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On a Class of Lattice Ordered Inverse Semigroups

✍ Scribed by Gracinda M.S. Gomes; Emilia Giraldes; Donald B. McAlister

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 150 KB
Volume: 230
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8324

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✦ Synopsis

It is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saitô has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.

These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure.

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