Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 2, 461-466.

Pappus’ Theorem for Ring-geometries

Thorsten Pfeiffer

Zelterstrasse 6, D-55545 Bad Kreuznach, Germany

Abstract. A famous result, due to Hilbert, states for a projective plane over a division ring K the equivalence of Pappus’ theorem and the commutativity of K. In this paper we extend this result to the projective plane over an arbitrary ring. MSC 1991: 06A15, 06C, 51A05, 51A10 Keywords: ring-geometry, Pappus’ theorem

1. Introduction

Let P = P(K) a projective plane over a field K. In [1] E. Artin writes: “One of the simple but fascinating results in foundations of geometry is the fact that one can find a simple geometric configuration which is equivalent with the commutative law for multiplication of our field K.” Let us take a closer look at this configuration: Given distinct lines l and m of the projective plane which intersect in a unique point, say Z. With i = 1, 2, 3 let L_{i} and M_{i} be pairwise distinct points on l and m, respectively, which are all distinct from Z as well. With 1 ≤ i < j ≤ 3 consider the three distinct points P_{i,j} = (L_{i} + M_{j}) ∩ (L_{j} + M_{i}). We say Pappus’ theorem holds iff P_{1,2}, P_{1,3}, and P_{2,3} are collinear (see Fig. 1). The crucial point of the proof of the above stated equivalence is identifying the group of multiplications of K with the subgroup Γ(Z, P_{1,2} + P_{1,3}) of the group of collineations of P, i.e. σ ∈ Γ(Z, P_{1,2} + P_{1,3}) iff σ has centre Z (i.e. every line through Z remains fix under σ) and axis P_{1,2} + P_{1,3} (i.e. every point of P_{1,2} + P_{1,3} is a fix point of σ). The challenge in extending this well-known equivalence to ring-geometries is finding a representation of the ring’s additive and multiplicative operation in terms of appropriate geometric mappings. But let us not get ahead of our story. First, let us review our concept of ringgeometry.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

[figure: Z; L_{1}; L_{2}; L_{3}; M_{1}; M_{2}; M_{3}; m; l; P_{1,2}; P_{1,3}; P_{2,3}; P_{1,2} + P_{1,3} = P_{1,3} + P_{2,3} = P_{2,3} + P_{1,2}] Fig. 1

2. Planar geometry over a ring

Preliminaries. Our approach to ring-geometries is a lattice-theoretic one. A given arbitrary ring^{1} R induces the planar projective lattice geometry P G(_{R}R^{3}) := (L, P), where L := L(_{R}R^{3}) denotes the lattice of all submodules of _{R}R^{3} (partially ordered by set-inclusion) and P := P (_{R}R^{3}) denotes the set of freely one generated submodules of _{R}R^{3}. The elements of P are called points, i.e. a point is a cyclic submodule of _{R}R^{3} which is (module-)isomorphic to the underlying ring R. The elements of L are called geometric objects.

By abuse of notation we use “≤” for the order of L and “1_{L}” (“0_{L}”) for the top (bottom) element of L, respectively, i.e. 1_{L} =_{R}R^{3} and 0_{L} = {0}. The meet of an arbitrary family of submodules is given by their set-theoretic intersection and the meet of two geometric objects x, y is denoted by xy. The join of an arbitrary family of submodules U_{i}, i ∈ I in L is given by ∑_{i∈I} U_{i}, i.e. the smallest submodule which contains all moduls U_{i}. For geometric objects x, y ∈ L we write x + y := ∑{x, y} and x ⊕ y if x and y are disjoint, i.e. xy = 0_{L}. A line l is the span of two disjoint points, i.e. l = x ⊕ y for points x, y. Three geometric objects x, y, z are directly balanced iff x ⊕ y = y ⊕ z = z ⊕ x. A geometric object y is called complement of x if x ⊕ y = 1_{L} and x, y are perspective via z if x ⊕ z = y ⊕ z for some z ∈ L.

For any geometric object x we obtain the subgeometry of x as P G(x) := (L(x), P (x)) where L(x) denotes the sublattice {t ∈ L | t ≤ x} and P (x) = {p ∈ P | p ≤ x}. Note that P (x) = ∅ is possible for some x ∈ L.

It is well-known that L is a modular lattice, i.e. (x + y)z = x + yz holds for all x, y, z ∈ L with x ≤ z (where x + yz := x + (yz)).

