely the composite of π_{QL} : Z → Rd_{1} + Rd_{2} and the basic projector of the free module Rd_{1} + Rd_{2} onto Rd_{2}. For the same reason Rd_{1} + Rh_{12} is also convex in Z. We verify directly that Rd_{1} is convex in Rd_{1} + Rh_{12} and so is convex in Z. Indeed, if αd_{1} + βh_{12} ⪯ γd_{1} with α, β, γ ∈ R, then βh_{12} ⪯ γd_{1}. A fortiori βh_{12} ≤ γd_{1}, which means that βx_{1}x_{2} ≤ γx_{1}^{2} for all x_{1}, x_{2} ∈ R. In particular for x_{1} = 1 we obtain βx_{2} ≤ γ for all x_{2} ∈ R, which forces β = 0 by multiplicative unboundedness. But Z is not the direct sum of the convex submodules Rd_{1} and Rd_{2} + Rh_{12}. In fact d_{1} + d_{2} = d_{1} + d_{2} + h_{12}, while d_{2} ≠ d_{1} + h_{12}. (Insert x_{1}ε_{1} + ε_{2} with some x_{1} >_{ν} 1.) Neither is Z the direct sum of Rd_{2} and Rd_{1} + Rh_{12}. We conclude that Z is indecomposable.

Remark 4.5. Hypothesis (†) in Convention 3.8 is not needed in this example. If (†) holds in addition to multipicativity unboundedness, then it is immediate from Theorem 3.10 that Rd_{2} + Rh_{12} is free. Also Rd_{1} + Rh_{12} is free, and so all proper basic submodules of Z are free.

Definition 4.6.

(a) We call a basic module Z linked if (i) for every h_{ij} ∈ Z both d_{i} and d_{j} are in Z; (ii) for each d_{i} ∈ Z there exists some j ≠ i such that h_{ij} ∈ Z, and so

d_{i} is linked in Z to some d_{j} ∈ Z (cf. Definition 3.11).

(b) We call

E_{ij} := Rd_{i} + Rd_{j} + Rh_{ij} (i, j ∈ I, i ≠ j)

an elementary linked module. Thus Z is linked iff Z is the sum of all elementary linked modules contained in Z.

(c) Given any basic module Z we denote the sum of all E_{ij} ⊂ Z by Z_{link}, which is the maximal linked submodule of Z. We call it the linked core of Z.

Example 4.4 shows that every elementary linked module is indecomposable. The formation of linked cores behaves well with respect to direct sums.

Proposition 4.7. If (Z_{α} | α ∈ A) is any family of basic R-modules, then

( ⨁_{α∈A} Z_{α})_{link} = ⨁_{α∈A} (Z_{α})_{link}.

This is an immediate consequence of the fact that every elementary linked module is indecomposable, together with the following easy lemma.

Lemma 4.8. Assume that Z is a basic module and (X_{α} | α ∈ A) is a family of submodule such that

Z = ⨁_{α∈A} X_{α}.

Then for any convex submodule W of Z we have

W = ⨁_{α∈A′} W ∩ X_{α}

where A′ := {α ∈ A | W ∩ X_{α} ≠ 0}. In particular, when W is indecomposable, W ⊂ X_{α} for some α ∈ A′.

Proof. We have a family (π_{α} | α ∈ A) of basic projectors π_{α} : Z → Z at hands with π_{α}(Z) = X_{α}. It follows that π_{α}π_{β} = π_{β}π_{α} = 0 for α ≠ β and ∑_{α∈A} π_{α} = id_{Z} (which means that, given q ∈ Z, almost all values π_{α}(q) are zero and ∑_{α∈A} π_{α}(q) = q). Now π_{α}(z) ⪯ z for every z ∈ Z, α ∈ A, and thus π_{α}(W) ⊂ W for every α ∈ A. By restriction we obtain a family of projectors

π′_{α} = π_{α}|W : W → W, α ∈ A′.

Since π′_{α}π′_{β} = π′_{β}π′_{α} = 0 for α ≠ β and ∑_{α∈A′} π′_{α} = id_{W}, it is immediate that W = ⨁_{α∈A′} π′_{α}(W) = ⨁_{α∈A′} W ∩ X_{α}. ^{6} □

Proposition 4.9. Assume that Convention 3.8 is in force. Then Z is free iff Z_{link} = 0.

Proof. When Z is free, all its basic submodules are free, and so Z cannot contain any elementary linked module, whence Z_{link} = 0. (N.B. In this argument hypothesis (†) is not needed.) Conversely, if Z_{link} = 0, we know by Theorem 3.12 that Z ∩ Rig(V) is a basic projection of Z with kernel Z ∩ QL(V). (All d_{i} ∈ 𝔇_{0}(Z) may be discarded from the list 𝔅_{0}(Z).) But also Z ∩ QL(V) is a basic projection of Z, namely the image of the restriction π_{QL} : Quad(V) ↠ QL(V) to Z. Thus

6In this proof our standard assumption that eR is multiplicatively unbounded is not needed.

Z = (Z ∩ QL(V)) ⊕ (Z ∩ Rig(V)).

Since both QL(V) and Rig(V) are free, Z is also free. □

Given a basic submodule X of Z, we call the unique basic module Y with X + Y = Z, X ∩ Y = {0}, the basic complement Y of X in Z.

Theorem 4.10. Assume that Convention 3.8 is in force. Then, the basic complement of Z_{link}, Y, in Z is free and Z = Z_{link} ⊕ Y.

Proof. Y does not contain any elementary linked module, and thus is free by the preceding proposition. We will be done by verifying that both Z_{link} and Y are basic projections of Z. By mapping all the h_{ij} ∈ Y to zero we obtain a basic projector Z ↠ Z_{1} with

ℌ_{0}(Z_{1}) = ℌ_{0}(Z) \ ℌ_{0}(Y) = ℌ_{0}(Z_{link}), 𝔇_{0}(Z_{1}) = 𝔇_{0}(Z),

cf. Corollary 3.6. We have Z_{1} ⊃ Z_{link} and

𝔇_{0}(Z_{1}) = 𝔇_{0}(Y) ˙∪ 𝔇_{0}(Z_{link}),

but no d_{i} ∈ 𝔇_{0}(Y) is linked to any d_{j} ∈ 𝔇_{0}(Z_{1}), and thus Theorem 3.10 gives us a basic projector Z_{1} ↠ Z_{link}. Composing the two projectors we obtain a basic projector π_{1} : Z ↠ Z_{link}. On the other hand by mapping all h_{ij} ∈ ℌ_{0}(Z_{link}) to zero we obtain a basic projector Z ↠ Z_{2} with

ℌ_{0}(Z_{2}) = ℌ_{0}(Z) \ ℌ_{0}(Z_{link}) = ℌ_{0}(Y), 𝔇_{0}(Z_{2}) = 𝔇_{0}(Y) ˙∪ 𝔇_{0}(Z_{link}).

As no d_{i} ∈ 𝔇_{0}(Z_{link}) is linked to any d_{j} ∈ 𝔇_{0}(Y), we obtain a basic projector Z_{2} ↠ Y, again by Theorem 3.10, which together with Z ↠ Z_{2} yields a basic projector π_{2} : Z ↠ Y. The projectors π_{1} and π_{2} together entail Z = Z_{link} ⊕ Y.

□

We denote the basic complement Y of Z_{link} in Z by Z_{free}, and obtain

Z = Z_{link} ⊕ Z_{free}.

Corollary 4.11. Suppose Z is a basic module and Convention 3.8 is in force, then Z_{free} is the unique maximal basic free submodule of Z which is a direct summand of Z.

Proof. Let Z = X ⊕ T with X free. Then (cf. Proposition 4.7)

Z_{link} = X_{link} ⊕ T_{link} = T_{link},

and T = T_{link} ⊕ T_{free}. We conclude that

Z = X ⊕ T_{link} ⊕ T_{free}

and also Z = Z_{link} ⊕ Z_{free} = T_{link} ⊕ Z_{free}, whence Z_{free} = X ⊕ T_{free}. □

Definition 4.12. We call any indecomposable direct summand X ≠ 0 of a basic module Z a component of Z.

We start out to determine the components of a basic R-module Z under Convention 3.8. First an easy preliminary lemma, valid over any supertropical semiring R.

Lemma 4.13. Assume that ⨁_{α∈A} Z_{α} is a direct decomposition of Z, where

each Z_{α} is indecomposable and ≠ 0. Then these Z_{α} are precisely all components of Z.

Proof. Let X be a basic nonzero submodule of Z. By Lemma 4.8

X = ⨁_{α∈A} X ∩ Z_{α}.

If X is indecomposable then X = X∩Z_{α} for a unique α ∈ A, whence X ⊂ Z_{α}. If Y is the basic complement of X in Z so that Z = X⊕Y, then Z_{α} = X⊕(Y ∩Z_{α}). Since Z_{α} is indecomposable, this implies that X = Z_{α}. □

Definition 4.14.

(a) We call a basic module Z graphic if for any h_{ij} ∈ ℌ_{0}(Z) both d_{i} and d_{j} are in Z.

(b) When Z is graphic, we define the graph Γ(Z) (simple, undirected, without loops) as follows. Γ(Z) has the sets of vertices and edges

Ver(Γ(Z)) = 𝔇_{0}(Z) and Edg(Γ(Z)) = ℌ_{0}(Z).

An edge h_{ij} connects the vertices d_{i}, d_{j}. ^{7}

(c) A basic module Z has a unique maximal submodule which is graphic, denoted Z_{graph}. 𝔅_{0}(Z_{graph}) is obtained from 𝔅_{0}(Z) by omitting every h_{ij} ∈ Z with d_{i} ∉ Z or d_{j} ∉ Z.

