Page 89 - Textos de Matemática Vol. 45

7.2. Special Schreier short five lemma 81 Proposition 7.1.5. Given any pullback in Mon with h a surjective homomor-
phism:
X f ////Y
g X′
h // // Y ′
f′
if f is a special Schreier (resp. special homogeneous) surjection, so is f′.
Proof. First, since the surjections in Mon are stable under pullbacks, we have that g is surjective. Let us then complete the previous diagram with the kernel equivalence relations, which produce the left hand side pullbacks:
// // Y
h
Suppose f is a special Schreier (resp. homogeneous) surjection. According to Corollary 2.3.6, since g is a surjective homomorphism and (p0,s0) is a Schreier (resp. homogeneous) split epimorphism, so is (p′0, s′0) and f′ is a special Schreier (resp. homogeneous) surjection.
7.2 Special Schreier short five lemma
We will call special Schreier extension an exact sequence K[f]//k //Xf////Y
such that f is a special Schreier surjection. The exact sequence will be called special homogeneous when f is special homogeneous.
The name Schreier extension was introduced, with a different meaning, by R´edei in [31] in the context of semigroups, semirings and semimodules over a semiring, and later developed by Patchkoria in [27, 29]. This is another reason why we prefer to use the name special Schreier extension. The study of the relationships between these two different concepts will be material for a future work.
p0 R[f]oo s0 // X
p1 R ( g ) p ′0
// f g
// ′
R[f]s0 //X ′ Y.
′ oo ′
p′ f
1
//// ′