THE EXISTENCE OF E-FREE OBJECTS IN E-VARIETIES OF REGULAR SEMIGROUPS

Abstract

An existence variety (or e-variety) of regular semigroups is a class of regular semigroups which is closed under [Formula: see text], and ℍ. This concept was introduced by T.E. Hall and independently for orthodox semigroups by J. Kadourek and M.B. Szendrei who called them bivarieties. In this paper we prove the existence of e-free objects in each e-variety of E-solid regular semigroups and in each e-variety of locally inverse regular semigroups. By contrast, we show that there is no e-free object in other e-varieties.