Note that Z is linked iff Z is graphic and Γ(Z) has no isolates vertices. When Convention 3.8 holds, clearly Z is graphic iff ℌ_{0}(Z_{free}) is empty.

Remark 4.15. Γ(Quad(V)) is the complete graph over the vertex set {d_{i} | i ∈ I}, and thus may be seen as the graph (I, I^{[2]}). For any subgraph Γ′ of Γ(Quad(V)) there exists a unique graphic module Z with Γ(Z) = Γ′.

Next we describe the components of graphic modules in graph theoretic terms. As before we tacitly assume that G is multiplicatively unbounded.

Proposition 4.16. Assume that Z is a graphic module and (X_{α} | α ∈ A) is a family of submodules of Z. For each α ∈ A let J_{α} := {i ∈ I | d_{i} ∈ X_{α}}. The following are equivalent:

(i) Z = ⨁_{α∈A} X_{α};

(ii) Each X_{α} is graphic and Γ(Z) is the disjoint union of the graphs Γ(X_{α}), for which we write

7It would be more precise to consider 𝔇(Z) and ℌ(Z) respectively as sets of vertices and edges, but the present setting has proved to be more convenient.

Γ(Z) = ⨆_{α∈A} Γ(X_{α}).

When (i) and (ii) hold, the projection Z ↠ X_{α} is given, for q ∈ Z, by (cf. Notation 2.4)

(4.1)  π_{α}(q) = (q|V_{J_{α}})^{I}.

Proof. a) Assume that Z is the direct sum of the X_{α}. If h_{ij} ∈ X_{α} for some i, j ∈ I, then the d_{i} and d_{j} are elements of Z, since Z is graphic. The elementary linked submodule E_{ij} of Z is indecomposable (cf. Example 4.4) and E_{ij} ∩ X_{α} ≠ {0}. By the (essentially trivial) Lemma 4.8 we conclude that E_{ij} ⊂ X_{α}. Thus X_{α} is graphic. It now follows directly from Definition 4.14 that

Γ(Z) = ⨆_{α∈A} Γ(X_{α}).

b) We now assume that Γ(Z) is the disjoint union of the graphs Γ(X_{α}), α ∈ A. Firstly this implies that Z = ∑_{α∈A} X_{α}. For each α ∈ A we define a map π_{α} : Z → Z by formula (4.1). Let J := {i ∈ I | d_{i} ∈ Z}, so that J is the disjoint union of the sets J_{α}. Given i ∈ J we have (d_{i}|V_{J_{α}})^{I} = d_{i} for i ∈ J_{α} and (d_{i}|V_{J_{α}})^{I} = 0 otherwise. Given different i, j ∈ J we have a similar story: (h_{ij}|V_{J_{α}})^{I} = h_{ij} when {i, j} ⊂ J_{α} and zero otherwise. This proves that π_{α} is a basic projector on Z with image X_{α}, and furthermore that π_{α}π_{β} = 0 if α ≠ β and ∑_{α∈A} π_{α} = id_{Z}. It is now obvious that the sum ∑_{α∈A} X_{α} is direct with associated projectors π_{α}. □

Corollary 4.17. A graphic module Z is indecomposable iff the graph Γ(Z) is connected.

Proof. We argue by contradiction. If Z = X_{1} ⊕ X_{2} with X_{1} ≠ 0, X_{2} ≠ 0, then by Proposition 4.16 both X_{1}, X_{2} are graphic and Γ(Z) = Γ(X_{1}) ⊔ Γ(X_{2}); so Γ(Z) is not connected. Conversely, assume Γ(Z) = Γ_{1} ⊔Γ_{2} and let X_{i} denote the graphic submodule of Z with Γ(X_{i}) = Γ_{i} (i = 1, 2). Then Z = X_{1} ⊕ X_{2} by Proposition 4.16, and thus Z is decomposable. □

Theorem 4.18. Assume that Z is a graphic module. Let (Γ_{γ} | γ ∈ C) denote the set of components of the graph Γ(Z), arbitrarily indexed, and for every γ ∈ C let Z_{γ} denote the graphic module with Γ(Z_{γ}) = Γ_{γ}. Then

Z = ⨁_{γ∈C} Z_{γ}

and the Z_{γ} are precisely all components of the basic module Z.

Proof. We know by Proposition 4.16, that Z is the direct sum of the Z_{γ} and by Corollary 4.17 that the Z_{γ} are indecomposable. It follows from Lemma 4.8 that the Z_{γ} are all components of Z. □

We emphasize that in our study of the components of a basic module Z up to now hypothesis (†) has not been needed. But if (†) holds, then we know that Z = Z_{link} ⊕ Z_{free}. Applying Theorem 4.18 to Z_{link}, we obtain the following corollary.

Corollary 4.19. Assume that Convention 3.8 is in force. Let Z be a basic module, and let (Z_{γ} | γ ∈ C) denote the set of components of Z indexed in some way. Then

Z = ⨁_{γ∈C} Z_{γ}.

Furthermore, Z_{link} is the direct sum of those components Z_{γ}, which are not free, while Z_{free} is the direct sums of all others. They are free of rank one.

5. Basic endomorphisms and their associated projectors

As before R is a supertropical semiring. Let Z be a basic module over R, i.e., a basic submodule of Quad(V), for V a fixed free R-module V, cf. Definitions 3.2 and 3.3. While in §3 and §4 we studied basic projectors on Z, we now proceed to the more general “basic endomorphisms” of Z. We denote the set of endomorphisms of Z by End_{R}(Z) or simply End(Z). This is an R-algebra in the obvious sense. As before we work with the set of generators 𝔅_{0} = 𝔇_{0} ∪ ℌ_{0},

𝔇_{0} := {d_{i} | i ∈ I}, ℌ_{0} := {h_{ij} | {i, j} ∈ I^{[2]}}

of Quad(V) derived from a fixed base of V, cf. (3.1). By intersection with Z it gives a “basic set of generators” 𝔅_{0}(Z) = 𝔇_{0}(Z) ∪ ℌ_{0}(Z), cf. Notation 3.5.

Definition 5.1. An endomorphism φ of Z is called basic, if φ(Rq) ⊂ Rq for every q ∈ 𝔅_{0}, whence

φ(d_{i}) = μ_{i}d_{i}, φ(h_{ij}) = μ_{ij}d_{ij}

for all d_{i} ∈ Z, h_{ij} ∈ Z, with elements μ_{i}, μ_{ij} in R, which we name the coefficients of φ. The set of all basic endomorphisms of Z, denoted by End_{b}(Z), is a commutative subalgebra of End(Z).

We remark that for any basic submodule Z′ of Z, every φ ∈ End_{b}(Z) restricts to a basic endomorphism φ|Z′ of Z′. Note also that the basic projectors on Z are precisely the basic endomorphisms of Z with all coefficients in {0, 1}. They are idempotents of the R-algebra End_{b}(Z).^{8}

Example 5.2. If ρ is an endomorphism of the free R-module V, then for every quadratic form q : V → R, the composite q ∘ ρ : V → R is again a quadratic form on V, and so we obtain an endomorphism

ρ^{∗} : q ↦ q ∘ ρ,

of the R-module Quad(V). We call these ρ^{∗} the geometric endomorphisms of Quad(V). If ρ itself is “basic”, i.e., ρ(ε_{i}) = μ_{i}ε_{i} for every i ∈ I with some μ_{i} ∈ R, then an easy computation shows that

ρ^{∗}(d_{i}) = μ_{i}^{2}d_{i}, ρ^{∗}(h_{ij}) = μ_{i}μ_{j}h_{ij},

8There exist other idempotents, e.g. φ = e id_{Z}.

whence ρ^{∗} is a basic endomorphism of Quad(V). We denote this endomorphism by γ_{μ}, where μ := (μ_{i} | i ∈ I), and call the γ_{μ} the geometric basic endomorphisms of Quad(V).

These endomorphisms γ_{μ} are the “easy” basic endomorphisms of Quad(V). Every tuple μ = (μ_{i} | i ∈ I) ∈ R^{I} gives such an endomorphism with system of coefficients

(μ_{i}^{2} | i ∈ I) ∪ (μ_{i}μ_{j} | {i, j} ∈ I^{[2]}).

By restriction we obtain for γ_{μ} a basic endomorphism γ_{Z,μ} := γ_{μ}|Z, and then have the following upshot of Example 5.2.

Proposition 5.3. Let Z be a basic module and set I(Z) := (i ∈ I | d_{i} ∈ Z). Every tuple μ = (μ_{i} | i ∈ I(Z)) yields a unique basic endomorphisms γ_{Z,μ} of Z with

γ_{Z,μ}(d_{i}) = μ_{i}^{2}d_{i}

for all d_{i} ∈ Z and

γ_{Z,μ}(h_{ij}) = μ_{i}μ_{j}h_{ij}

for all h_{ij} ∈ Z.

We call the γ_{Z,μ} the geometric basic endomorphisms of Z. They form a subset of End_{b}(Z), closed under multiplication. Note that this set does not depend on the choice of the base {ε_{i} | i ∈ I} of V. In particular we have the geometric basic projectors of Z at hands. These are the endomorphisms γ_{Z,μ} with μ_{i} ∈ {0, 1} for all i ∈ I(Z), and thus they correspond uniquely to the subsets J = {i ∈ I(Z) | μ_{i} = 1} of I(Z).

Notations 5.4. We denote the basic projection coming from such a set J by π_{Z,J}. In the most important case, namely Z = Quad(V), we write π_{J} instead of π_{Quad(V),J} and so

π_{Z,J} = π_{J}|Z

for J ⊂ I(Z).

Proposition 5.5.

(a) For any J ⊂ I the geometric basic projector π_{J} on Quad(V) can be also described by the formula (q ∈ Quad(V))

(5.1)  π_{J}(q) = (q|V_{J})^{I} =: q_{J},

cf. Notation 2.4.

