Page 102 - Textos de Matemática Vol. 45

94 Chapter 7. Special Schreier and special homogeneous surjections
It produces a relation S on X defined by xSz if and only if xR[f]z and q(xR[f]z) ∈ K[h]. This relation S is clearly reflexive. The domain of the kernel of d0 : S → X is the ring K[h] and the reflexive relation S is a Schreier one with the Schreier retraction qS : S → K[h] defined by qS(xSz) = q(xR[f]z). Accordingly S is a Schreier equivalence relation on X in SRng. Let us define X′ as the quotient semiring of this equivalence relation S:
d0 // h¯ Soos0//X ////X′
d1
Since the kernel of d0 : S → X is the ring K[h], K[h] is also the kernel of h¯. On the other hand, since S is included in R[f], h¯ coequalizes p0 and p1, hence there exists a unique surjective homomorphism f′ : X′ Y making the following lower right hand side square commutative:
K[k(h¯)] //
//K[h] //1
// R[f] oo s0 // X p1
K(h¯)
(0,kf ) R(h¯)
// // Y
k¯
A //
kh¯
p0 // f
K[f′] // (0,kf′)
p′0 / / h¯ // R[f′] oo s′0 // X′
// // Y.
p′ f′ 1
Completing in SRng the diagram above with the kernels of the vertical maps produces the upper horizontal kernel diagram, which shows that the kernel of K(h¯) is, up to isomorphisms, the ring K[h]; consequently the ring K[f′] is, up to isomorphisms, the given ring A′, and makes the map k′ : A′ X′, defined by k′(a′) = h¯(a) for any a such that a′ = h(a), a kernel for f′. We have to show now that f′ is a special Schreier surjection. If it is the case, the associated Schreier retraction q′ is necessarily defined by q′(x¯R[f′]z¯) = hq(xR[f]z) for any x, z ∈ X such that h¯(x) = x¯ and h¯(z) = z¯. Let us check that this definition does not depend on the choice of x and z. Suppose that h¯(x) = h¯(x′) and h¯(z) = h¯(z′). Then xSx′ and zSz′, and this implies that xR[f]x′ and zR[f]z′. Then, by Proposition 3.1.5, we get
q(x′R[f]z′) + q(xR[f]x′) = q(xR[f]z′) = q(zR[f]z′) + q(xR[f]z) Hence
hq(x′R[f]z′) + hq(xR[f]x′) = hq(zR[f]z′) + hq(xR[f]z).
But xSx′ and zSz′ imply also q(xR[f]x′) ∈ K[h] and q(zR[f]z′) ∈ K[h], and so hq(xR[f]z) = hq(x′R[f]z′).