Page 20 - Textos de Matemática Vol. 45

12 Chapter 2. Schreier and homogeneous split epimorphisms in monoids Proof. Given a right homogeneous split epimorphism (A, B, f, s), the unique α
that appears in Definition 2.1.2 is given by α = μ−1 (a). Conversely, given a f (a)
Schreier split epimorphism, for any b ∈ B the inverse of the map μb : K[f] → f−1(b) is defined in the following way: if a ∈ f−1(b), μ−1(a) is the unique
b
α ∈ K[f] such that a = α · sf(a).
Proposition 2.1.4. A split epimorphism (A, B, f, s) is a Schreier split epimor-
phism if and only if there exists a set-theoretical map q: A K[f] such that:
q(a) · sf(a) = a q(α · s(b)) = α
for every a ∈ A, α ∈ K[f] and b ∈ B. Dually, a split epimorphism is left homogeneous if and only if there exists a set-theoretical map q¯: A K[f] such that:
sf(a)·q¯(a) = a q¯(s(b) · α) = α
for every a∈A, α∈K[f] and b∈B.
Proof. Suppose that for every a ∈ A, there exists a unique α ∈ K[f] such that a = α·sf(a). This property defines a map q: A → K[f], by q(a) = α such that a = q(a)·sf(a), for every a ∈ A. In order to prove that q(α·s(b)) = α for any α ∈ K[f], it suffices to observe that sf(α · s(b)) = s(b).
Conversely, given a set-theoretical map q : A → B satisfying the asserted identities, we can choose α = q(a) for every a ∈ A by the first identity; suppose now that a = α′ · sf(a), then we get:
q(a) = q(α′ · sf(a)) = α′
by the second identity. There is a similar proof for the left homogeneous split
epimorphisms.
We shall call the following diagram:
phism and q the associated Schreier retraction. In fact, we have: Proposition 2.1.5. Given a Schreier split epimorphism (A, B, f, s), we have:

// B,
the canonical Schreier split sequence associated with the Schreier split epimor-
oo q
K[f] // //A
oo s oo kf
(a) qk = 1K[f];