Page 60 - Textos de Matemática Vol. 45

52 Chapter 5. Schreier and homogeneous representability 5.2 Semidirect products
In analogy with the classical case of groups, given two monoids B and X, we will call action of B on X a morphism φ: B → End(X). If X is a commutative monoid, we will say that φ gives X a B-module structure. Given two B-modules φ: B → End(X) and ψ: B → End(X′), a B-module homomorphism is a monoid homomorphism h: X → X′ such that h(φ(b)(x)) = ψ(b)(h(x)) for all b ∈ B and all x ∈ X.
The following classical construction (see Section V4 in [19], or Section 1.2.2 in [32]) gives then a converse of Theorem 5.1.2:
Definition 5.2.1. Given an action φ: B → End(X) of a monoid B on a monoid X, the semidirect product of B and X w.r.t φ is the monoid X φ B whose underlying set is the cartesian product X × B, and whose operation is defined as follows:
(x1, b1) · (x2, b2) = (x1 · φ(b1)(x2), b1 · b2)
It is easy to see that the operation defined above gives X φ B a monoid
structure, and that the following diagram is a split exact sequence in Mon: X ⟨1,0⟩//X φBoo⟨0,1⟩ //B.
πB
Actually it is a Schreier split sequence; the Schreier retraction q is nothing but
theprojectionπX onX;indeed,foranyx∈X andanyb∈B: (x, b) = (x, 1) · (1, b) = ⟨1, 0⟩πX (x, b) · ⟨0, 1⟩πB (x, b),
and
This fact, together with Theorem 5.1.2 gives the following:
Proposition 5.2.2. Given two monoids B and X, there is a natural bijection between the set of Schreier split epimorphisms with codomain B and kernel X and the set of actions φ: B → End(X) of B on X.
Let us observe that, given an action φ: B → End(X) of B on X, the construction of the semidirect product X φ B can be obtained by building explicitly the pullback of the morphism θX : Hol(X) → End(X) along φ.
5.3 The homogeneous split extension classifier
We are now going to show that the homogeneous split epimorphisms also have their classifier, given by the group Aut(X) of automorphisms of a monoid X.
πX ((x, 1) · (1, b)) = πX (x, b) = x.