More generally, for any J ⊂ I and φ ∈ End_{b}(Quad(V)), q ∈ Quad(V), we have the formula

(5.2)  (φπ_{J})(q) = (φ(q)|V_{J})^{I} = φ(q)_{J}.

(b) Given subsets J, K of I, we have π_{J}π_{K} = π_{J∩K}. If J ∩ K is empty, then π_{J∪K} = π_{J} + π_{K}.

Proof. (a): (5.1) is the formula (5.2) in the special case that φ is the identity map. In order to verify (5.2) it is suffices to check this formula for every q ∈ 𝔅_{0} = 𝔇_{0} ∪ ℌ_{0}. Let (μ_{i}) ∪ (μ_{ij}) be the family of coefficients of φ. If i, j ∈ J, i ≠ j, then

(φπ_{J})(d_{i}) = φ(d_{i}) = μ_{i}d_{i},
(φπ_{J})(h_{ij}) = φ(h_{ij}) = μ_{ij}h_{ij},

while

(φπ_{J})(d_{k}) = 0 if k ∉ J,
(φπ_{J})(h_{kℓ}) = 0 if {k, ℓ} ⊄ J.

(φ(q)|V_{J})^{I} has exactly the same values for all q ∈ 𝔅_{0}.

(b): Obvious by considering the coefficients of the occurring projectors, or (better) the endomorphisms of V including these geometric endomorphisms of Quad(V). □

In the case of |J| ≤ 2 we simplify the notation by writing

(5.3)  π_{i} := π_{{i}},  π_{ij} := π_{{i,j}}.

These projectors will play a very helpful role later. All basic endomorphisms of Quad(V) can be built from basic endomorphisms of the elementary linked submodules of Quad(V) (cf. Definition 4.6) as follows.

Lemma 5.6 (Pasting Lemma). Let Z = Quad(V). Assume that (φ_{ij} | {i, j} ⊂ I) is a family of basic endomorphisms φ_{ij} ∈ End_{b}(E_{ij}), E_{ij} = Rd_{i} + Rd_{j} + Rh_{ij}. Assume further that φ_{ij}|Rd_{i} = φ_{ik}|Rd_{i} for any three distinct i, j, k ∈ I. Then there is a unique φ ∈ End_{b}(Z) with φ|E_{ij} = φ_{ij}, for all {i, j} ∈ I^{[2]}.

Proof. We have elements μ_{i}, μ_{ij} in R with

φ(d_{i}) = μ_{i}d_{i}, φ(h_{ij}) = μ_{ij}h_{ij}

for all {i, j} ∈ I^{[2]}. The basic endomorphism φ of Z with system of coefficients (μ_{i}) ∪ (μ_{ij}) has the required properties, and clearly is the unique one, provided that φ exists. For notational convenience we choose a total ordering on I. If q ∈ Quad(V), and two presentations of q are given,

(A)  q = ∑_{i∈I} α_{i}d_{i} + ∑_{i<j} α_{ij}h_{ij} = ∑_{i∈I} α_{i}d_{i} + ∑_{i<j} β_{ij}h_{ij}

(of course with only finitely many scalars α_{i}, α_{ij}, β_{ij} ≠ 0 ), we need to verify that

(B)  ∑_{i∈I} α_{i}μ_{i}d_{i} + ∑_{i<j} α_{ij}μ_{ij}h_{ij} = ∑_{i∈I} α_{i}μ_{i}d_{i} + ∑_{i<j} β_{ij}μ_{ij}h_{ij}.

For any pair i < j in I we apply the projector π_{ij} to (A), viewed as a map onto E_{ij}, and then the map φ_{ij}. We obtain

α_{i}μ_{i}d_{i} + α_{j}μ_{j}d_{j} + α_{ij}μ_{ij}h_{ij} = α_{i}μ_{i}d_{i} + α_{j}μ_{j}d_{j} + β_{ij}μ_{ij}h_{ij}.

Since this holds for all i < j, (B) is now evident. From B we conclude that there is a well defined map φ : Quad(V) → Quad(V) sending any q = ∑_{i∈I} α_{i}d_{i} + ∑_{i<j} α_{ij}h_{ij} to ∑_{i∈I} α_{i}μ_{i}d_{i} + ∑_{i<j} α_{ij}μ_{ij}h_{ij}. It is obvious that this map φ is a basic endomorphism and φ(d_{i}) = μ_{i}d_{i}, φ(h_{ij}) = μ_{ij}h_{ij} for all i < j. □

Remark 5.7. If I is finite, then

φ = ∑_{i<j} φ_{ij} ∘ π_{ij},

with π_{ij} the basic projector from (5.3), viewed as a map from Quad(V) onto E_{ij}, since π_{ij}|E_{ij} = id_{E_{ij}}, π_{kℓ}|E_{ij} = 0 for {k, ℓ} ≠ {i, j}.

Theorem 5.8 (Extension Theorem). Let φ be a basic endomorphism of a graphic module Z ⊂ Quad(V), and for every d_{i} ∈ 𝔇_{0} \ 𝔇_{0}(Z) choose a scalar v_{i} ∈ R. Then there exists a unique ψ ∈ End_{b}(Quad(V)) such that ψ|Z = φ, ψ(d_{i}) = v_{i}d_{i} for d_{i} ∉ Z, and ψ(h_{ij}) = 0 for all h_{ij} ∉ Z.

Proof. We choose a family (ψ_{ij} | i < j) of endomorphisms ψ_{ij} ∈ End_{b}(E_{ij}) as follows. When E_{ij} ⊂ Z we set ψ_{ij} = φ|E_{ij}. When E_{ij} ⊄ Z, whence h_{ij} ∉ Z, we define ψ_{ij} : E_{ij} → E_{ij} as ψ_{ij}(d_{i}) = μ_{i}d_{i} with μ_{i} = v_{i} if d_{i} ∉ Z, and ψ_{ij}(d_{i}) = φ(d_{i}) if d_{i} ∈ Z. We define ψ_{ij}(d_{j}) by the same rule, setting ψ_{ij}(h_{ij}) = 0. This map ψ_{ij} is the composite of the quasilinear projector π_{QL}|E_{ij} and an endomorphism of the free module Rd_{i} + Rd_{j} with prescribed values for d_{i}, d_{j}, and thus is well defined. By construction it is clear that for any three different indices i, j, k we have ψ_{ij}(d_{i}) = ψ_{ik}(d_{i}). Thus the pasting Lemma 5.6 applies and yields a basic endomorphism ψ of Quad(V), which clearly extends φ. □

In particular, we can choose in Theorem 5.8 all v_{i} = 0 to obtain an extension ψ = φ̃ of φ to Quad(V) with φ̃(q) = 0 for all q ∈ 𝔅_{0} \ 𝔅_{0}(Z). We call φ̃ the extension of φ by zero.

Convention 5.9. Up to end of §5 we assume, usually without explicitly stating it, that for R and Z one of the two following conditions holds.

Hypothesis A: G is multiplicatively unbounded and Z is a graphic module. Hypothesis B: G is multiplicatively unbounded and has property (†), cf.

Convention 3.8. Here Z can be any basic module.

Recall that, since G is assumed to be multiplicatively unbounded, the set R\{0} is closed under multiplication ([10, Remark 6.5]). We are ready for a central result of this section.

Theorem 5.10. Given a basic endomorphism φ of Z there exists a unique basic projector on Z, denoted by p_{φ}, such that any q ∈ 𝔅_{0}(Z) has the image p_{φ}(q) = q if φ(q) ≠ 0 and p_{φ}(q) = 0 if φ(q) = 0.

Proof. a) We assume Hypothesis A. We extend φ to a basic endomorphism ψ of Quad(V) in some way, which is possible by Theorem 5.8. It suffices to prove the theorem for ψ instead of φ. Restricting p_{ψ} to Z we then obtain the desired p_{φ}. Thus we furthermore assume that Z = Quad(V). We employ the Pasting Lemma 5.6. For any two indices i ≠ j in I let φ_{ij} := φ|E_{ij}. We define a basic projector p_{ij} on E_{ij} as follows. If φ(d_{i}) ≠ 0, φ(d_{j}) ≠ 0, φ(h_{ij}) ≠ 0 we take p_{ij} = id_{E_{ij}}. If φ(d_{i}) ≠ 0, φ(d_{j}) ≠ 0, but φ(h_{ij}) = 0 we choose for p_{ij} the quasilinear projection π_{QL}|E_{ij}, i.e., p_{ij}(d_{i}) = d_{i}, p_{ij}(d_{j}) = d_{j}, p_{ij}(h_{ij}) = 0. If φ(d_{i}) = φ(d_{j}) = 0, whence also φ(h_{ij}) = 0, we choose p_{ij} = 0. There remains the case that exactly one of the vectors φ(d_{i}), φ(d_{j}) is not zero, say φ(d_{i}) = μ_{i}d_{i}, μ_{i} ≠ 0, φ(d_{j}) = 0. Let φ(h_{ij}) = μ_{ij}h_{ij}. Applying φ to the relation

d_{i} + d_{j} + h_{ij} = d_{i} + d_{j},

we obtain μ_{i}d_{i}+μ_{ij}h_{ij} = μ_{i}d_{i}. Inserting the vector x = v_{i}+cv_{j} for any c ∈ R we obtain that μ_{i} + cμ_{ij} = μ_{i} for all c ∈ R. Since G is multiplicatively unbounded, this forces μ_{ij} = 0, i.e., φ(h_{ij}) = 0. Now define p_{ij} as the composite of π_{QL}|E_{ij} with the obvious projection from Rd_{i} +Rd_{j} to Rd_{i}. Then p_{ij}(d_{i}) = d_{i}, p_{ij}(d_{j}) = p_{ij}(h_{ij}) = 0. By construction p_{ij}(d_{i}) = p_{ik}(d_{i}) for any three different indices i, j, k ∈ I. Thus there exists a basic endomorphism p of Quad(V) with p|E_{ij} = p_{ij} for any i ≠ j. It is the projector p_{φ} we were looking for.

b) We assume now Hypothesis B. By Theorem 4.10 there is a decomposition Z = Y_{1} ⊕ Y_{2} with Y_{1} = Z_{link} and Y_{2} a free module. Given φ ∈ End_{b}(Z), we have φ(Y_{i}) ⊂ Y_{i} for i = 1, 2, and thus φ = φ_{1} ⊕ φ_{2} with φ_{i} ∈ End_{b}(Y_{i}). As just proved there exists a unique basic projector π_{1} = p_{φ_{1}} on Y_{1} with π_{1}(q) = q ⇔ φ_{1}(q) ≠ 0 for all q ∈ 𝔅_{0}(Y_{1}). Since Y_{2} is free we trivially also have a unique basic projector π_{2} on Y_{2} such that for all q ∈ 𝔅_{0}(Y_{2}) π_{2}(q) = q ⇔ φ_{2}(q) ≠ 0. The projector p_{φ} := π_{1} ⊕ π_{2} on Z has the required property addressed in the theorem. □

We call p_{φ} the basic projector associated to φ. We now strive for a characterization of the image and the kernel of p_{φ} in terms of the image and kernel of φ.

