(F,G)-abundant semigroups

Abstract

On a semigroup S, define the equivalence relation

F={(a,b)∈S×S∣∀x∈S:xa=x⇔xb=x}, and define G dually. We say S is F-abundant if there is an idempotent in every F-class, and similarly for G-abundance, and we say S is (F,G)-abundant if it is both F-abundant and G-abundant. These concepts are analogous to the notions of regularity and one- and two-sided abundance, defined in terms of Green’s relations L and R, and their generalisations L∗ and R∗, respectively. We relate this new form of abundance to the earlier ones, considering in particular the analogs of superabundance and amiability.