Page 22 - Textos de Matemática Vol. 45

14 Chapter 2. Schreier and homogeneous split epimorphisms in monoids Lemma 2.1.7. Given a Schreier split epimorphism (A,B,f,s), the following
diagram
// B,
makes s: B A the kernel of q in the category of pointed sets.
Proof. We already know that qs = 0 (Proposition 2.1.5 (b)), hence B is con- tained in the kernel of q. Conversely, if a belongs to the kernel of q, then
a = q(a) · sf(a) = 1 · sf(a) = sf(a),
and hence a is in the image of s.
2.2 Examples of Schreier and homogeneous split epi- morphisms
Proposition 2.2.1. Given any direct product diagram
oo q oo s oo K[f] // //A
ooπX ⟨1X ,0⟩
// X × B oo
πB // ⟨0,1B ⟩
kf
B,
the canonical split epimorphism (X × B, B, πB , ⟨0, 1B ⟩) is homogeneous.
Proof. The equality (x, 1) · (1, b) = (x, b) shows that it is right homogeneous, while (1, b) · (x, 1) = (x, b) shows that it is left homogeneous. Here the Schreier retraction πX is a monoid homomorphism.
Corollary 2.2.2. The terminal split epimorphism X oo // 1
is homogeneous.
Corollary 2.2.3. The identity split epimorphism
Proposition 2.2.4. If B is a group, then every split epimorphism (A, B, f, s) is homogeneous.
X
//
and more generally any isomorphism, is homogeneous.
X oo
X,
1X 1X