Theory and Applications of Categories, Vol. 8, No. 15, 2001, pp. 408–456.

PSEUDOGROUPOIDS AND COMMUTATORS

GEORGE JANELIDZE AND M.CRISTINA PEDICCHIO

ABSTRACT. We develop a new approach to Commutator theory based on the theory of internal categorical structures, especially of so called pseudogroupoids. It is motivated by our previous work on internal categories and groupoids in congruence modular varieties.

0. Introduction

The purpose of this paper is to develop a new approach to commutator theory.

Since J. D. H. Smith [S] introduced the notion of commutator for a pair of congruences in a congruence permutable (=Mal’tsev) variety of universal algebras, there were many attempts to extend this notion to more general varieties and to simplify it. Accordingly, the important work of J. Hagemann and C. Hermann [HH] and of H. P. Gumm [G] has to be mentioned. The conclusion, also well supported by R. Freese and R. McKenzie [FMK], seems to be that the right level of generality in Commutator theory is the level of congruence modular varieties, where various possible definitions coincide and the commutator has “all nice properties”. However there are various interesting investigations in non modular cases (see [K], P. Lipparini [L1], [L2] and references there).

Our viewpoint in this paper is that commutator theory should be based on the theory of internal categorical structures, this approach is motivated by the description of internal categories and groupoids in congruence modular varieties obtained in [JP]. In fact we use a new categorical structure which we call a pseudogroupoid — in contrast to pregroupoids (in the sense of A. Kock [Ko]) used in [P1] and [P2]; in the case of Mal’tsev varieties our approach is equivalent to the one developed in [P1].

Once the pseudogroupoids are introduced, the definition of the commutator becomes very simple.

The commutator [α, β] of congruences α and β on an algebra A is

[α, β] = {(x, y) ∈ A × A|η(x) = η(y)},  (0.1)

where η is the canonical homomorphism from A to the free pseudogroupoid on the span

\xymatrix{A/α & A \ar[l] \ar[r] & A/β.}  (0.2)

Received by the editors 2000 January 9 and, in revised form, 2001 April 30.

Transmitted by G. M. Kelly. Published on 2001 July 9.

2000 Mathematics Subject Classification: 18D99, 08B05.

Key words and phrases: variety, pregroupoid, commutator.

\begin{matrix}c \\ ◯ \end{matrix} George Janelidze and M.Cristina Pedicchio, 2001. Permission to copy for private use granted.

This definition has a powerful conclusion: all properties of commutators are just the properties of the free-forgetful adjunction

\xymatrix{(\begin{matrix}Spans \\ in ℂ \end{matrix}) \ar[r]^{free} & (\begin{matrix}Internal \\ pseudogroupoids \\ in ℂ \end{matrix}) , \ar[l]^{forgetful}}  (0.3)

where ℂ is a ground variety.

On the other hand the notion of a pseudogroupoids is quite natural: as we explain in the Section 3 below it was “almost” introduced by J. D. H. Smith [S], and then again by E. W. Kiss [Ki].

The paper is organized as follows:

Introduction

1. Rectangles and diamonds

2. Double equivalence relations

3. Pseudogroupoids

4. Free internal and algebraic pseudogroupoids

5. The commutator

6. Kiss, Gumm, Lipparini and abelianizable varieties

7. Abelian algebras and abelianization

8. Two characterizations of congruence modular varieties

In the first four sections we are trying to describe the language of internal categorical structures and to show that they are really needed and even unexplicitly used all the time in Commutator theory — although the commutator itself is introduced only in the fifth section, first for spans in general categories, and then for congruences on an algebra, in fact via (0.1).

Our notion of commutator coincides with the “usual” one for congruence modular varieties, but seems to be useful also for the larger classes of varieties which we call Kiss, Gumm and Lipparini varieties since their definitions were suggested by the results of these authors. Even the largest class which we consider — the class of “abelianizable” varieties admits the “fundamental theorem on abelian algebras” and a good description of the largest commutator (i.e. [∇_{A}, ∇_{A}], where ∇_{A} = A × A is the largest congruence on an algebra A). In the last section we show that a Kiss variety is congruence modular if (and only if) the commutator is preserved by surjective images, and deduce from the results of P. Lipparini [L1] that a Gumm variety is congruence modular if and only if the commutator is distributive (by “image” we mean the image in the categorical sense).

Since our commutator generalizes the “modular” one, it also generalizes the ordinary commutator for normal subgroups (and also [K, K′] = KK′ + K′K for ideals in a ring), however we give a simple direct proof — in order to show again that our generalization is very natural.

We would like to propose the following further questions and problems to be investigated:

1. Find out more about the relationship between the geometrical language of H. P. Gumm [G] and the categorical language.

2. What is the relationship with other known commutators in non-modular cases? (see also [KS])

3. Further development of commutator theory in general categories (we have introduced here only the definition and the very first properties — see Proposition 5.4).

4. Using the commutator investigate various congruence identities in the non-modular case and generalize them from varieties to exact, regular and possibly more general categories.

5. Further development of the theory of central extension [JK] in the case of pairs of varieties of the form (ℂ, Abelian algebras in ℂ), and extension of Homological methods of J. D. H. Smith [S] and J. Furtado-Coelho [F-C].

In the paper we use without proofs:

• elementary properties of limits, colimits, adjoint functors — much less that given in S. MacLane’s book [ML];

• some motivations from [JP] — so we are asking our readers to read at least the introduction of that paper;

• some results in Commutator theory — either well known, or proved by E. W. Kiss [Ki], or by P. Lipparini [L1].

1. Rectangles and diamonds

Let ℂ be a variety of universal algebras and α, β congruences on an algebra A in ℂ. The composition αβ is defined as

αβ = {(a, b) ∈ A × A|∃x : aαxβb},  (1.1)

and so αβ = βα can be expressed as

[figure: (∃x : aαxβb); (∃y : aβyαb).]  (1.2)

This tells us that it is useful to work with four-tuples (a, x, y, b) for which

a ^{β}_{α}  y_{α} _{,} x _{β} b  (1.3)

i.e. aαxβb and aβyαb. The picture (1.3) will be called an α-β-rectangle; H. P. Gumm [G] also says that (a, x, y, b) is an α-β-parallelogram, and E. W. Kiss [Ki] says that (ax, yb) is an α-β-rectangle.

An internal graph G =

\xymatrix{G_{1} \ar[r]^{d} \ar[r]_{c} & G_{0}}  (1.4)

in ℂ consists of two algebras G_{0} and G_{1} in ℂ and two homomorphisms d and c from G_{1} to G_{0}. The elements of G_{0} are called objects, or points, and the elements of G_{1} are called morphisms, or arrows (of G); if g ∈ G_{1} and d(g) = u, c(g) = v , then we write \xymatrix{g : u \ar[r] & v}.

If G is an internal graph in ℂ and

\begin{aligned}A & = & G_{1} \\ α & = & \{(f, g) ∈ G_{1}|d(f) = d(g)\}, \\ β & = & \{(g, h) ∈ G_{1}|c(g) = c(h)\}, \end{aligned}  (1.5)

then an α-β-rectangle

f ^{β}_{α}  k_{α} g _{β} h  (1.6)

becomes a “G-diamond”

\xymatrix{ & · \\ · \ar[ur]^{f} \ar[dr]_{g} &  & · \ar[ul]_{k} \ar[dl]^{h} \\  & ·}  (1.7)

— as in [JP]. Note that in this case A/α and A/β are canonically isomorphic to subalgebras in G_{0}, which is a useless additional condition on α and β. In order to avoid that condition we replace internal graphs by (internal) spans.

A span S =

\xymatrix{S_{0} & S_{1} \ar[l]_{π} \ar[r]^{π′} & S′_{0}}  (1.8)

in ℂ consists of three algebras S_{0}, S′_{0} and S_{1} in ℂ, and two homomorphisms \xymatrix{π : S_{1} \ar[r] & S_{0}} and \xymatrix{π′ : S_{1} \ar[r] & S′_{0}}; if g ∈ S_{1} and π(g) = u, π′(g) = v, we again write \xymatrix{g : u \ar[r] & v}.

Now instead of (1.5) we just say: given an algebra A in ℂ and congruences α and β on A, we obtain the span

\xymatrix{A/α & A \ar[l] \ar[r] & A/β}  (1.9)

of the canonical homomorphisms — and again an α-β-rectangle (1.6) is the same as an S-diamond (1.7).

Let S be an arbitrary span in ℂ. The set of all S-diamonds will be denoted by S_{4}, and for x ∈ S_{4} sometimes we will write x =

[figure: x_{11}; x_{21}; x_{12}; x_{22}; ,]  (1.10)

or just

x = ( \begin{matrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix} )  (1.11)

which suggest to consider the diagram

\xymatrix{S_{1} \ar[r]^{π′} \ar[d]_{π} & S′_{0} & S_{1} \ar[l]_{π′} \ar[d]^{π} \\ S_{0} & S_{4} \ar[ul]_{π_{11}} \ar[ur]^{π_{12}} \ar[dl]^{π_{21}} \ar[dr]_{π_{22}} & S_{0} \\ S_{1} \ar[u]^{π} \ar[r]_{π′} & S′_{0} & S_{1} \ar[l]^{π′} \ar[u]_{π}}  (1.12)

where π_{ij}(x) = x_{ij}. Note that this is a diagram in ℂ since S_{4} obviously is a subalgebra in S_{1}^{4} = S_{1} × S_{1} × S_{1} × S_{1} and the maps π_{ij} are homomorphisms. Moreover the diagram (1.12) represents S_{4} as the limits of

[figure: S_{1}; π′; π; S′_{0}; S_{1}; π′; π; S_{0}; S_{0}; .; S_{1}; π; π′; S′_{0}; S_{1}; π; π′]  (1.13)

Another way to present S_{4} as a limit is to form the pullbacks

[figure: S_{2}; π_{2}; π_{1}; S_{1}; π; ,; S′_{2}; π′_{2}; π′_{1}; S_{1}; π′; ,; S_{1}; π; S_{0}; S_{1}; π′; S_{0}]  (1.14)

and then the diagram

[figure: S_{1}; S′_{2}; π′_{1}; π′_{2}; S_{1}; S_{2}; π_{1}; π_{2}; S_{4}; S_{2}; π_{1}; π_{2}; ,; S_{1}; S′_{2}; π′_{1}; π′_{2}; S_{1}]  (1.15)

which presents S_{4} as the limit of

[figure: S_{1}; S′_{2}; π′_{1}; π′_{2}; S_{1}; S_{2}; π_{1}; π_{2}; S_{2}; π_{1}; π_{2}; .; S_{1}; S′_{2}; π′_{1}; π′_{2}; S_{1}]  (1.16)

Since we decided to write the elements of S_{4} as diamonds, it is convenient to write the elements of S_{2} as vertical diagrams

[figure]

and the elements of S′_{2} as horizontal diagrams

[figure: ;]

in this notation the diagram (1.15) can be described (in term of elements) as

[figure: x_{11}; x_{11}; x_{12}; π′_{1}; π′_{2}; x_{12}; x_{11}; x_{21}; π_{1}; π_{2}; x_{11}; x_{12}; x_{21}; x_{22}; x_{12}; x_{22}; π_{1}; π_{2}; .; x_{21}; x_{21}; x_{22}; π′_{1}; π′_{2}; x_{22}]  (1.17)

There is also a convenient matrix notation:

[figure: x_{11}; (x_{11}, x_{12}); x_{12}; _{(\begin{matrix}x_{11} \\ x_{21} \end{matrix})}; _{(\begin{matrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix})}; _{(\begin{matrix}x_{12} \\ x_{22} \end{matrix})}; ;; x_{21}; (x_{21}, x_{22}); x_{22}]  (1.18)

it simplifies (1.17) just as (1.11) simplifies (1.10).

2. Double equivalence relations

Let G be an internal graph in a category ℂ with finite limits; just as in the case of a variety, G consists of two objects G_{0} and G_{1} in ℂ and two morphisms d and c from G_{1} to G_{0}. Such an internal graph G is said to be a relation if d and c are jointly monic, i.e. the morphism \xymatrix{⟨d, c⟩ : G_{1} \ar[r] & G_{0} × G_{0}} is a monomorphism. Let us also recall

2.1. Definition. An internal graph G is said to be an (internal) equivalence relation if it is a relation and is

(a) reflexive, i.e. there exists a morphism \xymatrix{e : G_{0} \ar[r] & G_{1}} with de = 1_{G_{0}} = ce;

(b) symmetric, i.e. there exists a morphism \xymatrix{i : G_{1} \ar[r] & G_{1}} with di = c and ci = d; (c) transitive, i.e. there exists a morphism \xymatrix{m : G_{1} ×_{G_{0}} G_{1} \ar[r] & G_{1}}, where G_{1} ×_{G_{0}} G_{1} is

constructed as the pullback

[figure: G_{1} ×_{G_{0}} G_{1}; p_{2}; p_{1}; G_{1}; c; ,; G_{1}; d; G_{0}]  (2.1)

such that dm = dp_{2} and cm = cp_{1}.

2.2. Remark.

(a) It is often convenient to consider an equivalence relation as a special case of an (internal) groupoid, i.e. to describe it as a system (G_{0}, G_{1}, d, c, e, m, i), where e, m, i (as in Definition 2.1) are however uniquely determined by d and c since ⟨d, c⟩ is a monomorphism;

(b) if ℂ is a variety of universal algebras, then the internal equivalence relations in ℂ are the same as “congruences”.

