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44 Chapter 4. Mal’tsev aspects of Mon 4.2 First observations
First, we observed that, in the category Mon, any Schreier reflexive relation (d0,d1): R ⇒ X is only necessarily transitive (Proposition 3.1.5). Example 2.2.7 gives a Schreier reflexive relation which is not an equivalence relation. A Schreier reflexive relation R is an equivalence relation if and only if K[d0] is a group.
In a Mal’tsev category, on a reflexive graph there is at most one structure of internal category, which is necessarily an internal groupoid. In the previous chapter we showed that, on a Schreier reflexive graph, there is again at most one structure of internal category, but there are Schreier internal categories which are not internal groupoids. A Schreier internal category is a Schreier internal groupoid if and only if K[d0] is a group.
4.3 Centrality for Schreier relations
More importantly, the Mal’tsev context guarantees some nice centrality prop- erties of the equivalence relations. The equivalence relations R on an object X, coinciding with the reflexive relations on X, are just subobjects of the object (p0 , s0 ) : X × X X in the fibre P tX C:
R //(d0,d1)//X×X
cc
p0
OO
s0
s0 d0
## X
It appears that two equivalence relations R and S on X centralize each other in a Mal’tsev category C when the subobjects (d1,d0): R X × X and (d0,d1): S X × X commute in the fiber PtXC. Indeed, the cooperator R×X S→X×Xinthefibermustbeoftheformφ(xRySz)=(x,p(xRySz)), with p(xRxSy) = y and p(xRySy) = x. The map p: R×X S → X, with the two equations, is characteristic of the fact that R and S centralize each other (see [13], and also [30] and [33]); it is called the connector between the relations R and S. It is well known that, in the category Gp of groups, two equivalence relations R and S on a group G centralize each other if and only if the nor- mal subgroups 1R and 1S given by the equivalence classes of the unit element commute inside the group G.
In the category Mon, for any monoid B the fiber PtBMon is SPtBMon- unital. So we can keep the same definition, provided that one of the relations, let us choose S, is a Schreier reflexive relation:
Definition 4.3.1. Given a reflexive relation R and a Schreier reflexive relation S on the monoid B, we say that R and S centralize each other when there