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58 Chapter 6. Semirings and
0 = φ(0, 0) = φ((0, y)(x, 0)) = φ(0, y)φ(x, 0) = g(y)f (x) foranyx∈X andy∈Y,andhencewegetCondition6.0.1.
Given a semiring M, the identity 1M cooperates with itself if and only if x · y = 0 for all x, y ∈ M .
Definition 6.0.5. We shall call trivial a semiring with a trivial multiplication.
Although not very interesting by themselves, we shall see that the trivial semirings will allow to characterize some specific properties inside SRng.
Since the underlying additive structure of a semiring is a commutative monoid, the notions of right homogeneous and left homogeneous split epimor- phism coincide at this level. This allows us to give the following:
Definition 6.0.6. A split epimorphism (A,B,f,s) of semirings is said to be homogeneous when, for any element b ∈ B, the map μb : K[f] → f−1(b) defined by μb(k) = k + s(b), is bijective.
Definition 6.0.7. A split epimorphism (A,B,f,s) of semirings is said to be a Schreier split epimorphism if, for every a ∈ A, there exists a unique x ∈ K[f] such that a = x + sf(a).
Lemma 6.0.8. A split epimorphism (A, B, f, s) of semirings is a Schreier (resp. homogeneous) one if and only if the split epimorphism (UA,UB,Uf,Us) of monoids is. In other words the functor U : SRng → CMon preserves and re- flects the Schreier split epimorphisms.
Thanks to the lemma above, Propositions 2.1.3 and 2.1.4 have the follow- ing immediate consequences:
Corollary 6.0.9. A split epimorphism (A,B,f,s) of semirings is homogeneous if and only if it is a Schreier split epimorphism.
Corollary 6.0.10. A split epimorphism (A,B,f,s) is a Schreier split epimor- phism if and only if there exists a set-theoretical map
such that
q: A K[f]
q(a) + sf(a) = a q(x + s(b)) = x
for every a∈A, x∈K[f] and b∈B.