J Austral Math Soc Ser A 43 pp47--63, 1987.

Modular Inverse Semigroups

Katherine G. Johnston and Peter R. Jones

Abstract

An inverse semigroup S is said to be modular if its lattice LF(S) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is Archimedean" in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each D-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author of a simple modular inverse semigroup is almost" distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

1980 AMS Subject Classification: 20M10, 08A30

Browse the article

Authors

Katherine G. Johnston

Department of Mathematics, College of Charleston, Charleston, South Carolina 29424, U. S. A.

Peter R. Jones

Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisconsin 53233, U. S. A.