Uncountably many existence varieties of orthodox locally idempotent semigroups

Abstract

Existence varieties (briefly e-varieties) of regular semigroups were introduced in [1] as classes of regular semigroups closed under the formation of homomorphic images, regular subsemigroups and direct products. Existence varieties of orthodox semigroups, which are simply the sub-e-varieties of the e-variety $\hbox{\sp O}$ of all orthodox semigroups, were independently introduced in [4] under the name of bivarieties. Moreover, in [4], the notions of bifree objects, biidentities and biinvariant congruences were introduced within $\hbox{\sp O}$ in such a way that a theory properly generalizing the theory of varieties of inverse semigroups arose. Existence varieties of orthodox semigroups include all varieties of orthodox completely regular semigroups, and hence, in particular, all varieties of idempotent semigroups, that is, all subvarieties of the variety $\hbox{\sp B}$ of all bands. These varieties form a countable lattice whose full description is well known.