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6.2. The fibration of Schreier points in semirings 61
where the lower row is a Schreier split sequence of semirings, while the top row is any split epimorphism with the same kernel X. The following conditions are equivalent:
(a) the pair (u,f′) is jointly monomorphic;
(b) the commutative square fu = vf′ is a pullback square;
(c) the upper row (with the map q′ = qu) is a Schreier split epimorphism.
Proposition 6.1.10. Consider the following commutative diagram in SRng, where the two rows are Schreier split sequences:
Y
oo q′ l
// C
oo β α
// // D
hgt oo q oo s
if t and h are;
(ii) g is a monomorphism if and only if t and h are.
Proposition 6.1.11. Consider a commutative diagram in SRng of split se- quences of the form
f
// E // // B. kf
X
(i) g is a regular epimorphism (i.e. a surjective homomorphism) if and only
oog oor
// // B // // B,
if and only if the lower split sequence is a Schreier one.
6.2 The fibration of Schreier points in semirings
Let us consider the category Pt(SRng) of split epimorphisms in SRng and its full subcategory SP t(SRng) whose objects are the Schreier split epimorphisms. As for monoids, we have that, according to Proposition 6.1.8, the restriction ¶S of the fibration ¶: Pt(SRng) → SRng to the full subcategory SPt(SRng) is still a fibration, which we shall call the fibration of Schreier points. The following results follow then immediately from the analogous ones in the category of monoids:
X
// A φ
k
ooq oos
// E
where the upper split sequence is Schreier. The morphism φ is an isomorphism
X
p
l