A basis theorem for free inverse semigroups

Abstract

Using the description of the free inverse semigroup 1, on the set X given by Scheiblich [9], Reilly [7] h as shown that some inverse subsemigroups of IX are not free. This contrasts with the classical result of Schreier [lo] and Nielsen [6] that the subgroups of a free group are all free. Nevertheless some properties of inverse subsemigroups of 1, can be found. In this paper we prove a “basis” theorem for I, : any two bases (minimal, or “irredundant” generating sets) of an inverse subsemigroup of I, have the same cardinality. We call this the basis property (for IX). We in fact prove a rather more general result (the “strong” basis property). From this we can deduce that any two bases for an inverse subsemigroup of 1, must have the same number of elements in any y-class of 1, . In Section 1, we discuss sufficient conditions for the (strong) basis property, and list some relevant properties of 1x . These conditions (triviality of%, and complete semisimplicity) are used in Section 2 to prove a lemma which, in effect, is the first step in an induction proof for the main theorem, from which we deduce the strong basis property and thus the basis property. In Section 3, we prove that every inverse subsemigroup of Ix does indeed have a basis, and prove the result about &-classes stated above. We go on to use these theorems and the results of Section 1 to find in what ways an element of a basis for an inverse subsemigroup S of IX can be replaced by another element of I,, so as to yield a new basis for S. These results lead to necessary and sufficient conditions for two irredundant sets to generate the same inverse subsemigroup (Section 4). In Section 5, we give some examples, and show that if an inverse semigroup has the basis property, it must be completely semisimple.