Page 84 - Textos de Matemática Vol. 45

76 Chapter 6. Semirings Inspired by the definition of an action accessible category given in [15],
we can say that the category SRng of semirings is Schreier accessible.
6.6 Centralizers of Schreier reflexive relations
As in the case of the category Mon of monoids, we shall show here that any Schreier reflexive relation in SRng has a centralizer.
d0 //
Let S oo s0 // M be a Schreier reflexive relation on a semiring M. De-
d1
note by k: X S the kernel of d0. We can take the index (φ,φ¯) of the in- duced Schreier split extension (S, M, d0, s0, k) towards a faithful split extension (S(X),M(X),θX,ρX,ιX) and complete the diagram by the kernel equivalence relations:
R [ φ¯ ] o o OO
dS0 dS1
// φ¯ S
/ / S ( X ) OO
// OO
R(d0) R(s0) d0 s0 d1 dM0//
θX ρX //M φ //M(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¯] is the pullback R[φ] ×M S and d1dS0 is the connector
making the reflexive relations R[φ] and S centralize each other.
Proposition 6.6.1. The equivalence relation R[φ] is the largest equivalence re- lation on M commuting with the reflexive relation S; in other words it is the centralizer of the reflexive relation S in SRng.
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 double centralizing relation (lower left hand side part of the diagram below),
//S //
d0 M