Definition 5.11. Given an R-submodule M of Quad(V), let M_{b} denote the unique maximal basic submodule of Quad(V) contained in M, and let M^{b} denote the unique minimal basic submodule of Quad(V) containing M. M_{b} and M^{b} are respectively called the basic core and basic hull of M.

It is obvious that

𝔅_{0}(M_{b}) = 𝔅_{0} ∩ M,

but determining 𝔅_{0}(M^{b}) can be difficult. For example, if M = Rq with q ∈ Quad(V), then 𝔇_{0}(M^{b}) consists of all d_{i} showing up in q_{QL}, while ℌ_{0}(M^{b}) is the minimal subset of ℌ_{0} appearing in a rigid complement of q_{QL}, in other terms, the minimal subset Λ ⊂ I^{[2]} such that q is Λ^{c}-quasilinear. But there is an extended class of submodules M of Quad(V) at hands, for which the determination of 𝔅_{0}(M^{b}) is trivial.

Definition 5.12. We call a submodule M of Quad(V) diagonal if

(5.4)  M = ∑_{i∈I} 𝔞_{i}d_{i} + ∑_{i<j} 𝔞_{ij}h_{ij}

with ideals (= R-submodules) 𝔞_{i}, 𝔞_{ij} of R.

We read off from (5.4) that

𝔇_{0}(M^{b}) := {d_{i} | 𝔞_{i} ≠ 0}, ℌ_{0}(M^{b}) := {h_{ij} | 𝔞_{ij} ≠ 0}.

Examples 5.13.

(a) Every basic module Z is diagonal.

(b) Every submodule of Quad(V) that is convex in the minimal ordering ⪯ of Quad(V) is easily seen to be diagonal.

(c) If M is a diagonal submodule of a basic module Z, then for any φ ∈ End_{b}(Z) also φ(M) is diagonal.

(d) Intersections and sums of families of diagonal submodules of Quad(V) are diagonal.

(e) R(d_{i} + d_{j}) is not diagonal if i ≠ j. (This module has the basic hull Rd_{i} + Rd_{j}.)

Proposition 5.14. If φ is a basic endomorphism of a basic module Z then φ(Z) is a diagonal module and p_{φ}(Z) = φ(Z)^{b}. The kernel φ^{−1}(0) is itself basic and so

p_{φ}^{−1}(0) = φ^{−1}(0) = φ^{−1}(0)_{b}.

Proof. It is evident that the modules φ(Z) and φ^{−1}(0) are diagonal by Examples 5.13.(b) and (c). By definition of p_{φ} we have

\begin{aligned}𝔅_{0}(p_{φ}(Z)) & = \{q ∈ 𝔅_{0}(Z) | p_{φ}(q) = q\} \\ & = \{q ∈ 𝔅_{0}(Z) | φ(q) ≠ 0\} & = 𝔅_{0}(φ(Z)^{b}), \end{aligned}

which proves that p_{φ}(Z) = φ(Z)^{b}. In the same way

\begin{aligned}𝔅_{0}(p_{φ}^{−1}(0)) & = \{q ∈ 𝔅_{0}(Z) | p_{φ}(q) = 0\} \\ & = \{q ∈ 𝔅_{0}(Z) | φ(q) = 0\}, \end{aligned}

whence p_{φ}^{−1}(0) = φ^{−1}(0)_{b}. But φ^{−1}(0) is already basic. Indeed, if ∑_{1}^{n} λ_{i}q_{i} ∈ φ^{−1}(0) with q_{i} ∈ 𝔅_{0}, λ_{i} ≠ 0, then ∑_{1}^{n} λ_{i}φ(q_{i}) = 0, whence λ_{i}φ(q_{i}) = 0, and so all φ(q_{i}) = 0, since R has no zero-divisors. □

Therefore, p_{φ} = π can also be characterized by the property that, for all q ∈Z, π(q) ≠ 0 iff φ(q) ≠ 0.

Example 5.15. Let μ = (μ_{i} | i ∈ I) ∈ R^{I}. Then the geometric basic endomorphism γ_{μ} of Quad(V) (cf. Example 5.2) has the associated basic projector π_{J} (cf. Notation 5.4) with J = {i ∈ I | μ_{i} ≠ 0}.

We turn to the inverse problem of analyzing the set of basic endomorphisms φ with p_{φ} = π for a given basic projector π.

Definition 5.16. Let Z be a basic module.

(a) We denote the set of all basic projectors on Z by P_{b}(Z).

(b) If π ∈ P_{b}(Z), then we call an endomorphism φ ∈ End_{b}(Z) with p_{φ} = π a satellite of π; the set of these φ is denoted by Stl(π).

It will be helpful to work in End_{b}(Z) with a partial ordering finer than the minimal ordering, analogous to our setting for Z itself. Given φ, ψ ∈ End_{b}(Z), we define

φ ≤ ψ ⇔ ∀z ∈ Z : φ(z) ≤ ψ(z),

where on the right hand side “ ≤′′ stands for the function ordering on Z ⊂ Quad(V). We call this finer relation ≤ the function ordering on End_{b}(Z), in contrast to the minimal ordering which is denoted by ⪯ and defined as

φ ⪯ ψ ⇔ ∃χ ∈ End_{b}(Z) : φ + χ = ψ.

Given φ, ψ ∈ End_{b}(Z) with coefficients systems (μ_{i}) ∪ (μ_{ij}) and (υ_{i}) ∪ (υ_{ij}), respectively, then φ ≤ ψ iff μ_{i} ≤ υ_{i}, μ_{ij} ≤ υ_{ij} for all i with d_{i} ∈ Z and all {i, j} with h_{ij} ∈ Z.

Theorem 5.17. Assume that either Hypothesis A or B holds, and that there exists some ϑ ∈ G with ϑ < e. Let π be a nonzero basic projector on Z with 𝔅_{0}(π(Z)) finite (e.g. 𝔅_{0}(Z) is finite, i.e., Z is finitely generated). Then a basic endomorphism φ of Z is a satellite of π iff there exist α, β ∈ R \ {0} such that

απ ≤ φ ≤ βπ;

in other terms, Stl(π) = conv_{≤}((Rπ) \ {0}). As usual, conv_{≤} stands for the convex hull in End_{b}(Z) (equivalently, in End_{R}(Z)) with respect to the function ordering.

Proof. Let {q_{j} | j ∈ J} denote the set 𝔅_{0}(Z). Let

φ(q_{j}) = μ_{j}q_{j} (j ∈ J)

with μ_{j} ∈ R, and write

K := {j ∈ J | π(q_{j}) = q_{j}},

which by assumption is a finite set. By definition, φ is a satellite of π iff

K = {j ∈ J | μ_{j} ≠ 0}.

When this holds, set

\begin{aligned}eμ_{k} & = min\{eμ_{j} | j ∈ K\}, \\ eμ_{ℓ} & = max\{eμ_{j} | j ∈ K\}. \end{aligned}

Set β := eμ_{ℓ}. If eμ_{k} < e set α := eμ_{k}^{2}, otherwise take α = ϑ. For these parameters we have α ≤ μ_{j} ≤ β for all j ∈ K, and so απ ≤ φ ≤ βπ. Conversely, if this holds for α, β ≠ 0, then for any q ∈ 𝔅_{0}(Z) we have φ(q) ≠ 0 iff π(q) ≠ 0, i.e., π(q) = q. Thus φ is a satellite of π. □

The finiteness assumption in Theorem 5.17 is not essential; it can be easily extended to the case in which V has an infinite basis {ε_{i} | i ∈ I}, as follows:

Corollary 5.18. Assume that either Hypothesis A or B holds, and that there is some ϑ < e in G. Let π ∈ P_{b}(Z) and φ ∈ End_{b}(Z). Then φ is a satellite of π iff for every finite subset K of I there exist α_{K}, β_{K} ∈ R \ {0} such that

α_{K}(π|Z_{K}) ≤ φ|Z_{K} ≤ β_{K}(π|Z_{K}),

where Z_{K} := (Z ∩ V_{K})^{I} = π_{K}(Z) for V_{K} = ∑_{i∈K} Rε_{i}, and π_{K} is the (geometric) basic projector (5.1) on Quad(V) with image Quad(V_{K})^{I} (cf. Proposition 5.5).

Theorem 5.19. Given π_{1}, π_{2} ∈ P_{b}(Z), both the minimum π_{1} ∧ π_{2} and the maximum π_{1} ∨ π_{2} exist in (End_{b}(Z), ≤), namely

π_{1} ∧ π_{2} = π_{1}π_{2}, π_{1} ∨ π_{2} = p_{π_{1}+π_{2}}.

