Affine semigroups over an arbitrary field

Abstract

Let S£V denote the algebra of all linear transformations on an w-dimensional vector space V over a field <I>. A subsemigroup S of the multiplicative semigroup of JS?F will be said to be an affine semigroup over $ if S is a linear variety, i.e., a translate of a linear subspace of ££V.This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.Except in the appendix, V and hence also £PV will be supposed finite-dimensional. In §1 we show that some power of every element of an affine semigroup S lies in a subgroup of S and that S always contains a completely simple minimal ideal K. (For definitions see below.) We then obtain a decomposition of S into a group, an algebra, and two vector spaces.The minimal ideal K of an afflne semigroup is not in general a linear variety, as was observed by Cohen and Collins [1]. When K is not a linear variety, one turns naturally to M(K), the smallest linear variety containing K. We show in §2 that M(K) n -K for any integer n exceeding the dimension of M{K).UK^M(K) and every element ofKis idempotent, then we are able to find a subsemigroup of M{K) isomorphic to the example of Cohen and Collins (see 2.1). In this case we also show that M{K) 2 = K. The requirement that every element of K be idempotent is of some interest since this is always true for affine semigroups on locally convex linear topological spaces which are generated by compact convex subsemigroups (see Theorem 3 of [1] and 2.10 below).The author wishes to express his gratitude to Professor A. H. Clifford for his encouragement and for many useful suggestions during the preparation of this paper.

Weakly Semi-Simple Finite-Dimensional Algebras

“…Then D C fAf trivially. To show the converse we need only show / to be principal, for then by (1, §9, p. 25) we have dim(/^4/) = dim D. Let g = g 2 3.9. COROLLARY.…”