The Bifree Locally Inverse Semigroup on a Set

Abstract

A class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is an e(xistence) variety of regular semigroups. For an e-variety V of locally inverse or E-solid regular semigroups, the bifree object BFV(X) on a set X is the natural concept of a "free object" in V. Its existence has been proved by Y. T. Yeh. In this paper, the bifree locally inverse semigroup BFLJ(X) is described as a homomorphic image of the absolutely free algebra of type [2, 2] generated by X and the set of formal inverses X′, and equivalently as subsemigroup of a semidirect product of a suitable free semilattice by the bifree completely simple semigroup on X. This latter realization is used to show that BFLJ(X) is combinatorial, completely semisimple and satisfies several finiteness conditions. Furthermore, the approach of biidentities is used to formulate a Birkhoff-type theorem for e-varieties of locally inverse semigroups and to establish a one-one correspondence between locally inverse e-varieties and fully invariant congruences on BFLJ(X) for countably infinite X. As an application, it is shown that in each e-variety of locally inverse semigroups all free products exist.