Representations of a semigroup

Rebecca Ellen Slover
1965 Transactions of the American Mathematical Society  
The purpose of this paper is to develop for semigroups the notions of radical and semisimplicity similar to those which have been developed for rings. In 1961, E. J. Tully, Jr. published a paper dealing with transitive and O-transitive operands of a semigroup. We shall be using his definitions and notation as a starting point in our considerations. The author wishes to express gratitude to Ancel Mewborn for many valuable suggestions. In this paper we shall first define the radical of a
more » ... S and investigate some of its properties. Just as in the case of rings, one finds that the radical of a semigroup is a quasi-regular two-sided ideal which contains each quasi-regular right ideal of the semigroup and that the radical of a semigroup contains each nil right ideal of the semigroup. If a semigroup with zero satisfies the minimum condition on left ideals and right ideals then its radical is also the left radical of the semigroup. One also finds that if Sn is the semigroup of all row-monomial n x n matrices over 5 then the radical of Sn is the semigroup of all row-monomial n x n matrices over the radical of S. If T is a two-sided ideal of £ then the radical of T is the intersection of T and the radical of S. In the latter part of the paper we shall study semisimple semigroups. If s, t e S, let spt provided that, if m is contained in some O-transitive operand of S, then ms = mt. Thus p is a two-sided congruence called the radical congruence of S. If the radical congruence of 5 is the identity relation then S is semisimple. Since M is a O-transitive operand of S if and only if M is a O-transitive operand of S/p, then it is easily seen that S/p is semisimple. If S is semisimple and T is a two-sided ideal of S, then T is semisimple. Finally, if S contains more than one element, then S is semisimple if and only if S is isomorphic to a subdirect sum of a set of semigroups each of which has a faithful O-transitive operand. 1. The radical of a semigroup. If each of a and ß is a right congruence of an operand M, then a is contained in ß provided mßn whenever man. The right congruence a is maximal if a is not the universal relation and the only right congruence which properly contains a is the universal relation. If a is a right congruence on the operand M, x is a cr-class, and s is in S, let xs be the cr-class containing the set {ms: m ex}. With this definition the set of Received by the editors March 30, 1964.
doi:10.1090/s0002-9947-1965-0188326-4 fatcat:qg3zm5b5u5fvdpkrgdyfb3kyay