Bands of Semigroups

In studying a general semigroup S, a natural thing to do is to decompose 5 (if possible) into the class sum of a set {Sa; aCl} of mutually disjoint subsemigroups Sa such that (1) each Sa belongs to some more or less restrictive type 13 of semigroup, and (2) the product SaSfi of any two of them is wholly contained in a third: SaSpClSy, for some yCI depending upon a and /3. We shall then say that 5 is a band of semigroups of type 13. If, for every a and /3 in I, SaSp and SpSa are both contained in the same Sy, then we shall call S a semilattice of semigroups of type 13. We shall also be concerned with the following specialization of the notion of band of semigroups. Suppose that / is the direct product JXK of two classes / and K. The subsemigroups Sa are then described by two subscripts: 5,-< (iCJ, kCK). Suppose moreover that SitSjxCSiK for all i, jCJ and all k, X£i£. We shall then call 5 a matrix of semigroups of type 13. The primary purpose of the present paper is to show (Theorem 4) that a band of semigroups of type 13 is a semilattice of semigroups each of which is a matrix of semigroups of type 13. The rest of the paper is devoted to giving necessary and sufficient conditions on a semigroup 5 that it be a band or a semilattice of (1) simple semigroups, (2) completely simple semigroups, and (3) groups. (Throughout this paper we use the term simple to mean simple without zero, i.e. a simple semigroup is one containing no proper two-sided ideal whatever.) For (1), we have the elegant condition, aCSa2S for all aCS, due to1 Olaf Andersen [l ]. If a semigroup 5 is a class sum of [completely] simple semigroups, it is also a semilattice of [completely] simple semigroups. But a class sum of groups need not be a band of groups, nor need a band of groups be a semilattice of groups; these three categories are characterized by Theorems 6, 7, and 8, respectively. We note that a semigroup 5 is a "band of groups of order one" if and only if each element of S1 is idempotent. In this case we call 5 simply a "band," and consequently make the definition: a band is a semigroup every element of which is idempotent. By the same token, we define a semilattice to be a commutative band. A "matrix of groups