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Eventually inverse semigroups whose lattice of eventually inverse subsemigroups is semimodular. (English) Zbl 1020.20038

A semigroup is called eventually regular (or \(\pi\)-regular) if some power of each element is regular; and eventually inverse if each regular element has a unique inverse. (In contrast with inverse semigroups themselves, the idempotents need not then commute.) An eventually inverse subsemigroup of such a semigroup \(S\) is a subsemigroup \(A\) with the property that if any of its elements has an inverse in \(S\) then it has an inverse in \(A\). The collection of all such subsemigroups forms a lattice, denoted \(\text{Sub}\pi(S)\), in the usual way. It has long been known that the lattice of inverse subsemigroups of an inverse semigroup is (upper) semimodular if and only if it is an ordinal sum of \(UM\)-groups, that is, a chain of groups having semimodular subgroup lattice, with each structure homomorphism trivial.
This result is extended to eventually inverse semigroups, at the same time extending the first author’s characterization [Z.-J. Tian, J. Syst. Sci. Math. 17, No. 3, 226-231 (1997; Zbl 0901.20048)] of modularity of \(\text{Sub}\pi(S)\).

MSC:

20M18 Inverse semigroups
08A30 Subalgebras, congruence relations
20M10 General structure theory for semigroups
06B15 Representation theory of lattices

Citations:

Zbl 0901.20048
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