A regular triangle is a family of points p_{i}, i ∈ {1, 2, 3}, such that p_{3} (p_{1} ⊕ p_{2}) = 0_{L}. A 3-frame is a set of points p_{i}, i ∈ I = {1, 2, 3, 4}, such that for every J ⊆ I with |J| = 3 the family p_{j}, j ∈ J, is a regular triangle and ∑{p_{j} | j ∈ J} = ∑{p_{j′} | j′ ∈ J′} for any J′ ⊆ I with |J′| = 3. Due to modularity of L, it is easily to be shown that every regular triangle can be extended to a frame.

1A ring is associative with 1 ≠ 0

Projections and perspectivities. Our goal is the representation of the ring’s addition and multiplication with appropriate mappings. So let us now turn to our concept of mappings of a planar geometry G = P G(_{R}R^{3}) and its subgeometries:

A mapping between geometries G = (L, P) and G′ = (L′, P′) is a mapping φ : L → L′ of the underlying lattices L and L′. An isomorphism between geometries G and G′ is an orderisomorphism φ : L → L′ (i.e. φ is bijective and x ≤ y iff φ(x) ≤ φ(y) for all x, y ∈ L) with φ(P) = P′. Note that every order-isomorphism is join-preserving, i.e. φ(∑ X) = ∑ φ(X) for all X ⊆ L.

An important class of join-preserving mappings is given by projections:

Example 2.1. A projection of a geometry G = (L, P) is defined as a mapping of the form Π_{x,y} : L → L, t ↦ (t + x)y where x, y ∈ L satisfy x ⊕ y = 1_{L}.

The centre of a projection Π_{x,y} is the geometric object x, its image the geometric object y.

What about the points of the geometry G under a projection? It is easy to check that a projection maps the points which are disjoint to its kernel onto the points which are less or equal to its image, i.e. a projection Π_{x,y} is point-preserving in the following sense:

Given any geometric object z then Π_{x,y}({p ∈ P (z) | px = 0}) = P (Π_{x,y}(z)), due to modularity of L. Furthermore, projections are idempotent and Π_{x,y}(t) = Π_{x,y}(t′) iff t + x = t′ + x for all t, t′ ∈ L.

For disjoint points p, q and h ∈ L such that q ∈ P (p ⊕ h) the points p, q, Π_{p,h}(q) are directly balanced iff q ⊕ h = p ⊕ h.

The notion of projection gives rise to an important class of isomorphisms, namely perspectivities: Given two geometric objects x, y which are perspective via z, then the restriction Π_{z,y} | G(x) → G(y), t ↦ (t + z)y is an isomorphism of G(x) and G(y), called perspectivity of x onto y with centre z, abbreviated x →^{z} y.

The ring’s operations in geometric terms. We will now use projections and perspectivities for a geometric description of the ring’s operations. The most important paper for us is [2] where the authors A. Day and D. Pickering construct a ternary ring to find a coordinatization of a given arguesian lattice. One of the most important features of [2] is the introduction of geometric mappings which can be adapted to our framework as follows:

Construction 2.2. Let G = P G(_{R}R^{3}) a planar lattice geometry and (f_{0}, f_{1}, f_{2}) a regular triangle of G which we extend with f_{4} to a 3-frame of the subgeometry G(f_{0} ⊕f_{1} ⊕f_{2}). Define f_{3} = (f_{1} ⊕ f_{2})(f_{0} ⊕ f_{4}). Since f_{3} = Π_{f_{2},f_{0}⊕f_{4}}(f_{1}) = Π_{f_{0},f_{1}⊕f_{2}}(f_{4}) holds, the triples of points (f_{1}, f_{2}, f_{3}) and (f_{0}, f_{3}, f_{4}) are directly balanced.

Define a set D = {f ∈ P (f_{0} ⊕ f_{4}) | f ⊕ f_{3} = f_{0} ⊕ f_{3}}. We can write f_{i} = Rx_{i} for x_{i} ∈ _{R}R^{3}, i ∈ {0, 1, 2}, f_{3} = R(x_{1} + x_{2}), and f_{4} = R(x_{0} + x_{1} + x_{2}) and obtain D = {R(x_{0} + r(x_{1} + x_{2})) | r ∈ R}. Our goal is introducing an addition and multiplication on D. We therefore define the following perspectivities:

φ : f_{0} ⊕ f_{4} →^{f_{2}} f_{0} ⊕ f_{1}, t ↦ Π_{f_{2},f_{0}⊕f_{1}}(t);

[figure: f_{0}; p; f_{3}; q; f_{4}; f_{1}; f_{2}; φ(q); σ_{p}(φ(q)); p+′q; Fig. 2; f_{0}; q; f_{3}; p; f_{4}; f_{1}; f_{2}; ψ(q); p ⊙ q; τ_{q}(p); Fig. 3]