Proof. This follows from the fact that for any q ∈ 𝔅_{0}(Z) we have

(π_{1}π_{2})(q) = q iff π_{1}(q) = q and π_{2}(q) = q,

while

p_{π_{1}+π_{2}}(q) = q iff π_{1}(q) + π_{2}(q) ≠ 0 iff π_{1}(q) = q or π_{2}(q) = q.
□

Now the following is immediate.

Scholium 5.20. Suppose π_{1}, π_{2} ∈ P_{b}(Z). Then

(a)

π_{1} ≤ π_{2} ⇔ π_{1}π_{2} = π_{1} ⇔ π_{1}(Z) ⊂ π_{2}(Z);

(b)

\begin{matrix}(π_{1} ∧ π_{2})(Z) = π_{1}(Z) ∩ π_{2}(Z), \\ (π_{1} ∨ π_{2})(Z) = π_{1}(Z) + π_{2}(Z), \end{matrix}

and thus π ↦ π(Z) embeds P_{b}(Z) as a sublattice into the boolean lattice of all basic submodules of Z. □

Thus we have a restriction map

τ_{Z} : End_{b}(Z) ↠ P_{b}(Z), φ ↦ p_{φ}

from the commutative R-algebra End_{b}(Z) to the distributive lattice P_{b}(Z). Concerning the fibers of τ_{Z}, i.e., the satellite sets, note that if π_{1} ≤ π_{2} then π_{1} Stl(π_{2}) ⊂ Stl(π_{1}). More generally, for any π_{1}, π_{2} ∈ P_{b}(Z) we have

π_{1} Stl(π_{2}) ⊂ Stl(π_{1}π_{2}).

It turns out that each fiber of τ_{Z} is closed under addition and multiplication. More generally the following holds.

Proposition 5.21. Let π_{1}, π_{2} ∈ P_{b}(Z). Then

Stl(π_{1}) + Stl(π_{2}) ⊂ Stl(π_{1} ∨ π_{2}) and
Stl(π_{1}) · Stl(π_{2}) ⊂ Stl(π_{1}π_{2}).

Proof. Let φ_{1} ∈ Stl(π_{1}), φ_{2} ∈ Stl(π_{2}). Fixing q ∈ 𝔅_{0}(Z), we have

φ_{1}(q) = μ_{1}q, φ_{2}(q) = μ_{2}q

with μ_{1}, μ_{2} ≠ 0, and so π_{i}(q) = q if μ_{i} ≠ 0 (i = 1, 2). Now (π_{1} ∨ π_{2})(q) = q iff (π_{1} + π_{2})(q) ≠ 0 iff μ_{1} + μ_{2} ≠ 0 iff (φ_{1} + φ_{2})(q) ≠ 0. This proves that φ_{1} + φ_{2} ∈ Stl(π_{1} ∨ π_{2}). Furthermore, (π_{1}π_{2})(q) = q iff π_{1}(q) ≠ 0 and π_{2}(q) ≠ 0 iff μ_{1} ≠ 0 and μ_{2} ≠ 0 iff μ_{1}μ_{2} ≠ 0 iff (φ_{1}φ_{2})(q) ≠ q, since R has no zero divisors. This proves that φ_{1}φ_{2} ∈ Stl(π_{1}π_{2}). □

Moreover, it is evident, say by Corollary 5.18, that for every π ∈ P_{b}(Z) the set Stl(π) is convex in the function ordering on End_{b}(Z), and that (R \ {0}) Stl(π) ⊂ Stl(π).

6. Modifications of basic endomorphisms

Up to now the only explicit examples of basic endomorphisms, which we have met, are the basic projectors (§3, §4) and the geometric basic endomorphisms (cf. §5, Example 5.2 and Proposition 5.3). We now look for procedures to obtain new basic endomorphisms from old ones. We start with a definition and a lemma valid in any supertropical semiring R. We intensely use the ν-notation, cf. §1.4.

Definition 6.1. Given μ, υ ∈ R, we say that υ is obedient to μ if (1) υ ≤_{ν} μ,

(2) ∀x, y ∈ R: μx = μy ⇒ υx = υy.

Remarks 6.2.

(a) If υ is obedient to μ and ω is obedient to υ, then ω is obedient to μ.

(b) If υ is obedient to μ then, for any ω ∈ R, υω is obedient to μω.

(c) We conclude from a) and b) that if υ_{1} is obedient to μ_{1} and υ_{2} is obedient to μ_{2}, then υ_{1}υ_{2} is obedient to μ_{1}μ_{2}.

(d) Every λ ≤_{ν} 1 in R is obedient to 1.

(e) When R is a supersemifield, every υ ∈ R is obedient to any μ ≥_{ν} υ, where μ ∈ T .

(f) If υ is obedient to μ and υ ∈ T , then also eυ is obedient to μ, but most often υ is disobedient to eυ.

Lemma 6.3. Assume that υ ∈ R is obedient to μ ∈ R and that a + μb = a + μc for a, b, c ∈ R. Then a + υb = a + υc.

Proof. 1) Assume first that a <_{ν} μb. Then

a + μb = μb = a + μc

which implies a <_{ν} μc. Indeed, otherwise we would have a ≥_{ν} μc, whence

a + μb = a + μc ≅_{ν} a <_{ν} μb,

a contradiction. We conclude that a + μc = μc, which implies μb = μc and then υb = υc by Property (2) in Definition 6.1, whence a + υb = a + υc.

2) The remaining case is a ≥_{ν} μb, in which

a ≅_{ν} a + μb = a + μc,

whence a ≥_{ν} μc. Thus by Property (1) in Definition 6.1

(∗)  a ≥_{ν} υb,  a ≥_{ν} υc.

We run through three subcases.

2.a: Suppose a ∈ G. Then (∗) implies directly that a + υb = a = a + υc. 2.b: Suppose a ∈ T and a >_{ν} μb. Then

a = a + μb = a + μc,

which forces a >_{ν} μc, since a ∈ T . A fortiori a >_{ν} υb and a >_{ν} υc; thus

a + υb = a = a + υc.

2.c: Suppose a ∈ T and a ≅_{ν} μb. Then

a + μb = ea = a + μc,

which forces eμb = eμc, since a ∈ T . By Property (2) we obtain eυb = eυc. Recall from (∗) that υb ≤_{ν} a. Thus, either υb ≅_{ν} υc ≅_{ν} a, which gives a + υb = ea = a + υc, or υb ≅_{ν} υc <_{ν} a, which gives a + υb = a = a + υc.

We have proved that in all cases a + υb = a + υc. □

The lemma ensures the following fact about quadratic forms, valid for any module V over a supertropical semiring.

Lemma 6.4. Assume that υ ∈ R is obedient to μ ∈ R. Let q_{0}, q_{1}, q′_{1} be quadratic forms on the R-module V such that q_{0} + μq_{1} = q_{0} + μq′_{1} then

q_{0} + υq_{1} = q_{0} + υq′_{1}.

Proof. For every x ∈ V we have q_{0}(x)+μq_{1}(x) = q_{0}(x)+μq′_{1}(x) and we conclude by Lemma 6.3 that q_{0}(x) + υq_{1}(x) = q_{0}(x) + υq′_{1}(x). □

In the following we assume as before that V is a free quadratic R-module with base {ε_{i} | i ∈ I}, and that |I| > 1, discarding a trivial case.

Theorem 6.5. Assume that Z is a graphic submodule of Quad(V). As before, we write

(6.1)  𝔇_{0}(Z) := {d_{i} | i ∈ K},  ℌ_{0}(Z) := {h_{ij} | {i, j} ∈ M},

with K ⊂ I, M ⊂ I^{[2]}. Assume that φ is a basic endomorphism of Z, having the system of coefficients

(μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M),

and furthermore that a family (υ_{ij} | {i, j} ∈ M) of elements of R is given such that υ_{ij} is obedient to μ_{ij} for every {i, j} ∈ M. Then there exists a basic endomorphism ψ of Z having the coefficient system

(μ_{i} | i ∈ K) ∪ (υ_{ij} | {i, j} ∈ M).

Proof. a) We first prove the theorem for the special case that I = {1, 2} and

Z = Quad(V) = E_{12} = Rd_{1} + Rd_{2} + Rh_{12}.

In this case φ ∈ End_{b}(Z) has three coefficients μ_{1}, μ_{2}, μ, where μ = μ_{12}, and we are given an element υ = υ_{12} of R obedient to μ. Then

φ(d_{1}) = μ_{1}d_{1},  φ(d_{2}) = μ_{2}d_{2},  φ(h_{12}) = μh_{12},

and we claim that there is a basic endomorphism ψ of E_{12} with ψ(d_{i}) = φ(d_{i}), for i = 1, 2, but ψ(h_{12}) = υh_{12}. Given q ∈ Quad(V) with two presentations

(1) q = α_{1}d_{1} + α_{2}d_{2} + αh_{12} = α_{1}d_{1} + α_{2}d_{2} + βh_{12}, the existence of ψ means that in this situation (1) always

(2) μ_{1}α_{1}d_{1} + μ_{2}α_{2}d_{2} + υαh_{12} = μ_{1}α_{1}d_{1} + μ_{2}α_{2}d_{2} + υβh_{12} holds. Applying φ to (1) we obtain

(3) μ_{1}α_{1}d_{1} + μ_{2}α_{2}d_{2} + μαh_{12} = μ_{1}α_{1}d_{1} + μ_{2}α_{2}d_{2} + μβh_{12}. Using Lemma 6.4 with

q_{0} = μ_{1}α_{1}d_{1} + μ_{2}α_{2}d_{2},  q_{1} = αh_{12},  q′_{1} = βh_{12},

we see that indeed (2) is a consequence of (3).

b) We employ the Pasting Lemma 5.6 to prove the theorem for Z = Quad(V) in general. Given {i, j} ∈ I^{[2]}, let

φ_{ij} := φ|E_{ij} ∈ End_{b}(E_{ij}),

having the coefficients μ_{i}, μ_{j}, μ_{ij}. By step a), for every {i, j} ∈ I^{[2]}, there exists ψ_{ij} ∈ End_{b}(E_{ij}) with coefficients μ_{i}, μ_{j}, υ_{ij}. Suppose i, j, k are different indices, then

ψ_{ij}(d_{i}) = μ_{i}d_{i} = ψ_{ik}(d_{i}).