Let Eq(ℂ) be the category of equivalence relations in ℂ; a morphism \xymatrix{f : G \ar[r] & G′} in Eq(ℂ) is a diagram

\xymatrix{G_{1} \ar[r]^{d} \ar[r]_{c} \ar[d]_{f_{1}} & G_{0} \ar[d]^{f_{0}} \\ G′_{1} \ar[r]^{d′} \ar[r]_{c′} & G′_{0}}  (2.2)

in which d′f_{1} = f_{0}d and c′f_{1} = f_{0}c.

It is easy to see that since ℂ has finite limits, Eq(ℂ) also has finite limits. Therefore we can consider the equivalence relations in Eq(ℂ) — we will call them double equivalence relations in ℂ. A double equivalence relation D in ℂ can also be described as a diagram

\xymatrix{D_{4} \ar[r]^{q′_{1}} \ar[r]_{q′_{2}} \ar[d]^{q_{2}} \ar[d]_{q_{1}} & D_{2} \ar[d]^{p_{2}} \ar[d]_{p_{1}} \\ D′_{2} \ar[r]^{p′_{1}} \ar[r]_{p′_{2}} & D_{1}}  (2.3)

in which

\begin{matrix}p′_{1}q_{1} = p_{1}q′_{1}, & p′_{2}q_{1} = p_{1}q′_{2} \\ p′_{1}q_{2} = p_{2}q′_{1}, & p′_{2}q_{2} = p_{2}q′_{2}, \end{matrix}  (2.4)

and each pair of parallel arrows forms an equivalence relation.

The identities (2.4) are equivalent to the commutativity of the diagram

\xymatrix{D_{1} & D′_{2} \ar[l]_{p′_{1}} \ar[r]^{p′_{2}} & D_{1} \\ D_{2} \ar[u]^{p_{1}} \ar[d]_{p_{2}} & D_{4} \ar[l]_{q′_{1}} \ar[u]_{q_{1}} \ar[r]_{q′_{2}} \ar[d]_{q_{2}} & D_{2} \ar[u]_{p_{1}} \ar[d]^{p_{2}} \\ D_{1} & D′_{2} \ar[l]^{p′_{1}} \ar[r]_{p′_{2}} & D_{1}}  (2.5)

clearly similar to (1.15). Moreover, it is easy to check that any span S in ℂ determines, via (1.15), a double equivalence relation in ℂ which we will denote by Eq(S).

By the analogy with ordinary (internal) equivalence relations we introduce

2.3. Definition. A double equivalence relation in ℂ is said to be effective if it is of the form Eq(S) for some span S in ℂ.

If ℂ is a variety of universal algebras (or, more generally, an exact category) then every equivalence relation in ℂ is effective, i.e. is of the form Eq(φ) =

\xymatrix{X ×_{Y} X \ar[r]^{pr_{1}} \ar[r]_{pr_{2}} & X}  (2.6)

for some \xymatrix{φ : X \ar[r] & Y} in ℂ. The situation with double equivalence relations is much more complicated: in some sense commutator theory is a theory of noneffective double equivalence relations. This viewpoint is suggested by the results of [JP] and the following

2.4. Example. Let G = (G_{0}, G_{1}, d, c, e, m, i,) be an internal groupoid in ℂ. Recall that it is a diagram in ℂ of the form

[figure: G_{1} ×_{G_{0}} G_{1}; m; G_{1}; i; c; d; G_{0}; e]  (2.7)

in which de = 1_{G_{0}} = ce, dm = dp_{2}, cm = cp_{1}, di = c, ci = d (where G_{1} ×_{G_{0}} G_{1} together with the projections p_{1} and p_{2} is constructed as in 2.1), and the following diagrams commute:

[figure: G_{1} ×_{G_{0}} G_{1} ×_{G_{0}} G_{1}; m×1; 1×m; G_{1} ×_{G_{0}} G_{1}; m; ,; G_{1} ×_{G_{0}} G_{1}; m; G_{1}]  (2.8)

[figure: G_{1}; ⟨1,ed⟩; ⟨ec,1⟩; G_{1} ×_{G_{0}} G_{1}; m; ,; G_{1} ×_{G_{0}} G_{1}; m; G_{1}]  (2.9)

[figure: G_{1}; ⟨1,i⟩; ⟨i,1⟩; ec; ed; G_{1} ×_{G_{0}} G_{1}; m; .; G_{1} ×_{G_{0}} G_{1}; m; G_{1}]  (2.10)

If ℂ is a variety of universal algebras, we will use the same notation as in the Section 1 (and as for the ordinary groupoids), i.e. write \xymatrix{g : u \ar[r] & v} if g ∈ G_{1} and d(g) = u, c(g) = v, and also

m(f, g) = fg, e(u) = 1_{u}, i(g) = g^{−1}.  (2.11)

The diagrams (2.8), (2.9) and (2.10) express the associativity, the right and left unit law, and the right and left inverse law respectively.

A diamond (1.7) in G is said to be commutative if fg^{−1} = kh^{−1}. The set Comm(G) of all commutative diamonds form a subalgebra in G_{4} which can be defined as any of the following two pullbacks:

\xymatrix{Comm(G) \ar[r] \ar[d] & G_{2} \ar[d]^{m(1×i)} & Comm(G) \ar[r] \ar[d] & G′_{2} \ar[d]^{m(i×1)} \\ G_{2} \ar[r]_{m(1×i)} & G_{1} & G′_{2} \ar[r]_{m(i×1)} & G_{1}}  (2.12)

and therefore it is a well defined (sub-) double equivalence relation (of Eq(S), where S = (\xymatrix{G_{0} & G_{1} \ar[l]_{d} \ar[r]^{c} & G_{0}}) is the underlying span of G), also in the case of an abstract category ℂ with pullbacks.

It is easy to see that the following conditions are equivalent:

(a) the double equivalence relation determined by Comm(G) is effective;

(b) Comm(G) = G_{4};

(c) G is a relation.

3. Pseudogroupoids

For a given diamond (1.7) in a groupoid G let us write

m(f, g, k, h) = fg^{−1}h;  (3.1)

in particular

m(f, g, fg^{−1}h, h) = fg^{−1}h.  (3.2)

So m(f, g, k, h) does not depend on k; the reason why we involve k in the notation is that we are going to generalize the notion of groupoid in such a way that the composition above will be defined only if such a k does exist.

3.1. Definition. A pseudogroupoid is a pair (S, m) in which S is a span and \xymatrix{m : S_{4} \ar[r] & S_{1}} a map written as

m (\begin{matrix}f & k \\ g & h \end{matrix}) = m(f, g, k, h),  (3.3)

with:

(a) m(f, g, k, h) is parallel to k, i.e. πm(f, g, k, h) = π(k) (= π(h)) and π′m(f, g, k, h) = π′(k) (= π′(f));

(b) m(f, g, k, h) does not depend on k, i.e. m(f, g, k, h) = m(f, g, k′, h) if both sides are defined;

(c) if f = g then m(f, g, k, h) = h;

(d) if g = h then m(f, g, k, h) = f;

(e) m(m(x_{1}, x_{2}, y, x_{3}), x_{4}, t, x_{5}) = m(x_{1}, x_{2}, t, m(x_{3}, x_{4}, z, x_{5})) for every diagram in S of the form

[figure: ·; ·; t; x_{5}; z; ·; x_{1}; x_{2}; ·; ·; ·; x_{3}; y; x_{4}]  (3.4)

This definition can be “internalized via Yoneda”, i.e. we have an obvious notion of internal pseudogroupoid in an abstract category ℂ with finite limits.

For, given a span S and an object C in ℂ, we construct the span hom_{ℂ}(C, S) =

\xymatrix{hom_{ℂ}(C, S_{0}) & hom_{ℂ}(C, S_{1}) \ar[l]_{hom_{ℂ}(C,π)} \ar[r]^{hom_{ℂ}(C,π′)} & hom_{ℂ}(C, S′_{0})}  (3.5)

(in Sets). In this span (hom_{ℂ}(C, S))_{4} can be identified with hom_{ℂ}(C, S_{4}) and we introduce

3.2. Definition. An internal pseudogroupoid in a category ℂ with finite limits is a pair (S, m) in which S is a span in ℂ and \xymatrix{m : S_{4} \ar[r] & S_{1}} (where S_{4} is the limit of (1.13)) a morphism in ℂ such that (hom_{ℂ}(C, S), hom_{ℂ}(C, m)) is a pseudogroupoid for every object C in ℂ.

Note that if ℂ is a variety of universal algebras, then an internal pseudogroupoid in ℂ is just a pair (S, m) in which S is a span in ℂ and \xymatrix{m : S_{4} \ar[r] & S_{1}} a homomorphism (i.e. a morphism in ℂ) making (S, m) a pseudogroupoid (in Sets).

Consider examples:

3.3. Example. Any groupoid G can be considered as a pseudogroupoid (S, m) in which S is the span

\xymatrix{G_{0} & G_{1} \ar[l]_{d} \ar[r]^{c} & G_{0}}  (3.6)

and m is defined by (3.1). All the conditions of Definition 3.1 clearly hold; in particular (c), (d) and (e) become

gg^{−1}h = h, fg^{−1}g = f  (3.7)

and

(x_{1}x_{2}^{−1}x_{3})x_{4}^{−1}x_{5} = x_{1}x_{2}^{−1}(x_{3}x_{4}^{−1}x_{5})  (3.8)

respectively — which tells us that they play the roles of “Mal’tsev identities” and associativity.

Similarly any internal groupoid in a category ℂ (with finite limits) determines an internal pseudogroupoid in ℂ.

3.4. Example. A pregroupoid is a pair (S, l) in which S is a span and \xymatrix{l : S_{3} \ar[r] & S_{1}}, where

S_{3} = {(f, g, h) ∈ S_{1} × S_{1} × S_{1}|π(f) = π(g), π′(g) = π′(h)},  (3.9)

is a map with:

(a) πl(f, g, h) = π(h) and π′l(f, g, h) = π′(f);

(b) if f = g then l(f, g, h) = h;

(c) if g = h then l(f, g, h) = f;

(d) l(l(x_{1}, x_{2}, x_{3}), x_{4}, x_{5}) = l(x_{1}, x_{2}, l(x_{3}, x_{4}, x_{5})) for every diagram in S of the form

\xymatrix{· & · \ar[l]_{x_{1}} \ar[r]^{x_{2}} & · & · \ar[l]_{x_{3}} \ar[r]^{x_{4}} & · & · \ar[l]_{x_{5}} & .}  (3.10)

We immediately see that this is a special case of a pseudogroupoid: just put

m(f, g, k, h,) = l(f, g, h).  (3.11)

Conversely, if for every diagram in S of the form

\xymatrix{· & · \ar[l]_{f} \ar[r]^{g} & · & · \ar[l]_{h}}  (3.12)

there exists k in S_{1} with π(k) = π(h) and π′(k) = π′(f), then (3.11) can be used as the definition of l in terms of m. Thus \xymatrix{(S, l) \ar@{|->}[r] & (S, m)} (where m is as in (3.11)) is an isomorphism between the category of pregroupoids and the category of pseudogroupoids in which every diagram (3.12) can be completed as above.

Again, the same can be repeated in the “internal context”. The internal pregroupoids were introduced by A. Kock [Ko], and then used in [P1] to develop Commutator theory in Mal’tsev categories. Note that the notion of pregroupoid is clearly “between” the notions of groupoid and of pseudogroupoid: if G is a groupoid, we replace (3.1) by

l(f, g, h) = fg^{−1}h  (3.13)

and this clearly defines a pregroupoid.

3.5. Example. If S is a relation, i.e. the map ⟨π, π′⟩ : \xymatrix{S_{1} \ar[r] & S_{0} × S′_{0}} is injective, then S has a unique pseudogroupoid structure; it is defined by

m(f, g, k, h) = k.  (3.14)

In the internal context this can be written as

m = π_{12} : \xymatrix{S_{4} \ar[r] & S_{1}},  (3.15)

where π_{12} is as in (1.12).

3.6. Example. Any pseudogroupoid (S, m) has an opposite (=dual) pseudogroupoid (S, m)^{op} = (S^{op}, m^{op}), in which the opposite span S^{op} is

\xymatrix{S′_{0} & S_{1} \ar[l]_{π′} \ar[r]^{π} & S_{0},}  (3.16)

and m^{op} is defined by

m(f, g, k, h) = m^{op}(h, g, k, f)  (3.17)

— or

m = m^{op}⟨π_{22}, π_{21}, π_{12}, π_{11}⟩  (3.18)

in the internal context.

Clearly this notion of opposite contains the notions of opposite groupoid and opposite (=inverse) relation.

3.7. Example. Recall that a variety ℂ of universal algebras is said to be congruence modular if, for every C in ℂ, the lattice Cong(C) of congruences on C is modular. Let A be an algebra in a congruences modular variety ℂ and α, β congruences on A. The modular commutator (we take this expression from the title of [Ki]; it is just the commutator in the common sense) [α, β] is defined by

[α, β] = {(a, b) ∈ A × A|((a, a), (a, b)) ∈ Δ_{α,β}}  (3.19)

where Δ_{α,β} is the congruence on α generated by all

{((a, a), (b, b))|(a, b) ∈ β}.  (3.20)

A four variable term q is said to be a Kiss difference term (E. W. Kiss [Ki] says “a 4-difference term”) if

q(x, y, x, y) = x, q(x, x, y, y) = y  (3.21)

are identities in ℂ and

(q(a, b, c, d), q(a, b, c′, d)) ∈ [α, β]  (3.22)

whenever (\begin{matrix}a & c \\ b & d \end{matrix}) , (\begin{matrix}a & c′ \\ b & d \end{matrix}) ∈ S_{4} — where S is the span (1.9).