ψ : f_{0} ⊕ f_{4} →^{f_{2}} f_{1} ⊕ f_{4}, t ↦ Π_{f_{2},f_{1}⊕f_{4}}(t);

For any p ∈ D define the following perspectivities:

σ_{p} : f_{1} ⊕ f_{0} →^{f_{3}} f_{1} ⊕ p, t ↦ Π_{f_{3},f_{1}⊕p}(t) and

τ_{p} : f_{0} ⊕ f_{4} →^{f_{1}} f_{0} ⊕ ψ(p), t ↦ Π_{f_{1},f_{0}⊕ψ(p)}(t);

Using these perspectivities we introduce the following operations on D:

p+′q := Π_{f_{2},f_{0}⊕f_{4}}(σ_{p}(φ(q))) and

p ⊙ q = Π_{f_{2},f_{0}⊕f_{4}}(τ_{q}(p)).

Claim. Due to these operations D becomes a ring (D, +′, ⊙, f_{0}, f_{4}), which is isomorphic to (R, +, ·, 0, 1) via R → D, r ↦ R(x_{0} + r(x_{1} + x_{2})) (Fig. 2, 3). Especially, R is commutative iff p ⊙ q = q ⊙ p holds for all p, q ∈ D iff τ_{p}(q) + f_{2} = τ_{q}(p) + f_{2} holds for all p, q ∈ D.

Proof. The first part is given by straightforward calculations. The second part follows since Π_{f_{2},f_{0}⊕f_{4}} is a projection with kernel f_{2}. □

Note that in the above construction we do not claim that the regular triangle (f_{0}, f_{1}, f_{2}) spans the planar geometry, i.e. it is not necessary to claim that 1_{L} = f_{0} ⊕ f_{1} ⊕ f_{2}.

3. The extended Pappus’ theorem

We are almost finished. We only have to find an appropriate translation for the “classical” theorem. One of the major difficulties of the more general ring-case is that the intersection of two distinct lines need not be a point. In order to adapt the configuration to our ring-case, we define:

Definition 3.1. Let (p, q′, q) be a regular triangle of a planar projective geometry G = P G(_{R}R^{3}), which we extend to a 3-frame (p, q, q′, r) of the subgeometry G(p ⊕ q ⊕ q′). For points u, t ≤ p ⊕ q such that conditions

(1) t ⊕ q = u ⊕ q = p ⊕ q and

(2) p, t, u are pairwise distinct

hold, we set t′ = Π_{r,p⊕q′}(t) and u′ = Π_{r,p⊕q′}(u).

Since q′ ⊕ (q ⊕ t) = p ⊕ q ⊕ q′, the geometric object f_{t,u′} := Π_{q,q′⊕t}(u′) = (q′ ⊕ t)(q ⊕ u′) is a point. Similarily, q′ ⊕ (u ⊕ q) = p ⊕ q ⊕ q′, and f_{u,t′} := Π_{q,q′⊕u}(t′) = (q′ ⊕ u)(q ⊕ t′) is a point as well.

We say the 3-frame (p, q, q′, r) satisfies the extended Pappus’ theorem (EP), if f_{u,t′} + r = f_{t,u′} + r holds for all points u, t with condition (1) and (2) (cf. Fig. 4).

We say the regular triangle (p, q, q′) (in this order) satisfies the extended Pappus’ theorem (EP), if (p, q, q′, r) satisfies the extended Pappus’ theorem for every r where (p, q, q′, r) is a 3-frame of the subgeometry G(p ⊕ q ⊕ q′).

[figure: p; t′; u′; q′; u; t; q; r; f_{u,t′}; f_{t,u′}; f_{t,u′} + r = f_{u,t′} + r] Fig. 4

Now from 2.2 we derive our main result:

Theorem 3.2. Let G = P G(_{R}R^{3}) be a planar projective geometry. Then the following are equivalent:

(a) One regular triangle satisfies (EP);

(b) Every regular triangle satisfies (EP);

(c) The multiplication of R is commutative.

Thorsten Pfeiffer: Pappus’ Theorem for Ring-geometries 466

References

[1] Artin, E.: Geometric Algebra. Interscience Publishers, New York 1957.

[2] Day, A.; Pickering, D.: The coordinatization of arguesian lattices. Trans. Amer. Math. Soc. 278 (1983) (2), 507–522.

Received October 2, 1997