Infinite iteration of matrix semigroups I. Structure theorem for torsion semigroups

Abstract

Given these conditions, we must precisely define EGC, special, and line. We begin with EGC. If X is a finite nonempty set X*, the free monoid generated by X, is by definition EGC because, intuitively, concatenation of two finite strings is a transparent multiplication. However, subsemigroups, and especially homomorphisms of EGCs, need not be EGCs. Furthermore, if S is EGC, and some “simple” operations applied to S gives S’, it is “reasonable” to say S’ is also EGC. Thus, by definition, if S is EGC and R: B x A + S, then S’ = R(S) (the Rees’ matrix semigroup over S) (see Definition 4.16), is also EGC. Note that the local structure, e.g., Green relations, egg box pictures, etc., of R(S) can be “easily” computed from the local structure of S and from R. For similar reasons, if 2 and S are EGC, R: B x A + S, (Z, A) and (B, Z) are actions, and R(bz, a) = R(b, za) for all b E B, z E Z, a E A, then, by definition, Z + R(S) (see Definition 4.16) is also EGC. 422 0021-8693186 $3.00