Among other things E. W. Kiss [Ki] proves that every congruence modular variety has such a term q, and that [α, β] = Δ_{A}(= {(a, a)|a ∈ A}) if and only if the following conditions hold:

(a) the map m : S_{4} ⟶ S_{1} = A defined by

m (\begin{matrix}a & c \\ b & d \end{matrix}) = q(a, b, c, d)  (3.23)

is a homomorphism;

(b) q(a, b, c, d) = q(a, b, c′, d) whenever (\begin{matrix}a & c \\ b & d \end{matrix}) and (\begin{matrix}a & c′ \\ b & d \end{matrix}) are in S_{4}.

As follows from Lemma 3.8 below, in our language that criterion for [α, β] = Δ_{A} simply says: [α, β] = Δ_{A} if and only if (3.23) defines a pseudogroupoid structure on S.

3.8. Lemma. Let ℂ be a variety of universal algebras, q a four variable term in ℂ satisfying (3.21), S a span in ℂ and m : \xymatrix{S_{4} \ar[r] & S_{1}} a homomorphism satisfying Definition 3.1(b) and (3.23). Then (S, m) is an internal pseudogroupoid in ℂ.

Proof. We have to show that (S, m) satisfies the conditions of Definition 3.1.

3.1(a):

πm(f, g, k, h) = π(q(f, g, k, h))
= q(π(f), π(g), π(k), π(h)) =
= q(π(f), π(f), π(k), π(k)) = π(k);
π′m(f, g, k, h) = π′(q(f, g, k, h)) =
= q(π′(f), π′(g), π′(k), π′(h)) =
= q(π′(k), π′(g), π′(k), π′(g)) = π′(k).

3.1(c): if (\begin{matrix}f & k \\ g & h \end{matrix}) ∈ S_{4} has f = g, then k is parallel to h, and we have m(f, g, k, h) =

m(f, f, h, h) = q(f, f, h, h) = h.

3.1(d): if (\begin{matrix}f & k \\ g & h \end{matrix}) ∈ S_{4} has g = h, then k is parallel to f, and we have m(f, g, k, h) =

m(f, g, f, g) = q(f, g, f, g) = f.

3.1(e):

m(m(x_{1}, x_{2}, y, x_{3}), x_{4}, t, x_{5}) =
= m(m(x_{1}, x_{2}, y, x_{3}), m(x_{4}, x_{4}, x_{4}, x_{4}), m(x_{1}, x_{2}, t, z), m(x_{4}, x_{4}, x_{5}, x_{5})) =
= q(m(x_{1}, x_{2}, y, x_{3}), m(x_{4}, x_{4}, x_{4}, x_{4}), m(x_{1}, x_{2}, t, z), m(x_{4}, x_{4}, x_{5}, x_{5})) =
= m(q(x_{1}, x_{4}, x_{1}, x_{4}), q(x_{2}, x_{4}, x_{2}, x_{4}), q(y, x_{4}, t, x_{5}), q(x_{3}, x_{4}, z, x_{5})) =
= m(x_{1}, x_{2}, m(y, x_{4}, t, x_{5}), m(x_{3}, x_{4}, z, x_{5})) =
= m(x_{1}, x_{2}, t, m(x_{3}, x_{4}, z, x_{5})).

Now we are going to show that a pseudogroupoid can be described as a span equipped with an appropriate set of “commutative diamonds ”. We use the set-theoretic context just for simplicity — in fact everything can be repeated in the internal context as we will see.

First we need

3.9. Lemma. Given a diamond (1.7) in a pseudogroupoid, the following conditions are equivalent:

(a) m(f, g, k, h) = k;

(b) m(g, f, h, k) = h;

(c) m(h, k, g, f) = g;

(d) m(k, h, f, g) = f.

Proof. Since we deal with an arbitrary diamond in an arbitrary pseudogroupoid, it suffices to prove (a)⇒(b). If (a) holds, then using 3.1(e) for

[figure: ·; ·; h; h; k; ·; g; f; ·; ·; ·; f; g; g]  (3.24)

we obtain

m(g, f, h, k) = m(g, f, h, m(f, g, k, h)) =
= m(m(g, f, g, f), g, h, h) =
= m(g, g, h, h) = h.

This lemma suggests to introduce

3.10. Definition. A diamond (1.7) in a pseudogroupoid is said to be commutative if it satisfies the equivalent conditions of Lemma 3.9.

Consider the diagram

[figure: Comm(S, m); inclusion; S_{4}; S_{2}; ,; S′_{2}; S_{1}]  (3.25)

where Comm(S, m) is the set of commutative diamonds in a pseudogroupoid (S, m), and the square is the effective double equivalence relation Eq(S). This diagram determines a sub-double equivalence relation of Eq(S) — just as in the case of groupoids considered in 2.4. The proof is straightforward; however it is good to see how the “associativity” condition 3.1(e) helps to prove the transitivity of

\xymatrix{Comm(S, m) \ar[r] \ar[r] & S_{2}}  (3.26)

and

\xymatrix{Comm(S, m) \ar[r] \ar[r] & S′_{2}}.  (3.27)

Since (3.26) and (3.27) are dual to each other, let us consider only (3.26). The tran-

sitivity of (3.26) means that the commutativity of (\begin{matrix}f & k \\ g & h \end{matrix}) and (\begin{matrix}k & j \\ h & i \end{matrix}) implies the com-

mutativity of (\begin{matrix}f & j \\ g & i \end{matrix}), i.e.

\xymatrix{(m(f, g, k, h) = k, m(k, h, j, i) = j) \ar@{=>}[r] & m(f, g, j, i) = j.}  (3.28)

We have j = m(k, h, j, i) = m(m(f, g, k, h), h, j, i) = m(f, g, j, m(h, h, i, i)) = m(f, g, j, i) as desired; here 3.1(e) was applied to

[figure: ·; ·; j; i; j; ·; f; g; ·; .; ·; ·; h; k; h]  (3.29)

Note that, again just as in 2.4, Comm(S, m) determines an effective double equivalence relation if and only if S is a relation. In other words it determines a thin double equivalence relation in the sense of

3.11. Definition. A double equivalence relation D is said to be thin if it satisfies the following condition:

Let \overline{D}_{4} =

\xymatrix{D_{1} & D′_{2} \ar[l]_{p′_{1}} \ar[r]^{p′_{2}} & D_{1} \\ D_{2} \ar[u]^{p_{1}} \ar[d]_{p_{2}} & _{\overline{D}_{4}} \ar[l]_{_{\overline{q′}_{1}}} \ar[u]_{_{\overline{q}_{1}}} \ar[r]_{_{\overline{q′}_{2}}} \ar[d]_{_{\overline{q}_{2}}} & D_{2} \ar[u]_{p_{1}} \ar[d]^{p_{2}} \\ D_{1} & D′_{2} \ar[l]^{p′_{1}} \ar[r]_{p′_{2}} & D_{1}}  (3.30)

be the limit of the diagram (2.5) with removed D_{4}, and \xymatrix{i : D_{4} \ar[r] & \overline{D}_{4}} the canonical injection; let α, β, γ be any three out of four canonical maps \xymatrix{\overline{D}_{4} \ar[r] & D_{1}}. Then for every x ∈ \overline{D}_{4} there exists a unique y ∈ D_{4} such that x and i(y) have the same images under α, β, γ.

This in fact gives an alternative (equivalent) definition of a pseudogroupoid:

3.12. Theorem. Let S be a span and C a subset of S_{4} such that

\xymatrix{C \ar[r] \ar[r] \ar[d] \ar[d] & S_{2} \ar[d] \ar[d] \\ S′_{2} \ar[r] \ar[r] & S_{1}}  (3.31)

is a thin sub-double equivalence relation of Eq(S). Then there exists a unique pseudogroupoid structure \xymatrix{m : S_{4} \ar[r] & S_{1}} on S with Comm(S, m) = C.

Proof. The assertion that (3.31) is thin means that for every diamond (\begin{matrix}f & k \\ g & h \end{matrix}) in S there

exists a unique k′ ∈ S_{1} such that (\begin{matrix}f & k′ \\ g & h \end{matrix}) is in C — and so we must have m(f, g, k, h) = k′,

which proves the uniqueness of m (provided it exists). On the other hand if we define m in this way, the conditions (a)–(e) of Definition 3.1 are satisfied:

3.1(a) and 3.1(b) hold by the definition of m.

3.1(c): m(f, g, k, h) = m(f, f, h, h) by f = g and 3.1(b), and m(f, f, h, h) = h since

(\begin{matrix}f & h \\ f & h \end{matrix}) is in C because \xymatrix{C \ar[r] \ar[r] & S′_{2}} is reflexive.

3.1(d): m(f, g, k, h) = m(f, g, f, g) by g = h and 3.1(b), and m(f, g, f, g) = f since

(\begin{matrix}f & f \\ g & g \end{matrix}) is in C because \xymatrix{C \ar[r] \ar[r] & S_{2}} is reflexive.

3.1(e): Given a diagram (3.4), we have to show that

(\begin{matrix}m(x_{1}, x_{2}, y, x_{3}) & t′ \\ x_{4} & x_{5} \end{matrix}) ∈ [figure: C] (\begin{matrix}x_{1} & t′ \\ x_{2} & m(x_{3}, x_{4}, z, x_{5}) \end{matrix}) ∈ C  (3.32)

(where t′ is an arrow parallel to t). That is, we have to show that

(\begin{matrix}x_{1} & y′ \\ x_{2} & x_{3} \end{matrix}) , (\begin{matrix}y′ & t′ \\ x_{4} & x_{5} \end{matrix}) , (\begin{matrix}x_{3} & z′ \\ x_{4} & x_{5} \end{matrix}) ∈ [figure: C](\begin{matrix}x_{1} & t′ \\ x_{2} & z′ \end{matrix}) ∈ C.  (3.33)

We have:

1^{∘} . Since \xymatrix{C \ar[r] \ar[r] & S′_{2}} is symmetric and (\begin{matrix}x_{3} & z′ \\ x_{4} & x_{5} \end{matrix}) is in C, (\begin{matrix}x_{4} & x_{5} \\ x_{3} & z′ \end{matrix}) also is in ℂ.

2^{∘} . Since \xymatrix{C \ar[r] \ar[r] & S′_{2}} is transitive and (\begin{matrix}y′ & t′ \\ x_{4} & x_{5} \end{matrix}), (\begin{matrix}x_{4} & x_{5} \\ x_{3} & z′ \end{matrix}) are in C, (\begin{matrix}y′ & t′ \\ x_{3} & z′ \end{matrix}) is in C.

3^{∘} . Since \xymatrix{C \ar[r] \ar[r] & S_{2}} is transitive and (\begin{matrix}x_{1} & y′ \\ x_{2} & x_{3} \end{matrix}), (\begin{matrix}y′ & t′ \\ x_{3} & z′ \end{matrix}) are in C, (\begin{matrix}x_{1} & t′ \\ x_{2} & z′ \end{matrix}) is in C

as desired.

3.13. Remark. If S is a span in a variety ℂ of universal algebras and C a subalgebra in S_{4} as in Theorem 3.12, then the corresponding m : \xymatrix{S_{4} \ar[r] & S_{1}} is a homomorphism and so we obtain an internal pseudogroupoid structure in ℂ.

Theorem 3.12 (+ Remark 3.13) suggests to consider the following example of the situation — in the original work of J. D. H. Smith [S] — where the notion of pseudogroupoid (in fact pregroupoid) was almost introduced.

3.14. Example. The Section 2.1 “Centrality in general” in [S] begins by:

“For this section, let \underline{\underline{T}} be any variety, not necessarily Mal’tsev.

211 Definition. Let A be a \underline{\underline{T}} -algebra, let β, γ be congruences on A, and let (γ|β) be a congruence on β. Then γ is said to centralizeβ by means of the centralizing congruence (γ|β) iff the following conditions are satisfied:

(C0): (x, y)(γ|β)\xymatrix{(x′, y′) \ar@{=>}[r] & xγx′}.

(C1): ∀(x, y) ∈ β, π^{∘} : \xymatrix{(x, y)^{(γ|β)} \ar[r] & x^{γ}}; \xymatrix{(x′, y′) \ar@{|->}[r] & x′} bijects.

(C2): The following three conditions are satisfied:

(RR): ∀(x, y) ∈ γ, (x, x)(γ|β)(y, y).

(RS): (x, y)(γ|β)\xymatrix{(x′, y′) \ar@{=>}[r] & (y, x)(γ|β)(y′, x′)}.

(RT): {(x, y)(γ|β)(x′, y′) and \atop (y, z)(γ|β)\xymatrix{(y′, z′) \ar@{=>}[r] & (x, z)}(γ|β)(x′, z′).}

Conditions (RR), (RS) and (RT) respectively are known as respect for the reflexivity, symmetry, and transitivity of β. (C2) is called respect for equivalence.

Intuitively, one thinks of the relation (x, y)(γ|β)(x′, y′) as a parallelogram

[figure: ·; y; γ; ·; y′; .”; ·; x; β; γ; ·; β; x′]

Let us translate this in our language. First we note that \underline{\underline{T}} plays the same role as our ℂ which now is supposed to be an arbitrary variety of universal algebras. Since the definition above begins with congruences β, γ (instead of our α, β in (1.9)), we take S to be the span

\xymatrix{A/β & A \ar[l] \ar[r] & A/γ}  (3.34)

(in ℂ) instead of (1.9). Then β and γ are the same as our \xymatrix{S_{2} \ar[r] \ar[r] & S_{1}} and \xymatrix{S′_{2} \ar[r] \ar[r] & S_{1}} respectively. Since (γ|β) is a congruence on β, i.e. on S_{2}, let us write it as (γ|β) = \xymatrix{C \ar[r] \ar[r] & S_{2}}. Since C is a subalgebra in S_{2} × S_{2}, the condition (C0) says that C is a subalgebra in S_{4}; together with (C2) this says that C determines a sub-double equivalence relation of Eq(S) (in particular (RR), (RS), and (RT) are just reflexivity, symmetry, and transitivity of \xymatrix{C \ar[r] \ar[r] & S′_{2}}). Accordingly the parallelogram above corresponds to

[figure: ·; ·; y; x; ·; y′; x′; ∈ C.; ·]  (3.35)

The condition (C1) translates now as

[figure: ·; ·; ·; ∀; y; x; ·; x′; ∃! y′; ·; y; x; ·; y′; x′; ,; ·; ·]  (3.36)

i.e. it says that for every (y, x, x′) ∈ S_{3} (see (3.9)) there exists a unique y′ with (3.35). That is, it says that C is thin and, moreover, the corresponding (internal) pseudogroupoid is a pregroupoid.

