Inverse semigroups determined by their lattices of inverse subsemigroups

Abstract

In a previous paper ([14]) the author showed that a free inverse semigroup W9 X is determined by its lattice t(^i x ) of inverse subsemigroups, in the sense that for any inverse semigroup T, tCSi x ) =* £(7") implies 9$ x =s T. (In fact, the lattice isomorphism is induced by an isomorphism of < 3$ x upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and £-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

On Lattice Isomorphisms of Inverse Semigroups

“…We may extend the lemma to all completely semisimple inverse semigroups of the appropriate type, without difficulty. It is known [7] that the class of such semigroups is closed under L-isomorphisms.…”

“…For instance this occurs whenever S is simple or is E-unitary with a greatest J -class which is nontrivial [7]. It also arises from external conditions.…”

On Lattice Isomorphisms of Inverse Semigroups, II

“…It is easily seen that since Φ restricts to an L-isomorphism between their respective semilattices of idempotents, E S and E T , it induces a bijection φ E between them. Continuing the context of the cited papers, we focus on the situation where φ E is an isomorphism, which occurs in a surprising number of contexts, for example (see [8]) when S is simple, when S is E-unitary with no trivial J -classes or, except in a singular case, when Φ is induced by an isomorphism between the partial automorphism semigroups of S and T .It has long been known that φ E extends to a bijection φ : E S ∪ N S → E T ∪ N T , where N S denotes the set of elements that belong to no subgroup of S. This is termed the "partial base bijection" in [14]. In the aperiodic (or "combinatorial") case where, by definition, all subgroups are trivial, φ is then a bijection between S and T .…”

“…It is easily seen that since Φ restricts to an L-isomorphism between their respective semilattices of idempotents, E S and E T , it induces a bijection φ E between them. Continuing the context of the cited papers, we focus on the situation where φ E is an isomorphism, which occurs in a surprising number of contexts, for example (see [8]) when S is simple, when S is E-unitary with no trivial J -classes or, except in a singular case, when Φ is induced by an isomorphism between the partial automorphism semigroups of S and T .…”

“…The existence of nontrivial isolated subgroups does not preclude positive outcomes, however. In [8] it was shown that the free product of two groups, in the variety of inverse semigroups, is determined by its lattice of inverse subsemigroups, despite the isolation of the original groups within the free product. (In fact, [9], any non-trivial free product of inverse semigroups is determined by its lattice of inverse subsemigroups.)…”