Inverse semigroups determined by their lattices of inverse subsemigroups

Peter R. Jones

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An inverse semigroup $S$ is pure if $e = e^{2}$ , $a \in S$ , $e < a$ 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...