Summarizing, we can simply say that γ centralizes β by means of (γ|β) in the sense of J. D. H. Smith [S] if and only if they form (as above) an internal pregroupoid.

4. Free internal and algebraic pseudogroupoids

The category P (ℂ) of internal pseudogroupoids in a variety ℂ of universal algebras has “all standard properties” plus the existence of the commutative triangle

[figure: P (ℂ); forgetful; S(ℂ); ,; R(ℂ); inclusion; inclusion]  (4.1)

where S(ℂ) is the category of spans in ℂ and R(ℂ) the category of (“homomorphic”) relations in ℂ; the left hand inclusion comes from Example 3.5, which says that every relation has a unique pseudogroupoid structure.

A morphism φ : \xymatrix{(S, m) \ar[r] & (T, n)} in P (ℂ) can be displayed as in S(ℂ), i.e. as

[figure: S_{0}; φ_{0}; S_{1}; π_{s}; π′_{s}; φ_{1}; S′_{0}; φ′_{0}; ;; T_{0}; T_{1}; π_{T}; π′_{T}; T′_{0}]  (4.2)

if φ_{0}, φ′_{0}, φ_{1} are inclusions we will say that(S, m) is a subpseudogroupoid in (T, n) and write (S, m) ⩽ (T, n).

The set of subpseudogroupoids in (T, n) forms a complete lattice which we will denote by Sub(T, n). For any subset X in T_{1} the subpseudogroupoid ⟨X⟩_{(T,n)} generated by X is defined as the smallest subpseudogroupoid (S, m) in (T, n) with X ⊂ T_{1}. It is often convenient to write

⟨X⟩_{(T,n)} = ⟨X⟩  (4.3)

and identify this pseudogroupoid, as well as other elements in Sub(T, n) with the corresponding subsets in T_{1} — as one usually does in Universal Algebra.

Of course there is a standard procedure to construct ⟨X⟩ as the union of

X = ⟨X⟩^{0} ⊂ ⟨X⟩^{1} ⊂ ⟨X⟩^{2} ⊂ …,  (4.4)

where ⟨X⟩^{i+1} is the subalgebra in ⟨X⟩ generated by

\{n(f, g, k, h) | f, g, k, h ∈ ⟨X⟩^{i}and (\begin{matrix}f & k \\ g & h \end{matrix}) ∈ T_{4}\}  (4.5)

for i = 0, 1, 2, …. In particular ⟨X⟩ is “as large as X”, i.e.

card X ⩽ card ⟨X⟩ ⩽ max{card X, card Ω, ℵ_{0}},  (4.6)

where Ω is the set of operators in the signature of ℂ.

Any morphism φ : \xymatrix{(S, m) \ar[r] & (T, n)} has an image φ(S, m) ∈ Sub(T, n). However in general it is not simply the set-theoretic image φ_{1}(S_{1}) ⊂ T_{1}, but

φ(S, m) = ⟨φ_{1}(S_{1})⟩.  (4.7)

Any X ⊂ T_{1} determines a relation Rel(X) ⊂ T_{0} × T′_{0} by

Rel(X) = {(t, t′) ∈ T_{0} × T′_{0} | ∃x ∈ X, π_{T} (x) = t, π′_{T} (x) = t′},  (4.8)

in particular so does T_{1} itself and we will write

Rel(T, n) = Rel(T) = Rel(T_{1}),  (4.9)

and similarly for spans.

4.1. Proposition.

(a) The category P (ℂ) is complete and cocomplete;

(b) the forgetful functor \xymatrix{P (ℂ) \ar[r] & S(ℂ)} has a left adjoint F : \xymatrix{S(ℂ) \ar[r] & P (ℂ)} such that the diagram

\xymatrix{S(ℂ) \ar[rr]^{F} \ar[dr]_{Rel} &  & P (ℂ) \ar[dl]^{Rel} \\  & R(ℂ)}  (4.10)

commute.

Proof. The completeness is obvious, and the cocompleteness and existence of the left adjoint follows from the completeness, (4.6), and (4.7) (although it is also a special case of a well known results for so-called essentially algebraic theories). The commutativity of (4.10) (up to an isomorphism) follows from the commutativity of (4.1) which consists of the right adjoints of the functors involved in (4.10).

Note also that since the diagram

\xymatrix{P (ℂ) \ar[r]^{forgetful} \ar[d]_{( )^{op}} & S(ℂ) \ar[d]^{( )^{op}} \\ P (ℂ) \ar[r]_{forgetful} & S(ℂ)}  (4.11)

commutes (see Example 3.6), so does (up to a canonical isomorphism) also the diagram

[figure: S(ℂ); F; ( )^{op}; P (ℂ); ( )^{op}; .; S(ℂ); F; P (ℂ)]  (4.12)

Given a span S in ℂ, we will write

F (S) = (S^{⋆}, m)  (4.13)

and S^{⋆}=

\xymatrix{S_{0} & S_{1}^{⋆} \ar[l]_{π^{⋆}} \ar[r]^{π′^{⋆}} & S′_{0}}  (4.14)

— since we can take

S_{0}^{⋆} = S_{0}, (S^{⋆})′_{0} = S′_{0}  (4.15)

as follows from Proposition 4.1(b). The canonical morphism \xymatrix{S \ar[r] & S^{⋆}} will be written as

\xymatrix{ & S_{1} \ar[dl]_{π} \ar[dd]^{η=η_{S}} \ar[dr]^{π′} \\ S_{0} &  & S′_{0} \\  & S_{1}^{⋆} \ar[ul]^{π^{⋆}} \ar@{-->}[ur]_{π′^{⋆}}}  (4.16)

In the pseudogroupoid F (S) we have

⟨η(S_{1})⟩ = S_{1}^{⋆}  (4.17)

and so

S_{1}^{⋆} = ⋃_{n=0}^{∞}⟨η(S_{1})⟩^{n}  (4.18)

It might happen that ⟨η(S_{1})⟩^{n+1} = ⟨η(S_{1})⟩^{n} for some n, and then also S_{1}^{⋆} = ⟨η(S_{1})⟩^{n}, but if this is the case for every span S in ℂ, then we could say that ℂ has the dimension dim(ℂ) ⩽ n — and dim(ℂ) = ∞ if there is no such n. However we “understand” only the cases dim(ℂ) = 0 and dim(ℂ) = ∞. In order to describe the first of them we introduce 4.2. Definition. An internal pseudogroupoid (S, m) in a variety ℂ is said to be algebraic if ℂ has a four variable term q such that

m(f, g, k, h) = q(f, g, k, h)  (4.19)

for every S-diamond (\begin{matrix}f & k \\ g & h \end{matrix}). We will also say that (S, m) is q-algebraic.

We are going to use the free algebras in ℂ; the free algebra on a set {x_{1}, …, x_{n}} will be denoted by ℂ(x_{1}, …, x_{n}).

Consider the span 𝔻=

\xymatrix{ℂ(s_{1}, s_{2}) & ℂ(x_{11}, x_{21}, x_{12}, x_{22}) \ar[l]_{π} \ar[r]^{π′} & ℂ(t_{1}, t_{2})}  (4.20)

in ℂ in which

\xymatrix{ & t_{1} \\ s_{1} \ar[ur]^{x_{11}} \ar[dr]_{x_{21}} &  & s_{2} \ar[ul]_{x_{12}} \ar[dl]^{x_{22}} \\  & t_{2}}  (4.21)

is a diamond, i.e. π and π′ are defined by π(x_{ij}) = s_{j} and π′(x_{ij}) = t_{i} respectively. This span could be called the generic diamond in ℂ; it has the following obvious universal property.

4.3. Proposition. For every span S in ℂ and every S-diamond x, there exists a unique morphism \xymatrix{𝔻 \ar[r] & S} in S(ℂ) which sends the diamond (4.21) to x.

In other words there is a canonical bijection

S_{4} ≈ hom_{S(ℂ)}(𝔻, S),  (4.22)

and of course in the case ℂ = Sets, (4.21) is just the picture of 𝔻.

Using the universal property of 𝔻 we will prove the following theorem which gives various equivalent conditions for dim(ℂ) = 0:

4.4. Theorem.

(a) For a given four variable term q in a variety ℂ the following conditions are equivalent:

(a_{1}) every internal pseudogroupoid in ℂ is q-algebraic;

Theory and Applications of Categories, Vol. 8, No. 15 431

(a_{2}) the term q considered as an element in 𝔻_{1} = ℂ(x_{11}, x_{21}, x_{12}, x_{22}) via

q = q(x_{11}, x_{21}, x_{12}, x_{22}) satisfies

η_{𝔻}(q) = m(η_{𝔻}(x_{11}), η_{𝔻}(x_{21}), η_{𝔻}(x_{12}), η_{𝔻}(x_{22}));  (4.23)

(a_{3}) the identities (3.21) (recall they are q(x, y, x, y) = x and q(x, x, y, y) = y) hold,

and for every diagram of the form

\xymatrix{ & · \\ · \ar[ur]^{f} \ar[dr]_{g} &  & · \ar[ul]^{k} \ar[ul]_{k′} \ar[dl]^{h} \\  & ·}  (4.24)

in any span S in ℂ we have

η_{S}(q(f, g, k, h)) = η_{S}(q(f, g, k′, h)).  (4.25)

(b) For a given variety ℂ the following conditions are equivalent: (b_{1}) dim(ℂ) = 0; (b_{2}) the homomorphism η_{S} : \xymatrix{S_{1} \ar[r] & S_{1}^{⋆}} is surjective for every span S in ℂ; (b_{3}) for every span S in ℂ and every S-diamond (\begin{matrix}f & k \\ g & h \end{matrix}) there exists k′ ∈ S_{1} with

η_{S}(k′) = m (η_{S}(f), η_{S}(g), η_{S}(k), η_{S}(h)); (b_{4}) the homomorphism η_{𝔻} : \xymatrix{𝔻_{1} \ar[r] & 𝔻_{1}^{⋆}} (where 𝔻 is the span (4.20) as above) is

surjective; (b_{5}) ℂ has a four variable term q satisfying the equivalent conditions (a_{1})–(a_{3}).

Proof. In order to prove (a) we will prove (a_{1})⇔(a_{2}), and (a_{1})⇔(a_{3}).

(a_{1})⇒(a_{2}). Since the pseudogroupoid F (𝔻) = (𝔻^{⋆}, m) must be q-algebraic, we have m(η_{𝔻}(x_{11}), η_{𝔻}(x_{21}), η_{𝔻}(x_{12}), η_{𝔻}(x_{22})) =

= q(η_{𝔻}(x_{11}), η_{𝔻}(x_{21}), η_{𝔻}(x_{12}), η_{𝔻}(x_{22})) =
= η_{𝔻}(q(x_{11}, x_{21}, x_{12}, x_{22})) = η_{𝔻}(q).

(a_{2})⇒(a_{1}). Given a pseudogroupoid (S, m) and an S-diamond (\begin{matrix}f & k \\ g & h \end{matrix}), we take φ :

\xymatrix{𝔻 \ar[r] & S} with

(\begin{matrix}φ_{1}(x_{11}) & φ_{1}(x_{12}) \\ φ_{1}(x_{21}) & φ_{1}(x_{22}) \end{matrix}) = (\begin{matrix}f & k \\ g & h \end{matrix}) ,  (4.26)

which does exist by Proposition 4.3. Denoting the morphism \xymatrix{F (𝔻) \ar[r] & (S, m)} induced by φ by ψ, we obtain

q(f, g, k, h) = q(φ_{1}(x_{11}), φ_{1}(x_{21}), φ_{1}(x_{12}), φ_{1}(x_{22})) =
= φ_{1}(q(x_{11}, x_{21}, x_{12}, x_{22})) = ψ_{1}η_{𝔻}(q(x_{11}, x_{21}, x_{12}, x_{22})) =
= ψ_{1}(m(η_{𝔻}(x_{11}), η_{𝔻}(x_{21}), η_{𝔻}(x_{12}), η_{𝔻}(x_{22}))) =
= m(ψ_{1}η_{𝔻}(x_{11}), ψ_{1}η_{𝔻}(x_{21}), ψ_{1}η_{𝔻}(x_{12}), ψ_{1}η_{𝔻}(x_{22})) =
= m(φ_{1}(x_{11}), φ_{1}(x_{21}), φ_{1}(x_{12}), φ_{1}(x_{22})) = m(f, g, k, h),

and so (S, m) is q-algebraic.

(a_{1})⇒(a_{3}). In order to prove the identity q(x, y, x, y) = x consider the span

[figure: 1; ℂ(x, y); ℂ(x, y),]  (4.27)

where 1 is a one element algebra. This is a relation in ℂ and therefore an internal

pseudogroupoid. Applying (4.19) to the diamond (\begin{matrix}x & x \\ y & y \end{matrix}) we obtain

q(x, y, x, y) = m(x, y, x, y) = x,

and since q(x, y, x, y) = x holds in the free algebra ℂ(x, y), it is an identity in ℂ.

