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64 Chapter 6.
(1′) lqw(vw′(u′1) · qw(u2)) = n(u′1) · lqw(u2), and
(2′) lqw(qw(u1) · vw′(u′2)) = lqw(u1) · n(u′2). We have that
n(u′1) · lqw(u2) = (nqw′ (u′1) + nv′w′(u′1)) · lqw(u2) = = nqw′ (u′1) · lqw(u2) + nv′w′(u′1) · lqw(u2) = (∗),
Semirings
but, since qw′ (u′1) ∈ K[w′], qw(u2) ∈ K[w] and K[w′] and K[w] commute, we have that nqw′ (u′1) · lqw(u2) = 0 and hence
(∗) = nv′w′(u′1) · lqw(u2) = lvw′(u′1)) · lqw(u2).
Then Condition (1′) is verified. Indeed, observe that vw′(u′1) · qw(u2) ∈ K[w],
and then
lvw′(u′1) · lqw(u2) = l(vw′(u′1) · qw(u2)) = lqw(vw′(u′1) · qw(u2)).
Condition (2′) can be proved similarly. We noticed that the fiber SPtB(SRng) is unital. Accordingly, there is,
inside this fiber, a natural notion of internal commutative object.
Corollary 6.2.6. A Schreier split epimorphism Aoo s ////B,
f
seen as an object in the fiber PtB(SRng) is commutative if and only if the kernel K[f] of f is a trivial ring. Accordingly any Schreier split epimorphism has at most one (commutative) monoid structure in this fiber.
Accordingly, the previous condition is the characteristic condition which makes the application m : R[f] → A defined by m(a, a′) = q(a) + a′ a semiring homomorphism; it is the binary operation which gives the commutative monoid structure structure inside the fiber PtB(SRng).
6.3 Schreier internal structures in semirings
The following definition is inspired by the analogous one (Definition 3.0.10) in the category of monoids.