Thus Lemma 5.6 applies and gives us a basic endomorphism ψ of Quad(V) with ψ|E_{ij} = ψ_{ij} for every {i, j} ∈ I^{[2]}. This basic endomorphism has the desired system of coefficients (μ_{i} | i ∈ I) ∪ (υ_{ij} | {i, j} ∈ I^{[2]}). c) Finally, we prove the theorem in its full generality. Given an endomorphism φ ∈ End_{b}(Z), we extend it to a basic endomorphism φ̃ ∈ End_{b}(Z) with coefficients (μ̃_{i} | i ∈ I) ∪ (μ̃_{ij} | {i, j} ∈ I^{[2]}), where μ̃_{i} = μ_{i} when i ∈ K, otherwise μ̃_{i} = 0, and μ̃_{ij} = μ_{ij} for {i, j} ∈ M, otherwise μ̃_{ij} = 0 (extension by zero, cf. Theorem 5.8). We further extend the family (υ_{ij} | {i, j} ∈ M) to a family (υ̃_{ij} | {i, j} ∈ M) by setting υ̃_{ij} = υ_{ij} when {i, j} ∈ M and υ̃_{ij} = 0 otherwise. It is evident that υ̃_{ij} is obedient to μ̃_{ij} for any {i, j} ∈ I^{[2]}. By step b) there exists a basic endomorphism ψ̃ of Quad(V) with coefficients (μ̃_{i} | i ∈ I) ∪ (υ̃_{ij} | {i, j} ∈ I^{[2]}). The restriction ψ := ψ̃|Z has the desired system of coefficients (μ_{i} | i ∈ K) ∪ (υ_{ij} | {i, j} ∈ M). □

Definition 6.6. The above endomorphism ψ ∈ End_{b}(Z) is called an Hmodification of φ ∈ End_{b}(Z).

Comment. We may interpret the pair (Z, φ) as a weighted graph by assigning weights to the edges and to the vertices of the graph Γ(Z) of Z, namely weight μ_{i} to the vertex d_{i} ∈ VerΓ(Z) and weight μ_{ij} to the edge h_{ij} ∈ EdgΓ(Z). For example, for Z = Quad(V) and I = {1, 2, 3} we have the weighted triangle

[figure: \begin{matrix}μ_{3} \\ • \end{matrix}; μ_{23}; μ_{13}; μ_{1}•; μ_{12}; •μ_{2}]

If, say, R is cancellative, we obtain all H-modifications of φ by lowering the ν-values of the edges.

Definition 6.7. Let Z be a graphic submodule of Quad(V). Let K ⊂ I, and let φ be a basic endomorphism of Z, having the system of coefficients (μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M), according to the notation in Theorem 6.5. We say, that a basic endomorphism ψ of Z with coefficients (υ_{i} | i ∈ K) ∪ (υ_{ij} | {i, j} ∈ M) is a D-modification of φ, if μ_{ij} = υ_{ij} for {i, j} ∈ M, but μ_{i} ≤_{ν} υ_{i} for i ∈ K, in other terms ψ(h_{ij}) = φ(h_{ij}) for all h_{ij} ∈ Z, while ψ(d_{i}) ≥_{ν} φ(d_{i}) for all d_{i} ∈ Z.

Open Problem 6.8. In contrast to the situation of H-modifications we do not know whether for every list of coefficients (υ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M), where υ_{i} ≥_{ν} μ_{i} for all i ∈ K, a D-modification of φ exists .

In general we only have the following weaker result.

Theorem 6.9. Given a basic endomorphism φ on a graphic submodule Z with coefficients (μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M), as in the notation of Theorem 6.5, and a family (υ_{i} | i ∈ K) in R such that μ_{i} ≤ υ_{i} (minimal ordering instead of ν-dominance), there exists a basic endomorphism ψ of Z with coefficients (υ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M).

Proof. We prove the theorem for the case that I = {1, 2} and Z = Quad(V). Then, the general case can be obtained as in the proof of Theorem 6.5. Let

Z = Quad(V) = E_{12} = Rd_{1} + Rd_{2} + Rh_{12}.

and let φ(d_{1}) = μ_{1}d_{1}, φ(d_{2}) = μ_{2}d_{2}, φ(h_{12}) = μh_{12}. For any given ω_{1}, ω_{2} ∈ R we obtain a basic endomorphism χ of Z with coefficients (ω_{1}, ω_{2}, 0) by composing the quasilinear projection

π_{QL} : E_{12} → Rd_{1} + Rd_{2}

with the endomorphism d_{1} ↦ ω_{1}d_{2}, d_{2} ↦ ω_{2}d_{2} of the free module Rd_{1} + Rd_{2}. Then ψ := φ + χ has the coefficients (μ_{1} + ω_{1}, μ_{2} + ω_{2}, μ). The elements μ_{i} + ω_{i} with ω_{i} running through R, are precisely all υ_{i} ≥ μ_{i}. □

It is not difficult to provide D-modifications which are not covered by Theorem 6.9.

Examples 6.10. Set Z = E_{12} = Rd_{1} + Rd_{2} + Rh_{12}.

(a) Choose t_{1}, t_{2} ∈ T_{e} with t_{1} ≠ 1, t_{2} ≠ 1. Then (t_{1}^{2}, t_{2}^{2}, t_{1}t_{2}) is the coefficient system of a geometric basic endomorphism φ_{0} of Z, which has an H-modification φ with (t_{1}^{2}, t_{2}^{2}, 1). Now φ is a D-modification of id_{Z} ˆ=(1, 1, 1). It is easy to find a supertropical semiring R where such t_{i} exist with t_{i}^{2} ≠ 1 (i = 1, 2), cf. e.g. [3, Construction 3.16].

(b) Let μ ∈ T_{e} where μ ≠ 1. Then μ id_{Z} has the H-modification (μ, μ, 1), which is a D-modification of id_{Z}.

In the same vein we obtain the following interesting class of basic endomorphisms.

Proposition 6.11. Assume that Z = E_{12} = Rd_{1} + Rd_{2} + Rh_{12}. Given a triple (u_{1}, u_{2}, v) ∈ R^{3} with u_{1} ≅_{ν} u_{2} ≅_{ν} 1, v <_{ν} 1, there is a well defined basic endomorphism φ of Z which maps any quadratic form

q = \left[{α_{1} α \atop α_{2}}\right] to \left[{u_{1}α_{1} vα \atop u_{2}α_{2}}\right] .

Proof. Let ψ := u_{1}π_{1} + u_{2}π_{2}, where π_{i} is the basic projector onto Rd_{i}, and χ := v id_{Z}. Then ψ and χ map q to \left[{u_{1}α_{1} 0 \atop u_{2}α_{2}}\right] and [ _{vα_{2}}^{vα_{1} vα}], respectively, and so φ := ψ + χ maps q to [ _{u_{2}α_{2}}^{u_{1}α_{1} vα}], as desired, since u_{i} + v = u_{i}, for i = 1, 2. □

7. The characteristic projectors

We assume in the whole section that Z is a basic module over a supertropical semiring R such that for every φ ∈ End_{b}(Z) there exists a basic projector p = p_{φ} associated to φ, i.e., for any generator q ∈ 𝔅_{0}(Z) of Z we have p(q) = q if φ(q) ≠ 0 and p(q) = 0 if φ(q) = 0. We know from §5 that this holds under one of the hypotheses A, B listed in Convention 5.9, but there may be also other cases where it is true. Our goal is to associate a projector p̃ on the R-module Z to φ which better reflects the nature of φ than p_{φ}.

We will use the following simple fact.

Lemma 7.1. Let φ ∈ End_{b}(Z) and p = p_{φ}. Then pφ = φ.

Proof. Since End_{b}(Z) is commutative, pφ = φp. Let q ∈ 𝔅_{0}(Z). If φ(q) ≠ 0 the p(q) = q, whence φ(p(q)) = φ(q). If φ(q) = 0 then p(q) = 0, whence again φ(p(q)) = φ(q). Thus φp = φ. □

We define a function χ : R → R taking values in {0, 1, e}, as follows.

χ(x) = \{ \begin{matrix}1 & if x ∈ T , \\ e & if x ∈ G, \\ 0 & if x = 0. \end{matrix}

Note that for every x ∈ R

(7.1)  χ(x)x = x

and that

(7.2)  χ(x)^{2} = χ(x).

We emphasize that in this section it is not necessary to assume that T = R\eR or G = eR \ {0} is closed under multiplication. As previously (cf. (6.1)) we use the labeling

𝔇_{0}(Z) := {d_{i} | i ∈ K}, ℌ_{0}(Z) := {h_{ij} | {i, j} ∈ M},

with K ⊂ I, M ⊂ I^{[2]}.

Theorem 7.2. Let φ be a basic endomorphism of Z with coefficients

(μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M).

Then there exists a basic endomorphism p̃ of Z with coefficients

(χ(μ_{i}) | i ∈ K) ∪ (χ(μ_{ij}) | {i, j} ∈ M).

Proof. Assume that two presentations of a quadratic form q ∈ Z are given,

q = ∑_{i∈K} α_{i}d_{i} + ∑_{{i,j}∈M} α_{ij}h_{ij}

= ∑_{i∈K} α_{i}d_{i} + ∑_{{i,j}∈M} β_{ij}h_{ij}.