Similarly, using the diamond (\begin{matrix}x & y \\ x & y \end{matrix}) in

[figure: ℂ(x, y); ℂ(x, y); 1]  (4.28)

we obtain q(x, x, y, y) = y.

For (4.25) we have

η_{S}(q(f, g, k, h)) = q(η_{S}(f), η_{S}(g), η_{S}(k), η_{S}(h)) =
= m(η_{S}(f), η_{S}(g), η_{S}(k), η_{S}(h)) =
= m(η_{S}(f), η_{S}(g), η_{S}(k′), η_{S}(h)) =
= q(η_{S}(f), η_{S}(g), η_{S}(k′), η_{S}(h)) = η_{S}(q(f, g, k′, h)),

i.e. (4.24) immediately follows from 3.1(b).

(a_{3})⇒(a_{1}): For a given diamond (\begin{matrix}f & k \\ g & h \end{matrix}) in a pseudogroupoid (S, m) in ℂ, the iden-

tities (3.21) give π(q(f, g, k, h)) = π(h), π′(q(f, g, k, h)) = π′(f), and then

(\begin{matrix}f & q(f, g, k, h) \\ g & h \end{matrix}) = q ((\begin{matrix}f & f \\ g & g \end{matrix}) , (\begin{matrix}g & g \\ g & g \end{matrix}) , (\begin{matrix}f & k \\ g & h \end{matrix}) , (\begin{matrix}g & h \\ g & h \end{matrix})) ,  (4.29)

and so we have

m(f, g, k, h) = m(f, g, q(f, g, k, h), h) =

= m (q ((\begin{matrix}f & f \\ g & g \end{matrix}) , (\begin{matrix}g & g \\ g & g \end{matrix}) , (\begin{matrix}f & k \\ g & h \end{matrix}) , (\begin{matrix}g & h \\ g & h \end{matrix}))) =

= q(m(f, g, f, g), m(g, g, g, g), m(f, g, k, h), m(g, g, h, h)) = = q(f, g, m(f, g, k, h), h).

On the other hand

η_{S}(q(f, g, m(f, g, k, h), h)) = η_{S}(q(f, g, k, h))

by (4.25), and we obtain

η_{S}(m(f, g, k, h)) = η_{S}(q(f, g, k, h)).

Since (S, m) is an internal pseudogroupoid, η_{S} is a (split) monomorphism by a general property of adjoint functors. Therefore we conclude m(f, g, k, h) = q(f, g, k, h) as desired.

(b): The conditions (b_{1}), (b_{2}) and (b_{3}) are clearly equivalent, and (b_{2}) implies (b_{4}). Therefore it suffices to prove (b_{4})⇒(b_{5}) and (b_{5})⇒(b_{3}).

(b_{4})⇒(b_{5}). Since η_{𝔻} is surjective, there exists q ∈ 𝔻_{1} satisfying (4.23). This means that there exists a four variable term q satisfying (a_{2}).

(b_{5})⇒(b_{3}): If we take k′ = q(f, g, k, h), then

η_{S}(k′) = η_{S}(q(f, g, k, h)) =
= q(η_{S}(f), η_{S}(g), η_{S}(k), η_{S}(h)) =
= m(η_{S}(f), η_{S}(g), η_{S}(k), η_{S}(h))

— by (4.19).

4.5. Example. Recall that a three variable term p in a variety ℂ is said to be a Mal’tsev term if

p(x, y, y) = x, p(x, x, y) = y  (4.30)

are identities in ℂ. If such a term does exist, ℂ is said to be a Mal’tsev variety (=“congruence permutable” variety). In this case the four variable term q defined as

q(x, y, t, z) = p(x, y, z)  (4.31)

trivially satisfies 4.4(a_{3}) and so dim(ℂ) = 0.

The formula (4.31) should be compared with the formula (3.11) in Example 3.4. In the Mal’tsev case every diagram (3.12) can be completed as

\xymatrix{ & · \\ · \ar[ur]^{f} \ar[dr]_{g} &  & · \ar[ul]_{p(f,g,h)} \ar[dl]^{h} \\  & ·}  (4.32)

since π(p(f, g, h)) = p(π(f), π(g), π(h)) = p(π(f), π(f), π(h)) = π(h) and π′(p(f, g, h)) = p(π′(f), π′(g), π′(h)) = p(π′(f), π′(g), π′(g)) = π′(f) — and therefore the pseudogroupoids are the same as the pregroupoids (this property is in fact equivalent to ℂ being Mal’tsev). In the “pregroupoid version” of Theorem 4.4 we would have

l(f, g, h) = p(f, g, h)  (4.33)

instead of (4.19).

Note that the equivalence (a_{1})⇔(a_{3}) in Theorem 4.4 gives the following characterization of Mal’tsev varieties: ℂ is a Mal’tsev variety if and only if it has a three variable term p such that every internal pseudogroupoid in ℂ is q-algebraic with q defined by (4.31).

Since most of varieties studied in classical Algebra are Mal’tsev varieties, let us also point out the following:

(a) If ℂ is a variety of groups, possibly with an additional algebraic structure (say rings, modules, algebras) then we can take p(x, y, z) = xy^{−1}z and so in every internal pseudogroupoid (=pregroupoid) (S, m) in ℂ we have

m(f, g, k, h) = fg^{−1}h,  (4.34)

or

m(f, g, k, h) = f − g + h  (4.35)

if the notation is additive.

(b) More generally, if we have the quasigroup structure instead of the group structure, then

m(f, g, k, h) = (f/(g\g)) · (g\h),  (4.36)

where ·, /, \ are the multiplication, the right division, and the left division respectively.

4.6. Example. Suppose that ℂ has a two variable term u, written as u(x, y) = xy, such that

(xy)y = xy, xy = yx, xx = x  (4.37)

are identities in ℂ. Then every internal pseudogroupoid in ℂ is a relation. Indeed, if f and g are parallel arrows (i.e. π(f) = π(g) and π′(f) = π′(g)) in an internal pseudogroupoid (S, m) in ℂ, then the third identity in (4.37) tells us that every arrow obtained from f and g by u (and its iterations) is also parallel to f and g, and

\begin{aligned}f & = m(fg, fg, f, f) = m(fg, gf, ff, ff) = \\ & = m(f, g, f, f)m(g, f, f, f) = hg, \end{aligned}

where h = m(f, g, f, f), and then

fg = (hg)g = hg = f

— and similarly gf = g, which gives f = fg = gf = g.

From this and (3.14) we conclude that every internal pseudogroupoid in ℂ is q-algebraic with

q(x, y, t, z) = t;  (4.38)

in particular dim(ℂ) = 0.

This of course applies to semilattices (with u(x, y) = x ⋀ y or u(x, y) = x ⋁ y), again possibly with an additional structure: lattices, Boolean and Heyting algebras and many other related varieties (although Boolean and Heyting algebras at the same time form Mal’tsev varieties!)

Now consider an example of dim(ℂ) = ∞:

4.7. Example. Let A be a monoid and ℂ the variety of A-sets. The monoid ring ℤ[A] being an A-module is an internal abelian group in ℂ and so the span S =

\xymatrix{1 & ℤ[A] \ar[l] \ar[r] & 1}  (4.39)

with the usual m(f, g, k, h) = f − g + h is an internal pseudogroupoid in ℂ. Since A is a subset in ℤ[A] we can consider the sequence (4.4) for X = A. Clearly ⟨X⟩^{n+1} ≠ ⟨X⟩^{n} for each n = 0, 1, 2, … and so dim(ℂ) = ∞. In particular dim(Sets) = ∞.

5. The commutator

Let us recall the notion of subobject.

Let ℂ be a category and A an object in ℂ.

If u : \xymatrix{U \ar[r] & A} and v : \xymatrix{V \ar[r] & A} are monomorphisms then we write (U, u) ⩽ (V, v) if there exists a morphism ω : \xymatrix{U \ar[r] & V} with vω = u; note that in this case ω is a uniquely determined monomorphism. We say that (U, u) is equivalent to (V, v) if (U, u) ⩽ (V, v) and (V, v) ⩽ (U, u); in this case the ω above is an isomorphism. The equivalence class of (U, u) written as ⟨U, u⟩ is called a subobject in A. The collection of all subobjects in A will be denoted by Sub(A); it has the induced partial order ⩽, so that ⟨U, u⟩ ⩽ ⟨V, v⟩ if and only if (U, u) ⩽ (V, v).

Recall the following

5.1. Definition. A category ℂ is said to be finitely well-complete if it has

(a) finite limits;

(b) all (even large) limits of diagrams which are collections of monomorphisms with the same codomain.

If ℂ is finitely well-complete, then each Sub(A) is a (possibly large) complete lattice, and each morphism φ : \xymatrix{A \ar[r] & B} induces the adjoint pair

\xymatrix{Sub(A) \ar[r]^{φ_{⋆}} & Sub(B), \ar[l]^{φ^{⋆}}}  (5.1)

where the right adjoint φ^{⋆} sends ⟨V, v⟩ to the class of the pullback (=“inverse image”) of v along φ, and φ_{⋆} defined as the left adjoint of φ^{⋆} is called the direct image (i.e. φ_{⋆}⟨U, u⟩ is called the direct image of ⟨U, u⟩). Furthermore, each Sub(A × A) contains the complete Λ-subsemilattice ER(A) of (internal) equivalence relations on A; the elements of ER(A) are the classes ⟨E, e⟩, where e : \xymatrix{E \ar[r] & A × A} is an equivalence relation (see Definition 2.1). And again φ : \xymatrix{A \ar[r] & B} induces an adjunction

\xymatrix{ER(A) \ar[r]^{φ_{#}} & ER(B) \ar[l]^{φ^{#}}}  (5.2)

such that the diagram

[figure: Sub(A × A); ⋃; Sub(B × B); (φ×φ)^{⋆}; ⋃; ER(A); ER(B); φ^{#}]  (5.3)

commutes.

Recall also that every morphism φ : \xymatrix{A \ar[r] & B} determines an equivalence relation Eq(φ) on A (see (2.6)).

5.2. Definition. Let ℂ be a finitely well-complete category and

S = (\xymatrix{S_{0} & S_{1} \ar[l]_{π} \ar[r]^{π′} & S′_{0}})

a span in ℂ. The commutator [S] of S is a congruence on S_{1} defined as the intersection

[S] = Λ_{φ∈Φ_{S}}Eq(φ_{1})  (5.4)

where Φ_{S} is the collection of all morphisms from S to the underlying spans of pseudogroupoids.

In particular

[S] = Δ_{S_{1}}  (5.5)

if S has an internal pseudogroupoid structure.

Note also that if φ = φ′′φ′ (in ℂ) then Eq(φ′) ⩽ Eq(φ), and this gives

5.3. Proposition. If the forgetful functor \xymatrix{P (ℂ) \ar[r] & S(ℂ)} has a left adjoint written as \xymatrix{S \ar@{|->}[r] & (S^{⋆}, m)} and η is as in (4.16) then

[S] = Eq(η);  (5.6)

in particular, if ℂ is a variety of universal algebras, then

[S] = {(x, y) ∈ S_{1} × S_{1}|η(x) = η(y)}.  (5.7)

This proposition tells us that the properties of commutators, at least in the case of a variety of universal algebras should be deduced from the properties of the adjunction \xymatrix{S(C) \ar[r] & P (C) \ar[l]}. However some of them can be obtained directly from the Definition 5.2 without the existence of the left adjoint of the forgetful functor \xymatrix{P (C) \ar[r] & S(C)}.

5.4. Proposition. Let ℂ be a finitely well-complete category, and \xymatrix{S = (S_{0} & S_{1} \ar[l]_{π} \ar[r]^{π′} & S′_{0})} a span in ℂ. Then:

(a) For every morphism \xymatrix{χ : S \ar[r] & T} in S(ℂ),

[S] ⩽ χ_{1}^{#}[T], χ_{1#}[S] ⩽ [T];  (5.8)

in particular, if S_{1} = T_{1} and χ_{1} = 1_{S_{1}}, then

[S] ⩽ [T]  (5.9)

(b) [S^{op}] = [S].

(c) [S] ⩽ Eq(π) ∧ Eq(π′).

Theory and Applications of Categories, Vol. 8, No. 15 438

Proof.

(a) For every φ ∈ Φ_{T} we have

φχ ∈ Φ_{S}  (5.10)

and

Eq(φ_{1}χ_{1}) = χ_{1}^{#}Eq(φ_{1})  (5.11)

since the diagram

\xymatrix{S_{1} ×_{(φ_{1}χ_{1},φ_{1}χ_{1})} S_{1} \ar[d] \ar[r] & T_{1} ×_{(φ_{1},φ_{1})} T_{1} \ar[d] \\ S_{1} × S_{1} \ar[r]_{χ_{1}×χ_{1}} & T_{1} × T_{1}}  (5.12)

is obviously a pullback. From (5.11) we obtain [S] ⩽ χ_{1}^{#}Eq(φ_{1}) and, since χ_{1}^{#} preserves intersections, this gives the first inequality of (5.8). The second one is then obvious since χ_{1#} is the left adjoint of χ_{1}^{#}.

(b) Follows from the facts that (since the diagram (4.11) commutes) the correspondence \xymatrix{(φ : S \ar[r] & P) \ar@{|->}[r] & (φ^{op} : S^{op} \ar[r] & P^{op})} determines a bijection \xymatrix{Φ_{S} \ar[r] & Φ_{S^{op}}}, and obviously Eq(φ_{1}) = Eq(φ_{1}^{op}) for every φ ∈ Φ_{S}.

