Characterizing pure, cryptic and Clifford inverse semigroups

Abstract

An inverse semigroup S is pure if e = e 2, a ∈ S, e < a implies a 2 = a; it is cryptic if Green’s relation H on S is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and Clifford semigroups in a similar way by means of divisors. The paper also contains characterizations of completely semisimple inverse and of combinatorial inverse semigroups in a similar manner. It ends with a description of minimal non-V varieties, for varieties V of inverse semigroups considered.