On the Structure of Semigroups

Abstract

The purpose of this paper is to give the basis, and a few fundamental theorems, of a suggested systematic theory of semigroups. By a semigroup is meant a set S closed to a single associative, binary multiplication. Two elements of S are said to be left equivalent (I) if they generate the same left ideal in S. Similarly a right equivalence (r) can be defined. These equivalences commute (or are associable, Dubreil, 1), and their product equivalence is denoted by b. The equivalence b is to be compared with the two-sided analogue f of I and r; x 3 y(f) means that x and y generate the same two-sided ideal. In the case where S is finite, these equivalences b and f coincide, but this is not true in general. A b-class has the property that the 1-classes which it contains generate isomorphic left ideals in S; a theorem on the structure of b-classes is given (Theorem 1). Schwarz (1) and Clifford (1, 3) have both made use of minimal conditions on right and left ideals; and Clifford has shown (1) that the minimum ideal of a semigroup, if it contains both minimal right, and minimal left ideals, is a simple subsemigroup of the type called by Rees completely simple (Rees, 1, 2). In his paper On Semigroups (Rees, 1), Rees determined the structure of such simple semigroups. The minimal conditions we use are more stringent, they are the minimal conditions on the partially ordered sets of the right, left and two-sided ideals. If the right and left conditions are satisfied, then so is the twosided condition (a fact not obvious in our case, since a two-sided ideal is not in general a right or left ideal). Further, in this case we do have b = f; and the b-classes take on a simpler aspect. The f-classes of any semigroup S correspond to semigroups called factors (cf. Rees, 1, p. 391) of S; our definition does not, however, depend on the existence of a principal series of ideals of S. We say that S is semisimple if all these factors are non-nilpotent semigroups. Another, and probably more fruitful idea, is that of regularity. The element a e S is regular if aza = a for some z e S; this is the condition introduced, for rings, by J. v. Neumann (1).1 A semigroup is regular, if all its elements are regular. Regular semigroups are semisimple, but not conversely; however, any semisimple semigroup which satisfies the right and left minimal conditions, is regular. It seems that these last-mentioned semigroups may form the class which will most repay study; the simple ones are completely simple (and therefore of known structure), and already the extension theory of Clifford (2) suggests the possibility of building up more