Free inverse semigroups and the theorems of McAlister

Abstract

Abstract The existence of the free inverse semigroup FI x was established in Chapter 1 (Theorem 1.1.10). A canonical form for the elements of Fix is afforded by their representation as the birooted word trees of Munn (1974). This in tum allows us to construct an alternative representation of the members ofF Ix due to Scheiblich (1972, 1973). Consideration of the semilattice ofFix leads to a further description ofFix as a so-called P-semigroup. This motivates the study of arbitrary P-semigroups which are then shown to comprise a class of £-unitary inverse semigroups. The famous theorem of McAlister (1974b) is that the converse is true: any £-unitary inverse semigroup is isomorphic to a P-semigroup. In proving this theorem we follow Munn (1976). Finally, we prove that any inverse semigroup is the image under an idempotent-separating homomorphism of an £-unitary semigroup. Again we do not record the original proof of McAlister (1974a), but establish the result by using a semidirect product construction of Wilkinson (1983), although the result can also be quickly obtained from the knowledge gained concerning Fix (Exercise 2.2.3).