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16 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. The equality (x, y) = (0, y−x)+(x, x), for any (x, y) ∈ R proves that the split epimorphism in question is right homogeneous, while the commutativity implies that it is left homogeneous.
Notice that, in the last example, the split epimorphism (p1,s0) is not a Schreier one. Indeed, there is no natural number n such that (0, 1) = (n, 0) + (1, 1).
Example 2.2.8. We denote by Z∗ the monoid of non-zero integers with the usual multiplication, and by N∗ its submonoid whose elements are the numbers greater than 0. Then the split epimorphism
Z∗oo i //N∗, abs
where i is the inclusion and abs associates with any integer its absolute value, is a homogeneous split epimorphism. In fact K[abs] = {±1}, and it is immediate to see that any non-zero integer z can be written in a unique way as z = ±1 · |z| = |z| · ±1.
Example 2.2.9. Let us fix a natural number n. We can define on the cartesian product N × N the monoid structure given by
(x1, b1) · (x2, b2) = (x1 + nb1 x2, b1 + b2).
(when both n and b1 are 0, we use the convention 00 = 1). We denote this monoid by N n N. The projection π1 : N n N → N defined by π1(x,b) = b is a monoid homomorphism, in the same way as the section σ1 : N → N n N defined by σ1(b) = (0, b). For any n ∈ N, the split epimorphism (N n N, N, π1, σ1) is a Schreier split epimorphism: the (unique) map q: N × N K[π1] is just the first projection π0 : N × N N. If n = 0, this split epimorphism is not homogeneous (because it is not left homogeneous): for example, for b = 1, the map λ : K[π ] π−1(1) defined by
λ1(x, 0) = (0, 1) · (x, 0) = (0, 1)
is clearly not bijective.
The last example is an instance of the semidirect product construction, whose relationship with the Schreier and homogeneous split epimorphisms will be explored in Chapter 5.
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