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62 Chapter 6. Semirings Proposition 6.2.1. The change-of-base functors of the fibration ¶S are conser-
vative (i.e. they reflect isomorphisms).
Theorem 6.2.2. Given any semiring B, the fiber PtB(SRng) is SPtB(SRng)- unital.
Proposition 6.2.3. For any semiring B, the fiber SPtB(SRng) is closed under finite limits. Hence, the fiber SPtB(SRng) is a unital category.
As for the case of monoids, according to Section 1.3 we can define the commutativity of two morphisms (m,n) in PtB(SRng) provided that one of the domains of the pair (m, n) is a Schreier split epimorphism.
Proposition 6.2.4. Let us denote by (A, B, f, s) the Schreier split epimorphism which is the domain of m and by (C, B, g, t) the domain of n, as in the diagram below. The cooperator φ: A×B C → X of the pair (m,n) in PtB(SRng), when it exists, is necessarily (being unique) defined by φ(a, c) = mqf (a) + n(c). It is effectively a cooperator if and only if we have mqf (sg(c) · α) = n(c) · m(α) and mqf(α·sg(c))=m(α)·n(c) for α∈K[f] and c∈C.
A ×B C
φ
;;
cc
eA
eC
m // oo n AXC cc OO ;; fg
s
Proof. If the cooperator φ exists, then it is the cooperator of the two points (UA, UB, Uf, Us) and (UC, UB, Ug, Ut) of (additive) monoids. Hence, thanks to Proposition 2.4.3, we already know that the cooperator, if it exists, is given by φ(a, c) = mqf (a) + n(c). Moreover, since all the monoids involved are com- mutative, φ is always a morphism of (additive) monoids. It remains to prove that φ is a morphism of semirings if and only if
mqf (sg(c) · α) = n(c) · m(α) and mqf (α · sg(c)) = m(α) · n(c) (6.2.2) for any α ∈ K[f] and c ∈ C. We have that
φ((a1,c1)·(a2,c2))=φ(a1 ·a2,c1 ·c2)=mqf(a1 ·a2)+n(c1)·n(c2)=
= mqf (a1)·mqf (a2)+mqf (sf(a1)·qf (a2))+mqf (qf (a1)·sf(a2))+n(c1)·n(c2) = = mqf (a1)·mqf (a2)+mqf (sg(c1)·qf (a2))+mqf (qf (a1)·sg(c2))+n(c1)·n(c2), while
## {{ B
t
φ(a1, c1) · φ(a2, c2) = (mqf (a1) + n(c1)) · (mqf (a2) + n(c2)) =