Split epimorphisms as a productive tool in Universal Algebra

Abstract

By various examples, we show how, in Universal Algebra, the choice of a class of split epimorphisms $\Sigma$, defined by specific equations or local operations in their fibres, can be used as a productive and flexible tool determining $\Sigma$-partial properties. We focus, here, our attention on $\Sigma$-partial congruence modular and $\Sigma$-partial congruence distributive formulae.

Appendix: \(\Sigma \)-Mal’tsev and \(\Sigma \)-protomodular categories

Here we collect the main striking results concerning these two notions.

1.1 $\Sigma$-Mal’tsev categories

Let us recall from [6] that, given any fibrational class $\Sigma$ of split epimorphisms, a category ${\mathbb {E}}$ is a $\Sigma$-Mal’tsev one when any left hand side pullback:

of split epimorphisms is such that $Z\times _Y X$ is the supremum of the two “subobjects” ${\bar{s}}$ and ${\bar{t}}$ whenever (f, s) is in $\Sigma$. An equivalent condition is that, considering any commutative square of split epimorphisms as is the right hand square, the induced factorization $X'\rightarrow Z\times _Y X$ is a strong epimorphism (a surjective homomorphism when ${\mathbb {E}}$ is a variety ${\mathbb {V}}$) whenever (f, s) is in $\Sigma$. A category is a Mal’tsev one (i.e. a category in which any reflexive relation is an equivalence relation [16, 17]) if and only if this property is valid for any split epimorphism, see [2]. In the categorical Mal’tsev context, we get a conceptual notion of centralization of pair (S, T) of internal equivalence relations on an object X [10] (see [28] for instance for the varietal one), which remains valid in a $\Sigma$-Mal’tsev category whenever S is a $\Sigma$-equivalence relation. In this situation, we have $[S,T]=0$ as soon as $S\wedge T=\Delta _X$. When, moreover, ${\mathbb {E}}$ is regular, and the pair (S, T) is a pair of $\Sigma$-equivalence relations such that $[S,T]=0$, then we get $[f(S),f(T)]=0$ when f is a regular epimorphism.

For the $\Sigma$-Mal’tsev varieties the main properties are the following ones, see [6]:

• Any $\Sigma$-relation (resp. symmetric $\Sigma$-relation) is necessarily transitive (resp. a congruence).

• Given any pair (R, S) of a $\Sigma$-congruence R and a reflexive relation S on an algebra X, the two relations necessarily permute, namely we have $R\circ S=S\circ R$.

• There is a natural notion of centralization for the same kind of pairs of relations on an algebra X.

• In particular, a $\Sigma$-special homomorphism $f: X\rightarrow Y$ has an abelian kernel congruence when $[R[f],R[f]]=0$.

A direct consequence of the equivalent caracterization given with the right hand side commutative square of split epimorphisms above is the following one:

• The direct image of any congruence along a $\Sigma$-special surjective homomorphism is a congruence as well.

• The direct image of any $\Sigma$-congruence along a surjective homomorphism is a congruence as well; it is a $\Sigma$-congruence when $\Sigma$ is a Birkhof class.

• The congruence modular formula holds

  $$\begin{aligned} (R\vee S) \wedge T)=R\vee (S\wedge T); \mathrm{\; whenever \;} R\subset T \end{aligned}$$

  as soon as R is a $\Sigma$-congruence.

• When $\Sigma$ is a Birkhoff class, the modular formula holds as well as soon as S is a $\Sigma$-congruence.

• When moreover the class $\Sigma$ is 1-regular, namely such that pullbacks along regular epimorphisms reflect the split epimorphisms in $\Sigma$, there is a Baer sum for $\Sigma$-special extensions with abelian kernel congruence R[f], see [6].

All the examples of classes $\Sigma$ introduced in this article are 1-regular. When, in addition, the class $\Sigma$ is point congruous, the $\Sigma$-core is necessarily a Mal’sev variety, and more generally the full subcategory $\Sigma l_Y {\mathbb {V}}$ of any slice category ${\mathbb {V}} /Y$ whose objects are the $\Sigma$-special morphisms with codomain the algebra Y and whose morphisms are the commutative triangles above Y is a Mal’tsev category. Let us finish this section with the following:

Proposition 5.1

Let ${\mathbb {E}}$ be any $\Sigma$-Mal’tsev exact category with respect to a Birkhoff class $\Sigma$ and let $f: X \twoheadrightarrow Y$ be a regular epimorphism. The direct image f(S) of any $\Sigma$-equivalence relation S on X is a $\Sigma$-equivalence relation as well. Moreover, given any pair (S, T) of $\Sigma$-equivalence relations on X, the difference mapping induced by the inclusion $f(S\wedge T)\subset f(S)\wedge f(T)$ has necessarily an abelian kernel equivalence relation.

