Page 23 - Textos de Matemática Vol. 45

2.2. Examples of Schreier and homogeneous split epimorphisms 15 Proof. Given a split sequence of the form
k // oo s K[f] A
f
let us define, for any a ∈ A, q(a) = a · sf(a)−1 ∈ K[f]. This map q clearly satisfies the conditions of Proposition 2.1.4, showing that it is a Schreier split epimorphism. The map q¯(a) = sf(a)−1 · a ∈ K[f], which satisfies the dual conditions, shows that it is left homogeneous.
We will prove later (Corollary 3.1.7) that the converse is also true: if any split epimorphism with codomain B is a Schreier one, then B is a group.
Corollary 2.2.5. If B is a group and (A,B,f,s) a split epimorphism in Mon, A is a group if and only if K[f] is a group.
Proof. Since B is a group, the split epimorphism (A, B, f, s) is a Schreier one. If K[f] is a group and a = q(a) · sf(a), then a−1 = sf(a)−1 · q(a)−1. The other implication is obvious, since (A, B, f, s) turns out to be a split epimorphism of groups.
Example 2.2.6. From Proposition 2.3.4 below, given any split epimorphism be- tween groups and any submonoid M of its codomain B, the following pullback produces a homogeneous split epimorphism in Mon:
f−1(M) // //YOO OO j
f′s′ fs
// // MB i
Example 2.2.7. Consider the internal order in Mon given by the usual order between natural numbers:
p0 // ON oo s0 // N,
p1
// // B,
where
is a submonoid of the direct product N×N (with the usual sum), and the monoid
homomorphisms p0, p1 and s0 are given by:
p0(x,y) = x, p1(x,y) = y, s0(x) = (x,x).
Then the split epimorphism (ON, N, p0, s0) is homogeneous.
ON = {(x, y) ∈ N × N | x ≤ y}