(c) Consider the commutative diagram

[figure: S_{0}; 1_{S_{0}}; S_{1}; π; ⟨π,π′⟩; π′; S′_{0}; 1_{S′_{0}}; ;; S_{0}; S_{0} × S′_{0}; pr_{1}; pr_{2}; S′_{0}]  (5.13)

its bottom line being a product diagram is a relation and therefore an internal pseudogroupoid in ℂ.

Therefore it determines an element φ in Φ_{S} (with φ_{1} = ⟨π, π′⟩). Hence [S] ⩽ Eq(φ_{1}) = Eq⟨π, π′⟩ = Eq(π) ∧ Eq(π′).

If ℂ is a variety of universal algebras with dim(ℂ) = 0 then the free pseudogroupoid F (S) = (S^{⋆}, m) on a span S in ℂ can be described as S^{⋆} =

\xymatrix{S_{0} & S_{1}/[S] \ar[l]_{π^{⋆}} \ar[r]^{^{π′^{⋆}}} & S′_{0}}  (5.14)

with m defined by q (i.e. by (4.19)), where q is as in Theorem 4.4. From this and Lemma 3.8 we obtain the following

5.5. Theorem. Let S be a span in a variety ℂ with dim(ℂ) = 0, and q any four variable term in ℂ satisfying the equivalent conditions of Theorem 4.4. Then [S] is the smallest congruence on S_{1} such that :

(a) the composition

\xymatrix{S_{4} \ar[r]^{q|_{S_{4}}} & S_{1} \ar[r] & S_{1}/[S]}  (5.15)

of the restriction of q on S_{4} ⊂ S_{1} × S_{1} × S_{1} × S_{1} and the canonical homomorphism \xymatrix{S_{1} \ar[r] & S_{1}/[S]} is a homomorphism;

(b) (q(f, g, k, h), q(f, g, k′, h)) ∈ [S] for every diagram of the form (4.24) in S.

5.6. Corollary. Let S be a span in a Mal’tsev variety ℂ and p any Mal’tsev term in ℂ. Then [S] is the smallest congruence on S_{1} such that the composition

\xymatrix{S_{3} \ar[r]^{p|_{S_{3}}} & S_{1} \ar[r] & S_{1}/[S]}  (5.16)

of the restriction of p on S_{3} ⊂ S_{1} × S_{1} × S_{1} (see (3.9)) and the canonical homomorphism \xymatrix{S_{1} \ar[r] & S_{1}/[S]} is a homomorphism.

Consider two principal examples:

5.7. Example. Let ℂ be any variety of groups with p defined as in 4.5(a) and S a span in ℂ. Let K and K′ be the kernels

\begin{matrix}K = \{x ∈ S_{1}|π(x) = 1\}, \\ K′ = \{x ∈ S_{1}|π′(x) = 1\}. \end{matrix}  (5.17)

Since K and K′ are normal subgroups in S_{1}, so is their (ordinary) commutator [K, K′]. We claim that

[S] = [K, K′],  (5.18)

i.e. [S] is the congruence on S_{1} corresponding to the normal subgroup [K, K′]. In order to prove this we have to show that [K, K′] is the smallest normal subgroup H in S_{1} such that

f_{1}f_{2}(g_{1}g_{2})^{−1}h_{1}h_{2}H = f_{1}g_{1}^{−1}h_{1}f_{2}g_{2}^{−1}h_{2}H  (5.19)

for every (f_{1}, g_{1}, h_{1}), (f_{2}, g_{2}, h_{2}) ∈ S_{3}.

Note that (5.19) is equivalent to

f_{2}g_{2}^{−1}g_{1}^{−1}h_{1}H = g_{1}^{−1}h_{1}f_{2}g_{2}^{−1}H  (5.20)

and therefore to

[f_{2}g_{2}^{−1}, g_{1}^{−1}h_{1}] ∈ H.  (5.21)

Since π(f_{2}) = π(g_{2}) and π′(g_{1}) = π′(h_{1}), we know that f_{2}g_{2}^{−1} is in K and g_{1}^{−1}h_{1} is in K′. Therefore (5.21) holds for H = [K, K′].

Conversely, suppose that H satisfies (5.21). If k is an element in K and k′ in K′, then we can take (f_{1}, g_{1}, h_{1}) = (1, 1, k′), (f_{2}, g_{2}, h_{2}) = (k, 1, 1) — and (5.21) gives [k, k′] ∈ H. That is [K, K′] is contained in H.

5.8. Example. Let ℂ be a variety of rings (not necessarily associative, with or without identity) with p defined by p(x, y, z) = x − y + z and S a span in ℂ. Let K, K′ be the kernels

\begin{aligned}K & = \{x ∈ S_{1}|π(x) = 0\}, \\ K′ & = \{x ∈ S_{1}|π′(x) = 0\}. \end{aligned}  (5.22)

We claim that

[S] = KK′ + K′K,  (5.23)

i.e. [S] is the congruence on S_{1} corresponding to the ideal KK′ + K′K. In order to prove that we have to show that KK′ + K′K is the smallest ideal I is S_{1} such that

f_{1} + f_{2} − (g_{1} + g_{2}) + (h_{1} + h_{2}) + I = f_{1} − g_{1} + h_{1} + f_{2} − g_{2} + h_{2} + I  (5.24)

and

f_{1}f_{2} − g_{1}g_{2} + h_{1}h_{2} + I = (f_{1} − g_{1} + h_{1})(f_{2} − g_{2} + h_{2}) + I  (5.25)

for every (f_{1}, g_{1}, h_{1}), (f_{2}, g_{2}, h_{2}) ∈ S_{3}. However since the addition is commutative, (5.24) holds trivially, and we have to consider only (5.25).

Note that (5.25) is equivalent to

−g_{1}g_{2} + I = −f_{1}g_{2} + f_{1}h_{2} − g_{1}f_{2} + g_{1}g_{2} − g_{1}h_{2} + h_{1}f_{2} − h_{1}g_{2} + I,  (5.26)

which itself is equivalent to

(f_{1} − g_{1})(g_{2} − h_{2}) + (g_{1} − h_{1})(f_{2} − g_{2}) ∈ I.  (5.27)

Since π(f_{i}) = π(g_{i}) and π′(g_{i}) = π′(h_{i}) we know that f_{i} − g_{i} is in K and g_{i} − h_{i} in K′(i = 1, 2). Therefore (5.27) holds for I = KK′ + K′K.

Conversely, suppose that I satisfies (5.27). If k is an element in K and k′ in K′, then we can take (f_{1}, g_{1}, h_{1}) = (k, 0, 0), (f_{2}, g_{2}, h_{2}) = (k′, k′, 0) — and (5.27) gives kk′ ∈ I; or, we can take (f_{1}, g_{1}, h_{1}) = (k′, k′, 0), (f_{2}, g_{2}, h_{2}) = (k, 0, 0) — and (5.27) gives k′k ∈ I. That is KK′ + K′K is contained in I.

These two examples together with Example 3.7 suggest to define the commutator [α, β] of two congruences α and β on an algebra A in a variety ℂ as

[α, β] = [S],  (5.28)

where \xymatrix{S = (A/α & A \ar[l] \ar[r] & A/β)} is the span (1.9); then we get the following:

5.9. Theorem. Let ℂ be a congruence modular variety of universal algebras. Then:

(a) If q is a Kiss difference term in ℂ, then every internal pseudogroupoid in ℂ is q-algebraic; in particular dim(ℂ) = 0.

(b) The commutator [α, β], of congruences α and β on any algebra A in ℂ, defined by (5.28) coincides with the modular (i.e. “ordinary”) commutator [α, β] defined by (3.19).

Proof.

(a) Let (S, m) be an arbitrary internal pseudogroupoid in ℂ and α = Eq(π), β = Eq(π′) the corresponding congruences on S_{1}; let γ be their modular commutator. Since the set C ⊂ S_{4} of commutative S-diamonds forms an (internal) sub-double equivalence relation of Eq(S), it contains Δ_{α,β}. Therefore by (3.19) we conclude

\xymatrix{(x, y) ∈ γ \ar@{=>}[r] & (\begin{matrix}x & x \\ x & y \end{matrix}) ∈ C \ar@{=>}[r] & x = m(x, x, x, y) \ar@{=>}[r] & x = y,}

i.e. γ = Δ_{S_{1}}. Together with (3.22) this tells us that

q(f, g, k, h) = q(f, g, k′, h)  (5.29)

for every diagram of the form (4.24). Since this is true for all pseudogroupoids, and in particular for the free ones, the pair (ℂ, q) satisfies the condition 4.4(a_{3}) — which proves (a).

(b) Given an algebra A in ℂ and congruences α and β on A consider the span

\xymatrix{A/α & A/[α, β] \ar[l] \ar[r] & A/β} ,  (5.30)

where [α, β] is the modular commutator of α and β. Denoting the canonical homomorphism \xymatrix{A \ar[r] & A/[α, β]} by φ, we can rewrite (5.30) as

\xymatrix{(A/[α, β])/φ_{#}α & A/[α, β] \ar[l] \ar[r] & (A/[α, β])/φ_{#}β}  (5.31)

and for the modular commutator [φ_{#}α, φ_{#}β] we have

[φ_{#}α, φ_{#}β] = Δ_{A/[α,β]},  (5.32)

as follows from [G, Corollary 6.17]. Hence, from the remark at the end of Example 3.7 we conclude that the span (5.30) has a (unique) internal pseudogroupoid structure.

By the universal property of the free internal pseudogroupoid we then conclude that the modular commutator [α, β] contains the one defined by (5.28).

In order to prove the converse we have to prove that

\xymatrix{(a, b) ∈ [α, β] \ar@{=>}[r] & η_{S}(a) = η_{S}(b)},  (5.33)

where \xymatrix{S = (A/α & A \ar[l] \ar[r] & A/β)} as above.

Let C_{α,β} be the set of potentially commutative S-diamonds, i.e. those S-diamonds

(\begin{matrix}f & k \\ g & h \end{matrix}) for which

(\begin{matrix}η_{S}(f) & η_{S}(k) \\ η_{S}(g) & η_{S}(h) \end{matrix})  (5.34)

is a commutative diamond in the free internal pseudogroupoid F (S). Then clearly C_{α,β} determines an (internal) sub-double equivalence relation of Eq(S) and so Δ_{α,β} ⊂

C_{α,β}. Hence if (a, b) is in [α, β], then (\begin{matrix}a & a \\ a & b \end{matrix}) is in C_{α,β} and so

η_{S}(a) = q(η_{S}(a), η_{S}(a), η_{S}(a), η_{S}(b)) = η_{S}(b).

5.10. Remark.

(a) If we assume the existence of a left adjoint to the forgetful functor \xymatrix{P (ℂ) \ar[r] & S(ℂ)} , then (5.6) can be used as the definition of the commutator; in this situation ℂ does not need to be finitely well-complete, but we still need the finite limits. If in addition ℂ has coequalizers of effective equivalence relations, then every commutator [S] can be written as [α, β] for appropriate effective equivalence relations α and β. For,

given a span \xymatrix{S = (S_{0} & S_{1} \ar[l]_{π} \ar[r]^{π′} & S′_{0})} in ℂ, we construct the new span S̃ =

\xymatrix{S_{1}/Eq(π) & S_{1} \ar[l] \ar[r] & S_{1}/Eq(π′),}  (5.35)

and it is easy to see that the free internal pseudogroupoid F (S̃) satisfies the universal property as need for F (S); therefore

[S] = [Eq(π), Eq(π′)].  (5.36)

More generally, for every morphism χ = (χ_{0}, χ_{1}, χ′_{0}): \xymatrix{S \ar[r] & T} in S(ℂ) in which S_{1} = T_{1}, χ_{1} = 1_{S_{1}}, and χ_{0} and χ′_{0} are monomorphisms (which gives S_{4} = T_{4}), we have

[S] = [T]  (5.37)

(b) The “new” approach to commutators described in (a) easily gives all properties listed in Proposition 5.4.

If we replace [S] by [α, β], they should be written as (we omit here (5.8)):

\xymatrix{(α_{1} ⩽ α_{2}, β_{1} ⩽ β_{2}) \ar@{=>}[r] & [α_{1}, β_{1}] ⩽ [α_{2}, β_{2}],}  (5.38)

[α, β] = [β, α],  (5.39)

[α, β] ⩽ α ∧ β.  (5.40)

(c) In the proof of 5.9(b), we used the important property (5.32) of the modular commutator. Does this property hold for our commutator also in non-modular cases? We know only two simple cases: the trivial case where all commutators are trivial (for example for ℂ = Sets), and the case of a variety ℂ with dim(ℂ) = 0, where (5.32) follows from (5.5) and is equivalent to the isomorphism

F F ≈ F  (5.41)

for the free functor F : \xymatrix{S(ℂ) \ar[r] & P (ℂ).}

(d) Let α, β be congruences on an algebra A in an arbitrary variety ℂ and C_{α,β} the algebra of potentially commutative diamonds as in the proof of 5.9(b). From (5.7) we conclude

[α, β] = \{(a, b) ∈ A × A| (\begin{matrix}a & a \\ a & b \end{matrix}) ∈ C_{α,β}\} ,  (5.42)

which is similar to (3.19) (in which (\begin{matrix}a & a \\ a & b \end{matrix}) was written as ((a, a), (a, b)) since Δ_{α,β}

was considered as a congruence on α. Another known formula for the modular commutator mentioned in [Ki] is

[α, β] = {(a, b) ∈ A × A|∃c ∈ A, ((a, c), (b, c)) ∈ Δ_{α,β}},  (5.43)

and again, we can replace it in the general case by

[α, β] = \{(a, b) ∈ A × A|∃c ∈ A, (\begin{matrix}a & b \\ c & c \end{matrix}) ∈ C_{α,β}\} .  (5.44)

One might notice that (5.43) and (5.44) would better agree with (3.19) and (5.42)

respectively if we replace (\begin{matrix}a & b \\ c & c \end{matrix}) by (\begin{matrix}c & a \\ c & b \end{matrix}) (and ((a, c), (b, c)) by ((c, c), (a, b)));

these modified formulas would also be true of course. All these suggest that in the modular case

Δ_{α,β} = C_{α,β}  (5.45)

— not just Δ_{α,β} ⊂ C_{α,β} as we observed in the proof of 5.9(b). In fact this is true: if

(\begin{matrix}a & c \\ b & d \end{matrix}) is in C_{α,β}, then η(q(a, b, c, d)) = η(c), i.e. (q(a, b, c, d), c) ∈ [α, β] and then

(\begin{matrix}a & c \\ b & d \end{matrix}) = ((a, b), (c, d)) is in Δ_{α,β} by [Ki, Theorem 3.8(ii)]. That is, (5.45) easily

follows from the results of [Ki] and Theorem 5.9.

