Page 62 - Textos de Matemática Vol. 45

54 Chapter 5. Schreier and homogeneous representability Definition 5.3.2. For any monoid X, the monoid Aut(X) will be called the
homogeneous split extension classifier of X.
Using again the construction of the semidirect product, we get a converse
of Theorem 5.3.1, in analogy with Proposition 5.2.2:
Proposition 5.3.3. Given two monoids B and X, there is a natural bijection between the set of homogeneous split epimorphisms with codomain B and kernel X and the set of actions φ: B → Aut(X) of B on X.
5.4 Centralizers of Schreier reflexive relations
d0 //
Let S oo s0 // M be a Schreier reflexive relation on the monoid M. Denote by
d1
k: X S the kernel of d0. Take the index (φ,φ¯) of the induced Schreier split extension (S, M, d0, s0, k) and complete the diagram with the kernel equivalence relations:
R[φ¯] oo OO
dS0 dS1
// φ¯ S
// Hol(X) OO
// OO
R(d0) R(s0) d0 s0 d1 dM0//
θX ρX
// M φ // End(X)
R[φ] oo
Since the right hand side square is a pullback, so are the two left hand side
dM1
ones, and in particular the following one:
R[φ¯] dS1 R(d0 )
R[φ]
dM1
which shows that R[φ¯] actually is the pullback R[φ] ×M S and d1dS0 is the
connector making the reflexive relations R[φ] and S centralize each other.
Proposition 5.4.1. The equivalence relation R[φ] is the largest equivalence re- lation on M centralizing the reflexive relation S; in other words it is the cen- tralizer of the reflexive relation S.
Proof. We just observed that we have [R[φ], S] = 0. Suppose now we have an equivalence relation R on M such that [R,S] = 0. Consider the associated
//S //
d0 M