Page 71 - Textos de Matemática Vol. 45

6.2. The fibration of Schreier points in semirings 63 = mqf (a1) · mqf (a2) + mqf (a1) · n(c2) + n(c1) · mqf (a2) + n(c1) · n(c2),
and it is easy to see that these two expressions are equal if and only if the conditions 6.2.2 are satisfied.
Theorem 6.2.5. The kernel functor reflects commutativity. Given any pair of maps in PtB(SRng) with their two domains in SPtB(SRng):
l
v
functor cooperate in SRng.
Proof. As in the case of monoids (see Theorem 2.4.6), since the kernel functor is left exact, it preserves the cooperating pairs. Conversely, suppose that the following pair cooperates in SRng:
K[w] K(l) // K[f] oo K(n) K[w′]
Let us denote by qw : U → K[w] (resp. qw′ : U′ → K[w′]) the Schreier retraction of the kernel kw (resp. kw′ ). Thanks to Theorem 2.4.6, we already know that the unique possible cooperator φ: U ×B U′ → A is given by φ(u,u′) = lqw(u)+ n(u′), and that it is a morphism of (additive) monoids. We only have to prove that it also preserves multiplication. We have that
φ((u1, u′1) · (u2, u′2)) = φ(u1 · u2, u′1 · u′2) = lqw(u1 · u2) + n(u′1) · n(u′2) =
= lqw(u1)·lqw(u2)+lqw(vw(u1)·qw(u2))+lqw(qw(u1)·vw(u2))+n(u′1)·n(u′2),
while
φ(u1, u′1) · φ(u2, u′2) = (lqw(u1) + n(u′1))(lqw(u2) + n(u′2)) =
= lqw(u1) · lqw(u2) + lqw(u1) · n(u′2) + n(u′1) · lqw(u2) + n(u′1) · n(u′2). So we have to show that
(1) lqw(vw(u1) · qw(u2)) = n(u′1) · lqw(u2), and
(2) lqw(qw(u1) · vw(u2)) = lqw(u1) · n(u′2).
Since (u1, u′1), (u2, u′2) ∈ U ×B U′, we have that w(u1) = w′(u′1) and w(u2) = w′(u′2), hence conditions (1) and (2) become
U
ff
// OOA oo
w v′
n 88 U′ && xx w′
sf
B
l and n cooperate in the fiber PtB(SRng) if and only their images by the kernel