Zero-Semidistributive Inverse Semigroups

Abstract

An inverse semigroup S is said to be 0-semidistributive if its lattice ℒF (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) ^m  = a ⁿ or (ab) ^m  = b ⁿ , where σ is the minimum group congruence on S.

Key Words:

• Full inverse subsemigroup
• Inverse semigroup
• Principal factor
• 0-semidistributive lattice

(1991) Mathematics Subject Classification:

• 20M18

ACKNOWLEDGMENT

The author thanks Prof. P. R. Jones for his suggestions in modifying of my article.

Notes

Communicated by P. Higgins.