Page 32 - Textos de Matemática Vol. 45

24 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. Consider the following pullback of split epimorphisms (which gives the product in the category P tB (M on)), where the lower horizontal one is a Schreier split epimorphism:
(2.4.1)
oo eC
A ×B C // COO
OO πC πAeA gt
oos
A
// B.
The homomorphisms eA and eC are defined by eA(a) = (a,tf(a)) and eC(c) = (sg(c),c). We have to show that they are jointly strongly epimorphic in the fiber PtB(Mon), which is equivalent to be jointly strongly epimorphic in Mon; in other words we have to show that the only submonoid of A ×B C containing these two classes of elements is A ×B C itself. This is a consequence of the following calculation for any element (a¯, c¯) ∈ A ×B C;
(a¯, c¯) = (q(a¯), 1) · (sf (a¯), c¯) = (q(a¯), 1) · (sg(c¯), c¯) = (q(a¯), tf q(a¯)) · (sg(c¯), c¯) since we have f(a¯) = g(c¯).
Then, according to Section 1.3, we can define the commutativity of two morphisms (m,n) in PtB(Mon) provided that the domain of one of the two maps is a Schreier split epimorphism.
Proposition 2.4.3. Suppose the domain (A, B, f, s) of m is a Schreier split epi- morphism and denote by (C, B, g, t) the domain of n, as in the diagram below. The cooperator φ: A ×B C → X of (m,n) in PtB(Mon), when it exists, is necessarily (being unique) defined by φ(a, c) = mqf (a) · n(c). It is actually a cooperator if and only if we have mqf (sg(c)·α)·n(c) = n(c)·m(α) for α ∈ K[f] and c ∈ C.
A ×B C
φ
;;
cc
f
eA
eC
m // oo n AXC cc OO ;; fg
s
Proof. We must have φeA = m and φeC = n. Moreover, for any (a, c) ∈ A×B C, we have
(a, c) = (qf (a) · sf (a), c) = (qf (a) · sg(c), c) = (qf (a), 1) · (sg(c), c).
If φ is a monoid homomorphism, we get: φ(a, c) = mqf (a) · n(c). When it is a monoid homomorphism, the identity (sg(c), c) · (α, 1) = (sg(c) · α, c), for
## {{ B
t