(e) It is of course well known that the modular commutator in the cases of groups and rings becomes the ordinary one as described in the examples 5.7 and 5.8 respectively. However our purpose in those examples was to show that such description can be easily deduced directly from our definition of the commutator.

6. Kiss, Gumm, Lipparini, and abelianizable varieties

In this section ℂ denotes an arbitrary variety of universal algebras, A an algebra in ℂ, α and β congruences on A, and [α, β] their commutator defined by (5.28) (see also (5.7)); ∇_{A} = A × A will denote the largest congruence on A.

6.1. Definition. A variety ℂ is said to be a

(a) Kiss variety if it has a four variable term q with

\begin{matrix}q(a, b, a, b) = a, q(a, a, b, b) = b, \\ (q(a, b, c, d), q(a, b, c′, d)) ∈ [α, β] \end{matrix}  (6.1)

for every a, b, c, c′, d ∈ A ∈ ℂ and congruences α, β on A with

\xymatrix{ & · \\ · \ar[ur]^{a} \ar[dr]_{b} &  & · \ar[ul]^{c} \ar[ul]_{c′} \ar[dl]^{d} \\  & ·}  (6.2)

in \xymatrix{A/α & A \ar[l] \ar[r] & A/β} (i.e. with aαb, cαd, c′αd, aβc, aβc′, bβd);

(b) Gumm variety if it has a three variable term p with

p(a, b, b) = a, (p(a, a, b), b) ∈ [α, α]  (6.3)

for every a, b ∈ A ∈ ℂ and congruence α on A with aαb;

(c) Lipparini variety if it has a three variable term p with

(p(a, b, b), a), (p(a, a, b), b) ∈ [α, α]  (6.4)

for every a, b, A, α as in (b);

(d) abelianizable variety if it has a three variable term p with

(p(a, b, b), a), (p(a, a, b), b) ∈ [∇_{A}, ∇_{A}]  (6.5)

for every a, b, ∈ A ∈ ℂ.

Of course ℂ is a Kiss variety if and only if it satisfies the equivalent conditions (b_{1})–(b_{5}) of Theorem 4.4, i.e. if and only if dim(ℂ) = 0.

In particular we can say that ℂ is a Kiss variety if and only if every internal pseudogroupoid in ℂ is q-algebraic — and we can take the same q as in 6.1(a). It turns out that each of the types of varieties introduced in Definition 6.1 can be described as a variety in which certain internal pseudogroupoids are algebraic.

6.2. Theorem. A variety in ℂ is a

(a) Gumm variety if and only if it has a four variable term q such that every internal

pseudogroupoid (S, m) = \xymatrix{(S_{0} & S_{1} \ar[l]_{π} \ar[r]^{π′} & S_{0}, m)}in ℂ with

Eq(π) ⩽ Eq(π′)  (6.6)

is q-algebraic;

(b) Lipparini variety if and only if it has a four variable term q such that every internal pseudogroupoid (S, m) in ℂ with

Eq(π) = Eq(π′)  (6.7)

is q-algebraic;

(c) abelianizable variety if and only if it has a four variable term q such that every internal pseudogroupoid (S, m) in ℂ with

Eq(π) = ∇_{S_{1}} = Eq(π′)  (6.8)

is q-algebraic.

Proof. We have to reconsider the proof of (a_{3})⇒(a_{1}) of Theorem 4.4. In fact the arguments used there prove the following (stronger) assertion:

(⋆) Let q be a four variable term in ℂ and (S, m) an internal pseudogroupoid in ℂ such that q(f, g, f, g) = f, q(f, f, g, g) = g, (q(f, g, k, h), q(f, g, k′, h)) ∈ [S] for every diagram of the form (4.24) in S. Then (S, m) is q-algebraic.

We will use this in the proofs of (a), (b) and (c).

(a) Let ℂ be a Gumm variety, and (S, m) an internal pseudogroupoid in ℂ satisfying (6.6). We define q by the usual formula (4.31) (although now p is just a three variable term satisfying (6.3)), and we have

\begin{aligned}q(f, g, f, g) & = p(f, g, g) = f, \\ q(f, g, k, h) & = p(f, g, h) = q(f, g, k′, h), \end{aligned}

where f, g, k, k′, h are as in (⋆). So in order to prove that (S, m) is q-algebraic, we need to prove only q(f, f, g, g) = g, i.e. p(f, f, g) = g.

Since (S, m) is an internal pseudogroupoid, we have [S] = Δ_{S_{1}} (see (5.5)), after that Eq(π) ⩽ Eq(π′) gives

[Eq(π), Eq(π)] ⩽ [Eq(π), Eq(π′)] = [S] = Δ_{S_{1}},

and since (p(f, f, g), g) ∈ [Eq(π), Eq(π)] by (6.3), we obtain p(f, f, g) = g.

Conversely, suppose that ℂ has a four variable term q such that every internal pseudogroupoid (S, m) in ℂ satisfying (6.6) is q-algebraic. We define p by

p(x, y, z) = q(x, y, x, z)  (6.9)

and we have to prove (6.3).

Since the span

[figure: A; A; 1]  (6.10)

is a relation, it has an internal pseudogroupoid structure; moreover, that pseudo-

groupoid satisfies (6.6) and so is q-algebraic. Since (\begin{matrix}a & b \\ a & b \end{matrix}) is a diamond in the span

(6.10), we obtain

p(a, b, b) = q(a, b, a, b) = m(a, b, a, b) = a.

The span S =

\xymatrix{A/α & A \ar[l] \ar[r] & A/α}  (6.11)

might not have an internal pseudogroupoid structure, but we can consider the free internal pseudogroupoid F (S) which still satisfies (6.6) since F (S)^{op} = F (S^{op}) =

F (S), and since (\begin{matrix}a & a \\ a & b \end{matrix}) is a diamond in (6.11) we have

η_{S}(p(a, a, b)) = p(η_{S}(a), η_{S}(a), η_{S}(b)) =
= q(η_{S}(a), η_{S}(a), η_{S}(a), η_{S}(b)) =
= m(η_{S}(a), η_{S}(a), η_{S}(a), η_{S}(b)) =
= m(η_{S}(a), η_{S}(a), η_{S}(b), η_{S}(b)) = η_{S}(b),

i.e. (p(a, a, b), b) ∈ [α, α].

(b) Let ℂ be a Lipparini variety and (S, m) an internal pseudogroupoid in ℂ satisfying (6.7). We define q by (4.31) again and we have q(f, g, k, h) = q(f, g, k′, h) as above. Since again [Eq(π), Eq(π)] = Δ_{S_{1}}, (6.4) gives

\begin{aligned}q(f, g, f, g) & = p(f, g, g) = f, \\ q(f, f, g, g) & = p(f, f, g) = g, \end{aligned}

and so (S, m) is q-algebraic by (⋆).

Conversely, suppose ℂ has a four variable term q such that every internal pseudogroupoid (S, m) in ℂ satisfying (6.7) is q-algebraic. We define p by (6.9) again and using the span (6.11) we obtain

η_{S}(p(a, b, b)) = p(η_{S}(a), η_{S}(b), η_{S}(b)) =
= q(η_{S}(a), η_{S}(b), η_{S}(a), η_{S}(b)) =
= m(η_{S}(a), η_{S}(b), η_{S}(a), η_{S}(b)) = η_{S}(a),

η_{S}(p(a, a, b)) = η_{S}(b) (as in the proof of (c) above), i.e. we obtain (6.4).

The proof (c) is again similar.

6.3. Remark.

(a) As we see from the proof, we could add “with q defined by q(x, y, t, z) = p(x, y, z), where p is as in the definition” to any of the conditions (a), (b), (c) of Theorem 6.2; in particular we could take q to be independent of the third variable. However if we require q to be independent of the third variable in the definition of Kiss variety, then we obtain the definition of Mal’tsev variety.

(b) Let M_{∼}, CM_{∼}, K_{∼}, G_{∼}, L_{∼}, A_{∼}, and V_{∼} be the collections of Mal’tsev, congruence modular, Kiss, Gumm, Lipparini, abelianizable, and all varieties respectively. Then clearly

M_{∼} ⊂ CM_{∼} ⊂ K_{∼} ⊂ G_{∼} ⊂ L_{∼} ⊂ A_{∼} ⊂ V_{∼}  (6.12)

and each inclusion seems to be strict. In particular: M_{∼} ≠ CM_{∼} is well known; the variety considered in Example 4.6 (and also, say, the variety of commutative idempotent semigroups) is in K_{∼}, but not in CM_{∼}; we do not know how to prove K_{∼} ≠ G_{∼}, and G_{∼} ≠ L_{∼}, although we think K_{∼} ≠ G_{∼} should be related to Problem 3.11

in [Ki]^{1}, and G_{∼} ≠ L_{∼} should follow from the results of [L1]; Example 7.5 below shows that L_{∼} ≠ A_{∼}; the category of sets is in V_{∼} but not in A_{∼}, and so A_{∼} ≠ V_{∼}.

In addition, let N_{∼} be the collection of varieties in which any algebra A is neutral, that is

[α, β] = α ∧ β  (6.13)

for every two congruences α and β on A. Consider the intersections of N_{∼} with the “classes” involved in (6.12). The classes N_{∼} ∩ M_{∼} and N_{∼} ∩ CM_{∼} are well known and important: a variety is said to be arithmetical if it belongs to N_{∼} ∩M_{∼}; a variety ℂ is in N_{∼} ∩CM_{∼} if and only if it is congruence distributive, i.e. the lattice ER(A) = Cong(a) of congruences on A is distributive for every algebra A in ℂ. The other intersections are just N_{∼} since N_{∼} ⊂ K_{∼}(with q defined by (4.38)). Note that N_{∼} ∩ CM_{∼} ≠ N_{∼} by 4.6.

(c) The conditions on p used in 6.1(b)–(d) say that it is an F -G-difference term in the sense of [L1] for certain F and G, and so various results of [L1] can be applied to what we call Gumm, Lipparini, and abelianizable varieties.

Let (S, m) be an internal pseudogroupoid (in a variety ℂ) satisfying (6.6). Then

for every (f, g, h) ∈ S_{3}, (\begin{matrix}f & f \\ g & h \end{matrix}) is an S-diamond, and therefore (S, m) is an internal

pregroupoid. Together with Remark 6.3(a) this suggests to introduce

6.4. Definition. Let p be a three variable term in a variety ℂ. An internal pregroupoid (S, l) in ℂ is said to be p-algebraic if (4.33) holds (i.e. l(f, g, h) = p(f, g, h)) for every (f, g, h) ∈ S_{3}.

— and rewrite Theorem 6.2 as

6.5. Theorem. A variety ℂ is a

(a) Gumm variety if and only if it has a three variable term p such that every internal pregroupoid in ℂ satisfying (6.6) is p-algebraic;

(b) Lipparini variety if and only if it has a three variable term p such that every internal pregroupoid in ℂ satisfying (6.7) is p-algebraic;

K_{∼} = G_{∼}.

1Lipparini gave a surprising solution of the Kiss’s problem, which translated in our language implies

(c) abelianizable variety if and only if it has a three variable term p such that every internal pregroupoid in ℂ satisfying (6.8) is p-algebraic.

Instead of Theorem 5.5 we have

6.6. Theorem. Let p be a three variable term in a variety ℂ and S a span in ℂ such that one of the following conditions hold:

(i) ℂ is a Gumm variety and S satisfies (6.6);

(ii) ℂ is a Lipparini variety and S satisfies (6.7);

(iii) ℂ is an abelianizable variety and S satisfies (6.8).

Then [S] is the smallest congruence on S_{1} such that:

(a) (p(f, g, g), f) ∈ [S] if π(f) = π(g) and (p(g, h, h), h) ∈ [S] if π′(g) = π′(h) (of course (p(f, g, g), f) ∈ [S] can be omitted in the case (i));

(b) the composition

\xymatrix{S_{3} & S_{1} \ar[l]_{p|_{S_{3}}} \ar[r] & S_{1}/[S]}  (6.14)

of the restriction of p on S_{3} ⊂ S_{1} × S_{1} × S_{1} and the canonical homomorphism \xymatrix{S_{1} \ar[r] & S_{1}/[S]} is a homomorphism.

6.7. Remark. If p was a Mal’tsev term, then 6.6(a) can be omitted, which agree with Corollary 5.6. If ℂ is, say, a variety of semilattices (see Example 4.6) and (S, m) satisfies (6.8) then we can take any three variable term as p, for example p(x, y, z) = x ∧ y ∧ z; in this case 6.6(a) is relevant, but 6.6(b) can be omitted!

