Page 80 - Textos de Matemática Vol. 45

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and, similarly,
Chapter 6.
Semirings
l(x) · tg(b) = l(x) = tg′(b). Conversely, suppose that the Schreier split extension
// oo q′ r // //
0 X //Coo D 0
lt
Condition (ii) then implies that g(b) = g′(b) for every b ∈ B.
is faithful. Fixed an element d ∈ D, we can construct another Schreier split extension
0 //Xoo q //Aoo p //B //0,
k
s
with a morphism (f,g) into the faithful one in the following way. Let B = F({z}) be the free semiring on a singleton {z} (which is isomorphic to the semiring N of natural numbers, with the usual sum and multiplication). Let g: B → D be the unique morphism such that g(z) = d. Let A be the commu- tative monoid X × B, with the multiplication defined by
(x1, b1) · (x2, b2) = (x1 · x2 + q′(tg(b1) · l(x2)) + q′(l(x1) · tg(b2)), b1 · b2).
It is easy to see (and it follows from the results in [26]) that A is a semiring
and the following is a Schreier split extension:
0 //Xoo q //Aoo p //B //0,
s
where q = πX, p = πB, k = ⟨1,0⟩ and s = ⟨0,1⟩. If we define f: A → C by
putting
we obtain the following commutative diagram:
0 // X oo q // A oo p // B // 0
s
1X f g
k
f (x, b) = l(x) + tg(b),
k
// oo q′ r //
0 X //Coo D 0.
lt
We still have to prove that f is a morphism of semirings: f((x1,b1)+(x2,b2))=f(x1 +x2,b1 +b2)=
//
=l(x1 +x2)+tg(b1 +b2)=l(x1)+l(x2)+tg(b1)+tg(b2)=