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.
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Petrich, M. Characterizing pure, cryptic and Clifford inverse semigroups. Czech Math J 64, 1099–1112 (2014). https://doi.org/10.1007/s10587-014-0155-0
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DOI: https://doi.org/10.1007/s10587-014-0155-0
Keywords
- inverse semigroup
- pure inverse semigroup
- cryptic inverse semigroup
- Clifford semigroup
- group-closed inverse semigroup
- pure variety
- completely semisimple inverse semigroup
- combinatorial inverse semigroup
- variety