On left C-U-liberal semigroups

Abstract

In this paper the equivalence Q U on a semigroup S in terms of a set U of idempotents in S is defined. A semigroup S is called a U-liberal semigroup with U as the set of projections and denoted by S(U ) if every Q U -class in it contains an element in U . A class of U-liberal semigroups is characterized and some special cases are considered.

Projectively condensed semigroups, generalized completely regular semigroups and projective orthomonoids

“…Analogous to Table 1, the following Table 2 gives the basic information of some classes of semigroups: [3,20] semi-superabundant H-surjective SeSuAb [1] P-semi-superabundant H U -surjective PSeSuA [4], [14] [16] super rpp (L * ∩ R)-surjective SuRpp [9] strongly rpp strongly L * -surjective StRpp [2,6,7] strongly semi-rpp strongly L-surjective StSeRpp strongly P-semi-rpp strongly ( L U , U )-surjective StPSeRpp [11] super lpp ( L ∩ R * )-surjective SuLpp strongly lpp strongly R * -surjective StLpp strongly semi-lpp strongly R-surjective StSeLpp strongly P-semi-lpp strongly ( L U , U )-surjective StPSeLpp Table 2 In what follows, by a generalized completely regular semigroup we mean a semigroup in the classes of semigroups from Table 2, and by a class of generalized completely regular semigroups we mean a class of semigroups from Table 2. In this section, we shall consider the relationship between the classes of generalized completely regular semigroups and the subquasivarieties of PC and C partially satisfying the following implications:…”