Page 19 - Textos de Matemática Vol. 45

Chapter 2
Schreier and homogeneous split epimorphisms in monoids
2.1 Definitions and first properties
From now on, C will be the category Mon of monoids. Let us introduce the following definition:
Definition 2.1.1. A split epimorphism (A, B, f, s) of monoids is said to be right homogeneous when, for any element b ∈ B, the map μb : K[f] → f−1(b) de- fined by the multiplication on the right by s(b), as μb(k) = k · s(b), is bijective. Similarly, by duality, we can define a left homogeneous split epimorphism. (A,B,f,s) is said to be homogeneous when it is both right and left homoge- neous.
In [26] (Definition 2.6) was introduced the following
Definition 2.1.2. A split epimorphism (A,B,f,s) of monoids is said to be a Schreier split epimorphism when, for any a ∈ A, there exists a unique α in the kernel K[f] of f such that a = α · sf(a).
In other terms, a Schreier split epimorphism is a split epimorphism (A, B, f, s) equipped with a unique set-theoretical map q: A K[f] with the property that, for any a ∈ A, we have:
a = q(a) · sf(a).
Proposition 2.1.3. A split epimorphism (A, B, f, s) is right homogeneous if and
only if it is a Schreier split epimorphism.
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