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Chapter 6
Semirings
In this chapter we will show that many of the results presented above in the category Mon of monoids are still valid in the category SRng of semirings. We recall that a semiring (A,+,·,0) is a commutative monoid (A,+,0) endowed with a binary operation ·: A × A → A which is associative and distributive w.r.t. +. We have then a forgetful functor
U : SRng → CMon
associating with any semiring (A, +, ·, 0) the commutative monoid (A, +, 0).
Since we do not require a unit element for the second operation (that will be called multiplication), the category SRng of semirings is pointed, the zero object being the semiring {0}; the category SRng is also unital (this can be seen in the same way as for the category Mon of monoids). Concerning the commutativity (or cooperation) of two morphisms with the same codomain (as in Definition 1.2.3), we have the following characterization:
Proposition 6.0.4. In SRng, two morphisms f and g, as in Definition 1.2.3, cooperate if and only if
f(x)g(y) = g(y)f(x) = 0 for all x ∈ X, y ∈ Y. (6.0.1) Proof. If Condition 6.0.1 is true, we can define the cooperator as
φ(x, y) = f (x) + g(y).
It is easy to show that φ is a morphism of semirings and φ⟨1, 0⟩ = f, φ⟨0, 1⟩ = g.
Conversely, if a cooperator φ exists, then we have
0 = φ(0, 0) = φ((x, 0)(0, y)) = φ(x, 0)φ(0, y) = f (x)g(y),
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