Then

φ(q) = ∑_{i∈K} μ_{i}α_{i}d_{i} + ∑_{{i,j}∈M} μ_{ij}α_{ij}h_{ij}

= ∑_{i∈K} μ_{i}α_{i}d_{i} + ∑_{{i,j}∈M} μ_{ij}β_{ij}h_{ij}.

We introduce the quadratic forms

q′ = ∑_{i∈K} χ(μ_{i})α_{i}d_{i} + ∑_{{i,j}∈M} χ(μ_{ij})α_{ij}h_{ij},
q′′ = ∑_{i∈K} χ(μ_{i})α_{i}d_{i} + ∑_{{i,j}∈M} χ(μ_{ij})β_{ij}h_{ij}.

Then

(∗)  φ(q′) = φ(q′′)

since μ_{i}(χ(μ_{i})) = μ_{i}, μ_{ij}(χ(μ_{ij})) = μ_{ij} in consequence of (7.1). Applying p to equation (∗), we infer from Lemma 7.1 that q′ = q′′, completing the proof. □

Employing property (7.2) of χ it follows that (p̃)^{2} = p̃, and so p̃ is a projector of the R-module Z. We call p̃ the characteristic projector of φ (or, associated to φ), and denote it by p̃_{φ}. Theorem 7.2 implies the following.

Scholium 7.3. For any q ∈ 𝔅_{0}(Z)

p̃_{φ}(q) = \{ \begin{matrix}q & if φ(q) ∈ T q, \\ eq & if φ(q) ∈ Gq, \\ 0 & if φ(q) = 0. \end{matrix}

Corollary 7.4. p̃_{φ} is a satellite of p_{φ}.

Proof. Let p̃:= p̃_{φ}, p := p_{φ}, q ∈ 𝔅_{0}(Z). Trivially p̃(q) ≠ 0 iff φ(q) ∈ T q ∪ Gq and p(q) ≠ 0 iff φ(q) ≠ 0. Since T ∪ G = R \ {0} and the R-module Rq is free, it is evident that φ(q) ∈ T q ∪ Gq iff φ(q) ≠ 0. Thus p̃(q) ≠ 0 iff p(q) ≠ 0. □

There remains the problem of finding all basic endomorphisms of Z with coefficients in {0, 1, e}. They then appear as characteristic projectors of basic endomorphisms, in particular of themselves. We denote the set of these projectors by P′_{b}(Z). Our goal is to determine all elements of P′_{b}(Z) in the case that Z is graphic, but first we have some general observations on P′_{b}(Z). Since {0, 1, e} is a subsemiring of R consisting of idempotents, it is evident that P′_{b}(Z) is a subsemiring of the commutative semiring End_{b}(Z) consisting of idempotents. In the good case that 0, 1, e are the only idempotents of R, P′_{b}(b) is the set of all idempotents of End_{b}(Z). In the degenerate case that e = 1, i.e., eR = R, we have P′_{b}(Z) = P_{b}(Z). We now search for the elements of P′_{b}(Z) in the case that Z = Quad(V) with I = {1, 2}, so

Z = E_{12} = Rd_{1} + Rd_{2} + Rh_{12}.

Identifying each p ∈ P′_{b}(Z) with its triple of coefficients, we have to find out which triples (μ_{1}, μ_{2}, μ) with entries in {0, 1, e} are elements of P′_{b}(Z). We may assume that e ≠ 1.

The set P_{b}(Z) of basic projectors of Z = E_{12} is a subset of P′_{b}(Z), closed under multiplication. It consists of the five triples

(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1).

The finite sums of these triples constitute the subsemiring P′′_{b}(Z) of P′_{b}(Z), which is generated by P_{b}(Z). It contains beside P_{b}(Z) the nine triples

(e, 0, 0), (0, e, 0), (e, 1, 0), (1, e, 0),
(e, e, 0), (e, 1, 1), (1, e, 1), (e, e, 1), (e, e, e).

But the triples (1, 1, e), (e, 1, e), (1, e, e) which are not in P′′_{b}(Z) (as long as e ≠ 1) are nevertheless in P′_{b}(Z), as a consequence of Theorem 6.5 on Hmodifications, since (1, 1, 1), (e, 1, 1), (1, e, 1) are in P′_{b}(Z) and e is obedient to 1. We have verified

Lemma 7.5. P′_{b}(E_{12}) contains all triples (μ_{1}, μ_{2}, μ) with entries in {0, 1, e} such that μ_{1} ≠ 0, μ_{2} ≠ 0 whenever μ ≠ 0, equivalently, such that μ^{2} ≤_{ν} μ_{1}μ_{2}.

Under very mild conditions on the supertropical semiring R we now see that the triples from Lemma 7.5 exhaust the set P′_{b}(Z).

Lemma 7.6. Assume that eR contains an element z > e. Then all triples (μ_{1}, 0, μ), (0, μ_{2}, μ) with μ ≠ 0 are not in P′_{b}(Z).

Proof. Suppose that (e, 0, e) ∈ P′_{b}(Z), where Z := E_{12}. We conclude from the relation d_{1} + d_{2} = d_{1} + d_{2} + h_{12} that ed_{1} = ed_{1} + eh_{12}, which means that

ex_{1}^{2} = ex_{1}^{2} + ex_{1}x_{2}

for any x_{1}, x_{2} ∈ R. Inserting here x_{1} = e and x_{2} = z ∈ G with z > e, we obtain that e = e + z = z, a contradiction. Thus (e, 0, 0) ∉ P′_{b}(Z). We conclude by Theorem 6.9 that the triples (0, 0, e) and (1, 0, e) both are not in P′_{b}(Z) since otherwise (e, 0, e) would be a 𝔇-modification of one of these triples (since 0 < e and 1 < e), and so an element of P′_{b}(Z). This proves that no triple (μ_{1}, 0, μ) with μ ≠ 0 is in P′_{b}(Z). By symmetry also no triple (0, μ_{2}, μ) with μ ≠ 0 is in P′_{b}(Z). □

Theorem 7.7. Assume that there exists some z ∈ eR with z > e. Furthermore assume that the basic module Z is graphic and (μ_{i} | i ∈ K) ∪ (μ_{ij} | {i, j} ∈ M) is a tuple of elements in {0, 1, e}. Then there exists a projector p ∈ P′_{b}(Z) with these coefficients iff, whenever {i, j} ∈ M and μ_{i} = 0 or μ_{j} = 0, also μ_{ij} = 0, in other terms iff, μ_{ij}^{2} ≤_{ν} μ_{i}μ_{j} for all {i, j} ∈ M.

Proof. Since the Pasting Lemma 5.6 and the Extension Theorem 5.8 hold without any restriction on the supertropical semiring R, we can repeat the first two paragraphs in the proof of Theorem 5.10, showing that it suffices to verify Theorem 7.7 in the very special case that Z = E_{12}, which has be done above (Lemmas 7.5 and 7.6). □

8. The basic endomorphisms over a tangible supertropical semifield

In this section we address the “absolute” existence problem for basic endomorphisms, focussing on the case of Z = Quad(V), I = {1, 2}, i.e.,

Z = Quad(V) = Rd_{1} + Rd_{2} + Rh_{12}.

At the moment R may be any supertropical semiring. Given a triple (μ_{1}, μ_{2}, μ) ∈ R^{3}, we enquire whether a basic endomorphism φ of Z with coefficients μ_{1}, μ_{2}, μ exists, i.e., with

φ(d_{1}) = μ_{1}d_{1}, φ(d_{2}) = μ_{2}d_{2}, φ(h_{12}) = μh_{12}.

Since d_{1} + d_{2} = d_{1} + d_{2} + h_{12}, we have the necessary condition for existence of φ that

(8.1)  μ_{1}d_{1} + μ_{2}d_{2} = μ_{1}d_{1} + μ_{2}d_{2} + μh_{12},

which means an equivalence of triangular schemes ^{9}

\left[{μ_{1} μ \atop μ_{2}}\right] ≅ \left[{μ_{1} 0 \atop μ_{2}}\right] .

Problem 8.1. For which supertropical semirings R is Condition (8.1) sufficient for the existence of a basic endomorphism of Z = Quad(V) = Rd_{1} + Rd_{2} + Rh_{12} with coefficients μ_{1}, μ_{2}, μ?

While in the last sections, starting from §3, we have been eager to impose only mild restrictions on the supertropical semiring R, we now solve this problem for the class of tangible supersemifields. We start with a general lemma, implementing a previous argument in different context.

Lemma 8.2. A triple (μ_{1}, μ_{2}, μ) ∈ R^{3} serves as the coefficients of a basic endomorphism of Z = Rd_{1} + Rd_{2} + Rh_{12} precisely if, for any q ∈ Z, any two different presentations

(A)  q = \left[{α_{1} α \atop α_{2}}\right] = \left[{α_{1} β \atop α_{2}}\right]

ensue an equivalence

(B)  \left[{μ_{1}α_{1} μα \atop μ_{2}α_{2}}\right] ≅ \left[{μ_{1}α_{1} μβ \atop μ_{2}α_{2}}\right] .

