The Minimal Degree of a Finite Inverse Semigroup

The minimal degree of an inverse semigroup S is the minimal cardinality of a set A such that S is isomorphic to an inverse semigroup of one-to-one partial transformations of A . The main result is a formula that expresses the minimal degree of a finite inverse semigroup S in terms of certain subgroups and the ordered structure of S . In fact, a representation of S by one-to-one partial transformations of the smallest possible set A is explicitly constructed in the proof of the formula. All known and some new results on the minimal degree follow as easy corollaries. If p is an isomorphic or a homomorphic representation of an inverse semigroup S by one-to-one partial transformations of a set A, then the cardinality of A is denoted by 6(p) and called the degree of p . Every inverse semigroup has a faithful (that is, isomorphic) representation by one-to-one partial transformations of a set. The minimal degree of a faithful representation of S is called the minimal degree of S and denoted by 3(S). If the generalized continuum hypothesis is assumed and S is infinite, then S(S) is either |5| or the predecessor of |5|. This follows from the fact that J¿ , the symmetric inverse semigroup of all one-to-one partial transformations of an infinite set A, has cardinality 2^1. In particular, if \S\ is a limit cardinal, the S(S) = \S\. While finding ô(S) for infinite S is not devoid of interest, we consider only finite inverse semigroups in this paper. Our main result is an exact formula for S(S) "modulo groups." Solving semigroup problems "modulo groups" (a semigroup problem reduced to a group problem is considered solved) may raise objections, but in our case a "modulo groups" solution may be the best one can expect. Indeed, if G is a group, then S(G), as is easily seen (Lemma 1), is the minimal degree of faithful representations of G by permutations. The minimal degree of groups has been considered for more than a century, no definitive formula for 0(G) has been found yet, and one can hardly expect that such a formula exists. If 77 is a subgroup of G, we define ôG(H), the minimal degree of 77 in G, as the minimal cardinality of a nonempty set A such that there exists a (homomorphic) representation P of G by permutations of A which induces a faithful representation of 77. Such a representation p is called a minimal representation of 77 in G. It seems that this concept has not been considered in the theory of groups, although it belongs to this theory. Observe that A above Received by the editors May 9, 1990 and, in revised form, August 1, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 20M30; Secondary 20M18, 20M20. ©1992 American Mathematical Society 0002-9947/92 $1.00+ $.25 per page