The structure of superabundant semigroups

Projectively condensed semigroups, generalized completely regular semigroups and projective orthomonoids

“…The structure of completely regular semigroups has been described in details by Petrich and Reilly in [19]. During the recent decades, the generalizations of completely regular semigroups in some classes of generalized regular semigroups have been investigated in a number of papers (see for example [2][9], [14], [15], [20]). The aim of this paper is to consider various generalizations of completely regular semigroups in the classes of generalized regular semigroups in a systematic way.…”

“…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:…”

“…Furthermore, by Lemma 4.3, we can easily see that Y ; (S α , * α ) is indeed the structure semilattice decomposition of (S, * ). Standard representation of semigroups is an important tool for constructing bands, completely regular semigroups, orthodox super rpp semigroups, superabundant semigroups and orthodox semigroups (see [18,19,9,20,10]). We now construct P-orthomonoids by using this method.…”

A generalized Clifford theorem of semigroups

“…A U -superabundant semigroup can be regarded as an analogue of a completely regular semigroup (or a union of groups). It can also be regarded as an analogue of an abundant semigroup which is superabundant (see [3,10]). According to [2, p. 126], Clifford proved in 1941 that a semigroup S is a union of groups if and only if it is a semilattice of completely simple semigroups.…”

On U-orthodox semigroups

“…For notations and terminologies not given in this paper, the reader is referred to Hall [13] , Ren [14] and the texts of Howie [11,15] .…”