Proof. If (A) implies (B), then the map φ : Z → Z given by

φ (\left[{α_{1} α \atop α_{2}}\right]) = \left[{μ_{1}α_{1} μα \atop μ_{2}α_{2}}\right] ,

or in other terms as

φ(α_{1}d_{1} + α_{2}d_{2} + αh_{12}) = α_{1}μ_{1}d_{1} + α_{2}μ_{2}d_{2} + αμh_{12},

9In the terminology of [8, §1]: The “formal” quadratic forms \left[{μ_{1} μ \atop μ_{2}}\right] , \left[{μ_{1} 0 \atop μ_{2}}\right] present the same “functional” quadratic form.

is well defined and obviously is an endomorphism of Z, which is basic with coefficients μ_{1}, μ_{2}, μ. Conversely, if such basic endomorphism exists, then (A) implies (B). □

From now on R is a tangible supersemifield, which we tacitly assume to be nontrivial.^{10} Condition (8.1) means that the quadratic form q = \left[{μ_{1} μ \atop μ_{2}}\right] is quasilinear, q = q_{QL}, equivalently 0 ∈ C_{12}(q). By [8, §7] (cf. there [8, Proposition 7.9], [8, Theorem 7.11]) this holds iff either μ^{2} ≤_{ν} μ_{1}μ_{2} or R is discrete and μ^{2} ≅_{ν} π^{−1}μ_{1}μ_{2}, μ_{1}, μ_{2} ∈ G. We are ready to prove

Theorem 8.3. Assume that R is a tangible supersemifield and (μ_{1}, μ_{2}, μ) ∈ R^{3}. Then a basic endomorphism of Z := Quad(V) = Rd_{1} +Rd_{2} +Rh_{12}, I = {1, 2}, with coefficients (μ_{1}, μ_{2}, μ) exists iff

μ^{2} ≤_{ν} μ_{1}μ_{2},

excluding the degenerate case that R is discrete and T_{e} = {1}, in which every (μ_{1}, μ_{2}, μ) ∈ G^{3} with μ^{2} = π^{−1}μ_{1}μ_{2} yields a basic endomorphism,

Proof. a) As commented above, we may restrict to the case of triples (μ_{1}, μ_{2}, μ) with either μ^{2} ≤_{ν} μ_{1}μ_{2} or R is discrete and μ^{2} ≅_{ν} π^{−1}μ_{1}μ_{2}, where μ_{1}, μ_{2} ∈ G.

b) Assume that μ^{2} ≤_{ν} μ_{1}μ_{2}. Given q ∈ Z with two presentations (A) and α ≠ β, we verify that (B) is valid, assisted again by [8, Theorems 7.11 and 7.12]. We start with the case that α^{2} ≤_{ν} α_{1}α_{2} and β^{2} ≤_{ν} α_{1}α_{2}. Then

(μα)^{2} ≤_{ν} (μ_{1}α_{1})(μ_{2}α_{2}) and (μβ)^{2} ≤_{ν} (μ_{1}α_{1})(μ_{2}α_{2}),

and thus both forms in (B) are quasilinear, whence

\left[{μ_{1}α_{1} μα \atop μ_{2}α_{2}}\right] ≅ \left[{μ_{1}α_{1} 0 \atop μ_{2}α_{2}}\right] ≅ \left[{μ_{1}α_{1} μβ \atop μ_{2}α_{2}}\right] .

By [8, Theorem 7.11] the remaining case to be considered is that R is discrete and α^{2} ≅_{ν} π^{−1}α_{1}α_{2}, possibly with α and β interchanged. Then, by the same theorem, β ≅_{ν} α if α_{2} ∈ T or α_{1} ∈ T , while β ≤_{ν} α if both α_{1}, α_{2} are in G. If μ^{2} <_{ν} μ_{1}μ_{2}, then

(μα)^{2} ≤_{ν} (μ_{1}α_{1})(μ_{2}α_{2}) and (μβ)^{2} ≤_{ν} (μ_{1}α_{1})(μ_{2}α_{2}),

and as before we conclude that (B) is valid. Let μ^{2} ≅_{ν} μ_{1}μ_{2}, then

(μα)^{2} ≅_{ν} π^{−1}(μ_{1}α_{1})(μ_{2}α_{2}).

If β <_{ν} α then μβ <_{ν} μα, and μ_{1}α_{1}, μ_{2}α_{2} ∈ G, since α_{1}, α_{2} ∈ G. Thus (B) is valid, cf. [8, Theorem 7.12]. If β ≅_{ν} α then μβ ≅_{ν} μα, and (B) holds again by [8, Theorem 7.12] (regardless whether μ_{i}α_{i} is ghost or tangible). We conclude that, whenever μ^{2} ≤_{ν} μ_{1}μ_{2}, the triple (μ_{1}, μ_{2}, μ) yields a basic endomorphism. Finally, we consider the case that R is discrete and μ^{2} ≅_{ν} π^{−1}μ_{1}μ_{2}, where μ_{1}, μ_{2} ∈ G. We choose α_{1}, α_{2}, α, β in R such that

α^{2} ≅_{ν} π^{−1}α_{1}α_{2} ≅_{ν} β^{2},

10This means that G ≠ {e}. If G = {e}, all problems discussed here seem to be trivial.

whence α ≅_{ν} β, but, if possible, μα ≠ μβ. Then

(μα)^{2} ≅_{ν} π^{−2}(μ_{1}α_{1})(μ_{2}α_{2}) ≅_{ν} (μβ)^{2}.

Thus (B) fails when μα ≠ μβ, and we conclude that (μ_{1}, μ_{2}, μ) does not provide a basic endomorphism. For μ ∈ T we may choose α ∈ T and β = eα. When μ ∈ G, we need the group T_{e} to be nontrivial, since otherwise α ≅_{ν} β implies α = β. Therefore we choose α, β ∈ T with α ≠ β, α ≅_{ν} β, α^{2} ≅_{ν} π^{−1}α_{1}α_{2}. Thus only in the subcase that μ ∈ G, T_{e} = {1} the triple (μ_{1}, μ_{2}, μ) yields a basic endomorphism. □

We now obtain a partial answer, to Problem 8.1 as follows.

Theorem 8.4. Assume that R is a tangible supersemifield. Let Z = Quad(V) with I = {1, 2}, i.e.,

Z := Quad(V) = Rd_{1} + Rd_{2} + Rh_{12}.

Assume that (μ_{1}, μ_{2}, μ) ∈ R^{3} is a triple with

(8.2)  μ_{1}d_{1} + μ_{2}d_{2} + μh_{12} = μ_{1}d_{1} + μ_{2}d_{2}.

(a) If R is dense, then there exists a basic endomorphism of Z with coefficients (μ_{1}, μ_{2}, μ).

(b) If R is discrete there always exist triples in R^{3} satisfying (8.2) which do not yield basic endomorphisms. These are the (μ_{1}, μ_{2}, μ) ∈ R^{3} with the following properties: I. If T_{e} ≠ {1}:

μ^{2} ≅_{ν} π^{−1}μ_{1}μ_{2}, μ_{1}, μ_{2} ∈ G;

II. If T_{e} = {1}:

μ^{2} ≅_{ν} π^{−1}μ_{1}μ_{2}, μ_{1}, μ_{2} ∈ G, μ ∈ T .

Proof. Recall that (8.2) holds iff either μ^{2} ≤_{ν} μ_{1}μ_{2} or R is discrete μ^{2} ≅_{ν} π^{−1}μ_{1}μ_{2}, and μ_{1}, μ_{2} ∈ G. Comparing this with the conditions in Theorem 8.3 under which (μ_{1}, μ_{2}, μ) yields a basic endomorphism of Z, we obtain the claim. □

Theorem 8.3 readily generalizes to an existence theorem for basic endomorphisms of a linked module Z, as follows. (We consider first the case that Z = Quad(V) and adhere to the standard notation (3.1) for 𝔇_{0} and ℌ_{0}.)

Theorem 8.5. Assume that R is a tangible supersemifield.

(a) If T_{e} ≠ {1}, then a family

(μ_{i} | i ∈ I) ∪ (μ_{ij} | i < j)

in R serves as the system of coefficients of a basic endomorphism of Quad(V) iff

(8.3)  μ_{ij}^{2} ≤_{ν} μ_{i}μ_{j}

for all i, j ∈ I with i < j. The same holds when T_{e} = {1} and R is dense.

(b) When R is discrete and T_{e} = {1}, Condition (8.3) has to be replaced by the more complicated condition, that for all i < j either μ_{ij}^{2} ≤_{ν} μ_{i}μ_{j} or

μ_{ij}^{2} ≅_{ν} π^{−1}μ_{i}μ_{j}, (μ_{i}, μ_{j}, μ_{ij}) ∈ G^{3}.

(c) Mutatis mutandis, all this remains true if we replace Quad(V) by any linked submodule Z of Quad(V).

Proof. This follows from Theorem 8.3 by the same line of thought as in parts (b) and (c) of the proof of Theorem 6.5, using the Pasting Lemma 5.6. □

When Z is a basic R-module, and R is a tangible supersemifield, then Z = Z_{link} ⊕ Y with Y a free R-module (cf. Theorem 4.10). Every basic endomorphism φ of Z is a direct sum φ = φ_{1} ⊕ φ_{2}, with φ_{1} and φ_{2} basic endomorphisms of Z_{link} and Y respectively. Since Y is free there are no restrictions on the coefficients system of φ_{2}, and therefore we know all the endomorphisms of Z. We omit the details.

Remark 8.6. As a consequence of Theorems 8.5 and 6.9, Problem 6.8 about Dmodifications has a positive answer when R is a tangible supersemifield. Every change of coefficients μ_{i} ↦ υ_{i} (i ∈ K), prescribed in Definition 6.7 for a given basic endomorphism φ is realized by a D-modification of φ. Here we do not need to bother about the case that T_{e} = {1}, since then Problem 6.8 vanishes: If υ_{i} ≤_{ν} μ_{i}, then υ_{i} ≤ μ_{i}.

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Zur Izhakian

Institute of Mathematics University of Aberdeen AB24 3UE Aberdeen UK zzur@abdn.ac.uk Manfred Knebusch Department of Mathematics NWF-I Mathematik Universität Regensburg 93040 Regensburg Germany manfred.knebusch@ mathematik.uniregensburg.de 1708

Documenta Mathematica 22 (2017)