7. Abelian algebras and abelianization

Let (S, m) be an internal pseudogroupoid in a category ℂ with finite limits, in which S has the form

\xymatrix{1 & A \ar[l] \ar[r] & 1},  (7.1)

where 1 is a terminal object and A an arbitrary object in ℂ. Of course such a pseudogroupoid is a pregroupoid and in fact simply an internal herd, i.e. an object A together with a morphism

\xymatrix{l : A × A × A \ar[r] & A}  (7.2)

satisfying the identities

\begin{matrix}l(x, y, y) = x,  l(x, x, y) = y, \\ l(l(x_{1}, x_{2}, x_{3}), x_{4}, x_{5}) = l(x_{1}, x_{2}, l(x_{3}, x_{4}, x_{5})) \end{matrix}  (7.3)

Theory and Applications of Categories, Vol. 8, No. 15 450

(written “in terms of elements”) as in 3.4. Here m and l are related by (3.11) and we will write

(S, m) = (A, l).  (7.4)

Let us recall from [FMK, Definition 5.4]: 7.1. Definition. An algebra A in a variety ℂ is said to be affine, if ℂ has a three variable term p and A has an abelian group structure such that

(a) p(a, b, c) = a − b + c for all a, b, c ∈ A

(b) t(a_{1} − b_{1} + c_{1}, …, a_{n} − b_{n} + c_{n}) = t(a_{1}, …, a_{n}) − t(b_{1}, …, b_{n}) + t(c_{1}, …, c_{n}) for each n variable (n = 0, 1, …) term t in ℂ and a_{1}, …, a_{n}, b_{1}, …, b_{n}, c_{1}, …, c_{n} ∈ A.

From this definition we easily conclude 7.2. Proposition. An algebra A in a variety ℂ is affine if and only if ℂ has a three variable term p such that the formula

l(a, b, c) = p(a, b, c)  (7.5)

(just the special case of (4.33)!) defines an internal herd structure on A.

Following the usual terminology we introduce 7.3. Definition. An algebra A in a variety ℂ is said to be abelian if [∇_{A}, ∇_{A}] = Δ_{A}.

The following theorem generalizes the “fundamental theorem on abelian algebras”: 7.4. Theorem.

(a) Every affine algebra in any variety is abelian;

(b) every abelian algebra in an abelianizable variety is affine with p as in 6.1(d). Proof.

(a) follows from Proposition 7.2 and the fact that [S] = Δ_{S_{1}} for every internal pseudogroupoid (S, m).

(b) follows from Theorem 6.5(c) and Proposition 7.2 (we fix any element e in A and define a+b = p(a, e, b); the commutativity of + follows from 7.1(b) applied to t = p).

For a given algebra A in a variety ℂ and congruences α and β on A we consider again the span S = (\xymatrix{A/α & A \ar[l] \ar[r] & A/β})(see (1.9)) and define

M_{α,β} = the subalgebra of S_{4} generated by all

(\begin{matrix}a & a \\ a′ & a′ \end{matrix}) and (\begin{matrix}b & b′ \\ b & b′ \end{matrix}) with aαa′and bβb′ (7.6)

That is, our M_{α,β} is the same M(α, β) in [FMK, Definition 3.2(1) and Proposition 3.3(1)] and [L1, 1.1], and clearly

M_{α,β} ⊂ Δ_{α,β} ⊂ C_{α,β}  (7.7)

(see also Remark 5.10(d), where the relationship between Δ_{α,β} and C_{α,β} is discussed). According [FMK] and [L1] we say that α centralizes β modulo a congruence γ on A if

(\begin{matrix}a & c \\ b & d \end{matrix}) ∈ M_{α,β} and aγc implies bγd  (7.8)

(we do not need to consider here the well known relationship between this notion and the one introduced in [S] see Example 3.14); we also write C(α, β; γ) in this case. The smallest congruence γ with this property is denoted by C(α, β).

This C(α, β) is one of “several commutators” in [FMK], which coincide in the case of a congruence modular variety; in [L1] C(α, β) is written as [α, β] (which we will not do in this paper!) and used as the commutator. Since M_{α,β} ⊂ C_{α,β} and our commutator [α, β] obviously satisfies (7.8), we have

C(α, β) ⩽ [α, β].  (7.9)

In particular this tells us that if a variety has a weak difference term in the sense of [L1, Definition 2.1], i.e. satisfies the condition similar to 6.1(c) but with C(α, α) instead of [α, α], then it is a Lipparini variety and so Theorem 7.4 above extends the corresponding result (namely 5.9(i)⇔(ii) of [L1]).

Consider

7.5. Example. Let ℂ be a variety determined by a binary operator written \xymatrix{(x, y) \ar@{|->}[r] & xy} and a 0-ary operator 1 with a set of identities including

1x = x = x1  (7.10)

If (G, l) is an internal herd in ℂ, then

a · b = l(a, 1, b)  (7.11)

defines a group structure on G with the identity 1 and

(ab) · (cd) = (a · c)(b · d)  (7.12)

(since l must be a homomorphism) for all a, b, c, d in G. It is well known that this gives

a · b = ab = b · a  (7.13)

for all a, b in G, and so (G, ·) = (G with \xymatrix{(x, y) \ar@{|->}[r] & xy}) is an abelian group.

Such a ℂ might not be abelianizable. For example the variety of all monoids is not, since there the free internal pseudogroupoid on a span (7.1) is just

\xymatrix{1 & K(A) \ar[l] \ar[r] & 1}  (7.14)

where K(A) is the Grothendieck group of A.

So let us require a new identity, say

u = v,  (7.15)

where u and v are one variable terms. In the case of monoids (7.15) would be equivalent to

x^{n} = x^{m}  (7.16)

for some n and m, and the same argument with the Grothendieck group tells us that ℂ is abelianizable if and only if n ≠ m. Furthermore, all abelian algebras are trivial if and only if |n − m| = 1.

This gives many varieties which are abelianizable but not Lipparini varieties (in particular they do not have a weak difference term in the sense of [L1]). For, we fix any identity (7.15) with, say n > m > 0 in the corresponding identity (7.16), and consider an algebra A in which

ab = \begin{cases}b & if a = 1, \\ a & if a ≠ 1; \end{cases}  (7.17)

clearly such A satisfies the identity above. Let α be the congruence on A defined by

α = {(1, 1)} ∪ ((A\{1}) × (A\{1})).  (7.18)

We claim that the span \xymatrix{S = A/α & A \ar[l] \ar[r] & A/α} has an internal pseudogroupoid structure: just note that S_{4} satisfies (7.17) and any pseudogroupoid structure (in Set) \xymatrix{m : S_{4} \ar[r] & S_{1} = A} with m(1) = 1 is a homomorphism. Therefore [α, α] = Δ_{A} and it is easy to see that there is no p satisfying (6.4) for all such A.

Consider the adjunction

\xymatrix{(Groups) \ar[r]^{abelianization} & (Abelian groups) \ar[l]^{inclusion}}  (7.19)

The abelianization functor can be described as

\xymatrix{G \ar@{|->}[r] & G/[G, G] = G/[∇_{G}, ∇_{G}]},  (7.20)

and [G, G] can be described as the smallest normal subgroup in G such that the composition

\xymatrix{G × G \ar[r]^{(x,y)↦xy} & G \ar[r] & G/[G, G]}  (7.21)

is a homomorphism — or, equivalently the composition

\xymatrix{G × G × G \ar[r]^{(x,y,z)↦xy^{−1}z} & G \ar[r] & G/[G, G]}  (7.22)

is a homomorphism, which is a special case of Theorem 6.6 of course.

In general, Theorem 6.6 describes the abelianization functor involved in the adjunction

[figure: ℂ; ≀; abelianization:A↦A/[∇_{A},∇_{A}]; (\begin{matrix}Abelian \\ Algebras \\ in ℂ \end{matrix}); inclusion; ≀; =(\begin{matrix}Affine \\ algebras \\ in ℂ \end{matrix}); (Internal herds in ℂ); ≀; .; _{(\begin{matrix}Spans S in ℂ \\ with S_{0} = 1 = S′_{0} \end{matrix})}; free; _{(\begin{matrix}Internal pseudogroupoids \\ (S, m) in ℂ with S_{0} = 1 = S′_{0} \end{matrix})}; forgetful]  (7.23)

That is, “the abelianizable varieties admit a good abelianization!”

8. Two characterizations of congruence modular varieties

The purpose of this section is to prove

8.1. Theorem.

(a) A Kiss variety ℂ is congruence modular if and only if the commutator in ℂ is preserved by surjective images, i.e. for any surjective homomorphism \xymatrix{φ : A \ar[r] & A′} in ℂ and congruences α, β on A,

φ_{#}[α, β] = [φ_{#}α, φ_{#}β].  (8.1)

(b) A Gumm variety ℂ is congruence modular if and only if the commutator in ℂ is distributive, i.e.

[α_{1} ∨ α_{2}, β] = [α_{1}, β] ∨ [α_{2}, β]  (8.2)

for any algebra A in ℂ and congruences α_{1}, α_{2}, β on A.

Proof. Both (8.1) and (8.2) in the modular case are well known. So we need to prove only the “if” parts in (a) and (b).

(a) It is well known that a variety ℂ is congruence modular if and only if it satisfies the following

Shifting Lemma. For any algebra A in ℂ and congruences α, β, γ on A with α ∧ β ⩽ γ,

\xymatrix{(a, b) ∈ γ \ar@{=>}[r] & (c, d) ∈ γ}  (8.3)

for every diamond (\begin{matrix}a & c \\ b & d \end{matrix}) in the span S = (\xymatrix{A/α & A \ar[l] \ar[r] & A/β}).

In order to prove that (8.1) implies the Shifting lemma we take \xymatrix{φ : A \ar[r] & A′} to be the canonical homomorphism \xymatrix{A \ar[r] & A/γ} . We have

[φ_{#}α, φ_{#}β] = φ_{#}[α, β] ⩽ φ_{#}(α ∧ β) ⩽ φ_{#}γ = Δ_{A′}.

On the other hand if (\begin{matrix}a & c \\ b & d \end{matrix}) is a diamond in S with (a, b) ∈ γ, then since

(q(a, b, c, d), c) ∈ α ∧ β ⩽ γ

(where q is as in 6.1(a)) and φ(a) = φ(b) we obtain

η(φ(c)) = η(φ(q(a, b, c, d))) =
= q(ηφ(a), ηφ(b), ηφ(c), ηφ(d)) =
= q(ηφ(a), ηφ(a), ηφ(c), ηφ(d)) =
= m(ηφ(a), ηφ(a), ηφ(c), ηφ(d)) =
= m(ηφ(a), ηφ(a), ηφ(d), ηφ(d)) = ηφ(d),

and so (φ(c), φ(d)) ∈ [φ_{#}α, φ_{#}β]; here η and m were as in Proposition 5.3, but using for the span

\xymatrix{A′/φ_{#}α & A′ \ar[l] \ar[r] & A′/φ_{#}β}  (8.4)

Since (φ(c), φ(d)) ∈ [φ_{#}α, φ_{#}β] = Δ_{A′}, we obtain φ(c) = φ(d), i.e. (c, d) is in γ as desired.

(b) is just a special case of [L1, Theorem 3.2(i)].

8.2. Remark.

(a) If φ is the canonical homomorphism \xymatrix{A \ar[r] & A/[α, β]} then (8.1) holds in any Kiss variety ℂ (see Remark 5.10(c)).

(b) As we see from the proof it suffices to require (8.1) in the case [φ_{#}α, φ_{#}β] = Δ_{A′}. On the other hand (8.2) can be replaced by the stronger condition of infinite distributivity since it holds in congruence modular varieties.

References

[F-C] J. Furtado-Coelho, Homology and Generalized Baer Invariants, Journal of Algebra 40 n2, 1976, 596–600

[FMK] R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, London Math. Soc., Lecture Notes Series, vol. 125, Cambridge University Press, 1987

[G] H. P. Gumm, Geometrical methods in congruence modular algebra, Mem. AMS 45, No. 286, 1983

[HH] J. Hagemann and C. Harrmann, A concrete ideal multiplication for algebraic systems and its relationship to congruence distributivity, Arch. der Math. 32, 1979, 234–245

[JK] G. Janelidze and G. M. Kelly, Galois theory and a general notion of central extension, Journal of Pure and Applied Algebra 97, 1994, 135–161

[JP] G. Janelidze and M. C. Pedicchio, Internal Categories and groupoids in congruence modular varieties, Journal of Algebra, 193(1997), 552–570

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[Ki] E. W. Kiss, Three remarks on the modular commutator, Algebra Universalis 29, 1992, 455–476

[Ko] A. Kock, Fibre bundles in general categories, Journal of Pure and Applied Algebra 56, 1989, 233–245

[KS] K. Kearnes, A. Szendrei, The relationship Between Two Commutators, preprint 1997

[L1] P. Lipparini, Commutator theory without join-distributivity, Trans. AMS 346, No. 1, 1994, 177–202 [L2] P. Lipparini, A characterization of varieties with a difference term, Canad.

Math.Bull. 39(1996), 308–315

[L3] P. Lipparini, A Kiss 4-difference term from a ternary one, preprint

[ML] S. MacLane, Categories for the Working Mathematician, Springer-Verlag, New York-Heidelberg-Berlin, 1971

[P1] M. C. Pedicchio, A categorical approach to commutator theory, Journal of Algebra 177, 1995, 647–657

[P2] M. C. Pedicchio, Some remarks on internal pregroupoids, Communications in Algebra, 26(6), 1998, 1737–1744

[S] J. D. H. Smith, Mal’tsev Varieties, Lecture Notes in Mathematics 554, Berlin, 1976

Mathematical Institute of the Academy of Science, M. Alexidze str.1, 380093 Tbilisi, Georgia. and Dipartimento di Scienze Matematiche Università degli studi di Trieste Via Valerio, 12 34127 Trieste, Italy

Email: gjanel@rmi.acnet.ge and pedicchi@univ.trieste.it

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