Page 25 - Textos de Matemática Vol. 45

2.3. First properties of Schreier and homogeneous split epimorphisms 17 2.3 FirstpropertiesofSchreierandhomogeneoussplit
epimorphisms
Proposition 2.3.1. Consider a (vertical) map (h, l) in P tC:
oo s
X
// // Y
f lh ′oo s′ ′
X
Suppose that the two rows are Schreier split epimorphisms, then the Schreier retractions are compatible, i.e. the following leftward left hand side diagram commutes (in the category Set of sets):
ooq oos K[f] // X // // Y
kf
K(l) l h
′oo s′ ′
f′
// // Y .
′ ooq′ K[f ]
// X
Proof. We have to show that q′l(x) = lq(x) for any x in the monoid X. It is
true since we have:
lq(x) · s′f′l(x) = lq(x) · lsf(x) = l(q(x) · sf(x)) = l(x) = q′l(x) · s′f′l(x)

2.3.1 Stability properties
We are going to investigate here what are the main stability properties of the Schreier split epimorphisms.
Proposition 2.3.2. Schreier split epimorphims are stable under composition. When the composite (gf,st) of two split epimorphisms (f,s) and (g,t) is a Schreier one, so is (g, t). The same is true for homogeneous split epimorphisms.
Proof. Let be given a pair of composable split epimorphisms: Xoo s //W
YY f
gf t
st g
Y
k′
f′
// // Y .
EE