CHARACTERIZING PURE, CRYPTIC AND CLIFFORD
Inverse Semigroups · 2014
An inverse semigroup S is pure if e = e 2 , a 2 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 homo- morphism 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 homomor- phism 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 com- binatorial inverse semigroups in a similar manner. It ends with a description of minimal non-V varieties, for varieties V of inverse semigroups considered.