Split epimorphisms as a productive tool in Universal Algebra
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Algebra Universalis
Split epimorphisms as a productive tool in Universal Algebra Dominique Bourn Abstract. By various examples, we show how, in Universal Algebra, the choice of a class of split epimorphisms Σ, defined by specific equations or local operations in their fibres, can be used as a productive and flexible tool determining Σ-partial properties. We focus, here, our attention on Σ-partial congruence modular and Σ-partial congruence distributive formulae. Mathematics Subject Classification. 08B05, 08B10, 18A20, 18A32, 18C10, 18E13. Keywords. Split epimorphism, Congruence modular and congruence distributive varieties, Σ-Mal’tsev and Σ-protomodular categories.
1. Introduction The role of split epimorphisms was shown to be important in Categorical Algebra to study the notions of protomodular and Mal’tsev categories, see [2] and its bibliography, and because of their strong classification power, see [3,7]. We would like to show here how split epimorphisms could be used as a productive and flexible tool in Universal Algebra as well, but in a substantially different way. This idea emerged as we pointed out that an example of Σ-Mal’tsev category, where Σ is a class of split epimorphisms (see the precise definition in Section 5), was given by the variety Qnd of Quandles [5]. A quandle is a pair (X, ) of a set X and an idempotent binary operation such that, for any object x ∈ X, the application − x : X → X is bijective and preserves the binary operation . The notion was independently introduced in [22,25] in strong relationship with Knot Theory and Reidemeister moves. Any group G Presented by J. Ad´ amek. 0123456789().: V,-vol
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is endowed with a quandle structure for which the binary operation is defined by x y = yxy −1 . In this context, a relevant class Σ is given by the class of acupuncturing split epimorphisms, namely those split epimorphisms (f, s) : X Y such that any restriction of the application s(y) − to the fibre of (f, s) above y ∈ Y is bijective. Let us recall the following definition, where R(f ) denotes the kernel equivalence relation of a map f : X → Y (in any variety we get R[f ] = {(x, x ) ∈ X × X | f (x) = f (x )}): Definition 1.1 [14]. Given any class Σ of split epimorphisms in a category E, a graph X1 on an object X is said to be a Σ-graph when it is reflexive X1 o
d0 s0 d1
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and such that the split epimorphism (d0 , s0 ) belongs to the class Σ. Similar definitions can be given for the notions of Σ-relations, (internal) Σ-categories and Σ-groupoids. A morphism f : X → Y is called Σ-special when the equivalence relation R[f ] is a Σ-equivalence relation. An object X is said to be Σ-special when the terminal map τX : X → 1 is Σ-special; the full subcategory of Σ-special objects is called the core of Σ (or the Σ-core) in E. In the variety Qnd of quandles, we were then able to show that: (1) Any pair (R, S) of an acupuncturing congruence R and a reflexive relation S on a quandle (X, ) necessarily permute: that is to say R ◦ S = S ◦ R. (2) Any refl
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