Building New Semirings from Old

Abstract

We now consider a material from the previous chapter from a different angle. Let R be a, semiring and let A be a nonempty set which is either finite or countably-infinite. Then the set R ^A×A of functions from A × A to R is denoted by M _a (R), and such functions are called (A × A)-matrices on R. If A is a finite set of order n we write M _n (R) instead of M _a (R)] if A is countably-infinite we sometimes write M _Ω (R) instead of M _a (R) - If A is a finite or countably-infinite set we will often denote matrices in the usual matrix notation rather than in functional notation. In particular, we will sometimes employ “block notation” for such matrices. We have already noted that addition of such matrices, defined componentwise, turns M _a (R) into a commutative additive monoid, the identity element of which is the function which takes every element of A × A to 0.