Page 77 - Textos de Matemática Vol. 45

6.4. Mal’tsev aspects of SRng related to Schreier split epimorphisms 69 q′rk: K[d0] → K[d′0] (where q′ is the map given by the Schreier condition for
the split epimorphism d′0) is such that
iq′rk = qurk = qk = 1K[d0],
where q is the map given by the Schreier condition for the split epimorphism d0, and this proves that i is surjective. Similarly we prove that j is surjective, too. Now, given x ∈ K[d0] and y ∈ K[d1], there exist x′ ∈ K[d′0] and y′ ∈ K[d′1] such that i(x′) = x and j(y′) = y. Then we have:
x·y=i(x′)·j(y′)=u(x′ ·y′))=u(0)=0 y·x=j(y′)·i(x′)=u(y′ ·x′)=u(0)=0,
and
and this proves that k and l cooperate.
6.4 Mal’tsev aspects of SRng related to Schreier split epimorphisms
As for the category Mon of monoids, we can recover in the category SRng of semirings some partial aspects of Mal’tsev categories related to the Schreier split epimorphisms. The observations about Schreier reflexive graphs, categories and groupoids are the same.
In the same way, since the fiber PtBSRng is SPtBSRng-unital, we can introduce the following:
Definition 6.4.1. Given a reflexive relation R and a Schreier reflexive relation S on the semiring B, we say that R and S centralize each other, when there is a (necessarily unique) semiring homomorphism p: R ×B S → B such that p(xRxSy) = y and p(xRySy) = x. We denote this situation by [R, S] = 0.
Proposition 6.4.2. The reflexive relation R and the Schreier reflexive relation S centralize each other in SRng if and only if, for any 1St ∈ K[d0] and any xRy, we have q(yS(y·t)) = x·t and q(yS(t·y)) = t·x. In this circumstance we have p(xRySz) = x + q(ySz). When [R, S] = 0, we have necessarily xSp(xRySz) and p(xRySz)Rz.
Proof. The definition of p and the last assertion are a consequence of Proposi- tion 4.3.3. The condition on p is a straighforward consequence of Proposition 6.2.4.
We get also: