Page 68 - Textos de Matemática Vol. 45

60 Chapter 6. Semirings Lemma 6.1.2. Given a Schreier split epimorphism (A, B, f, s) of semirings, the
following diagram
ooq oos
K[f] // A // B,
kf
makes s: B → A the kernel of q in the category of pointed sets. Moreover, since q is a morphism of monoids, s is its kernel in the category of commutative monoids, too.
Proposition 6.1.3. Given a direct product diagram in semirings
ooπX ⟨1,0⟩
// X × B oo
πB // ⟨0,1⟩
X
the canonical split epimorphism (X × B, B, πB , ⟨0, 1⟩) is a Schreier split epi-
morphism.
Corollary 6.1.4. The terminal split epimorphism
X oo // 1 is a Schreier split epimorphism.
Corollary 6.1.5. The identity split epimorphism
1X
B,
//
and more generally any isomorphism, is a Schreier split epimorphism.
X,
split epimorphism (A, B, f, s) is a Schreier split epimorphism.
Proposition 6.1.7. Schreier split epimorphisms are stable under products, i.e.
the product of two Schreier split epimorphisms is a Schreier split epimorphism.
Proposition 6.1.8. Schreier split epimorphisms are stable under pullbacks along any morphism.
Theorem 6.1.9. Consider a commutative diagram
Xk′ //Coof′ //D s′
uv
ooq f//
X oo
Proposition 6.1.6. If B is a ring (i.e. (B, +, 0) is an abelian group), then every
X
// A oo
s
B
k