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70 Chapter 6. Semirings Proposition 6.4.3. The equivalence relation R and the Schreier equivalence rela-
tion S on the semiring M centralize each other in SRng as soon as R∩S = ∆X .
Proposition 6.4.4. Suppose both Rop and S are Schreier reflexive relations on a semiring M. Then the reflexive relations R and S centralize each other if and only if the subobjects dR0 kdR1 : K[dR1 ] X and dS1 kdS0 : K[dS0 ] X cooperate in the category SRng.
Proof. It is a straighforward consequence of Theorem 6.2.5.
Proposition 6.4.5. Let (A,B,f,s) be a Schreier split epimorphism such that R[f] is a Schreier equivalence relation. We have [R[f],R[f]] = 0 if and only if its kernel K[f] is a trivial ring.
Proof. Since R[f] is a Schreier equivalence relation, K[f] is a ring. By the reflection of commutativity, we have [R[f],R[f]] = 0 if and only if the ring K[f] is trivial.
When we have [R,S] = 0, the same diagram in the category SRng as the one described in Section 4.4 produces the associated double centralizing relation.
6.5 Schreier accessibility
We showed that in the category Mon there are classifiers for Schreier split extensions with a given kernel X. Some classification process for Schreier split extensions with a given kernel X in SRng does exist, but it is not so simple. Actually it is analogous to the one given for rings in [15] and related to the accessibility property. Again it will allow us to produce centralizers for Schreier reflexive relations.
Given two Schreier split epimorphisms of semirings (which, in particular, are split extensions, as observed in Lemma 1.2.8) (B,A,p,s) and (D,C,r,t) with the same kernel X, a morphism between them
(g,f) : (B,A,p,s) −→ (D,C,r,t) is a pair (g,f) of morphisms:
0 // X oo q // A oo p // B // 0
s
1X f g
(6.5.3)
(6.5.4)
k
// oo q′ r //
//
0 X //Coo D 0
lt