Proof

The direct image f(S) is an equivalence relation in any regular $\Sigma$-Malcev category as soon as S is a $\Sigma$-equivalence relation. It is a $\Sigma$-equivalence relation since $\Sigma$ is a Birkhoff class. Accordingly f(S), f(T) and $f(S\wedge T)$ are $\Sigma$ equivalence relations when f is any regular epimorphism. The proof now will mimick the one given in [4] for any exact Mal’tsev category. Consider the following diagram:

where $\rho$ is the quotient map of X by $S\wedge T$ and $\tau$ the quotient map of Y by $f(S\wedge T)$. They determine a unique factorization ${\bar{f}}$ which is a regular epimorphism. Since we have $S\wedge T\subset S$ and $S\wedge T\subset T$, we get $\rho (S)\wedge \rho (T)=\rho (S\wedge T)=\Delta _{X/S\wedge T}$, so that we have $[\rho (S),\rho (T)]=0$, the two equivalence relations being $\Sigma$-ones. Since ${\bar{f}}$ is a regular epimorphism, we get $0=[{\bar{f}}\rho (S),{\bar{f}}\rho (T)]=[\tau f(S),\tau f(T)]$, so that $[\tau f(S)\wedge \tau f(T),\tau f(S)\wedge \tau f(T)]=0$, which means that the $\Sigma$-equivalence realtion $\tau f(S)\wedge \tau f(T)$ is abelian. Now the difference mapping induced by the inclusion $f(S\wedge T)\subset f(S)\wedge f(T)$ is the regular epimorphism: $\delta : Y/f(S\wedge T) \twoheadrightarrow Y/f(S)\wedge f(T)$ such that $\delta .\tau$ is the quotient map of $f(S)\wedge f(T)$; accordingly the kernel equivalence relation $R[\delta ]$ is the direct image of $R[\delta .\tau ]=f(S)\wedge f(T)$ along $\tau$, namely $\tau (f(S)\wedge f(T))=\tau f(S)\wedge \tau f(T)$ (since both $f(S\wedge T)\subset f(S)$ and $f(S\wedge T)\subset f(T)$) which is abelian, as we just noticed. $\square$

1.2 $\Sigma$-protomodular categories

A category ${\mathbb {E}}$ is $\Sigma$-protomodular [6] when any pullback of split epimorphisms:

makes the pair $(s,{\bar{g}})$ jointly strongly epic as soon as the split epimorphism (f, s) is in $\Sigma$. Any $\Sigma$-protomodular category is a $\Sigma$-Mal’tsev one. A category is a protomodular one as soon this property is valid for any split epimorphism, see [2]; we specified examples of protomodular categories at the beginning of Section 4. In a pointed regular protomodular category, the classical homological lemmas hold: Short Five Lemma, Nine Lemma, Snake Lemma and Noether Isomorphism Theorems of split epimorphisms:.

For the $\Sigma$-protomodular varieties the main consequences are the following ones, see [6, 15]. When the variety ${\mathbb {V}}$ is pointed:

• A $\Sigma$-special morphism is a monomorphism if and only if its kernel is trivial.

• Any $\Sigma$-special surjective homomorphism is the cokernel of its kernel.

• A subalgebra is the equivalence class of at most one $\Sigma$-congruence.

Pointed or non-pointed, when moreover the class $\Sigma$ is point-congruous, the $\Sigma$-core is necessarily a protomodular variety, and more generally the full subcategory $\Sigma l_Y {\mathbb {V}}$ of any slice category ${\mathbb {V}} /Y$ is a protomodular category.

• When the variety ${\mathbb {V}}$ is pointed, the short five lemma holds for $\Sigma$-special short exact sequences.

• There are various versions of the $3\times 3$ lemma for the $\Sigma$-special short exact sequences.

• When the variety ${\mathbb {V}}$ is non-pointed, there are various versions of the denormalized $3\times 3$ lemma for the $\Sigma$-special extensions.