A New Construction for Free Inverse Semigroups
by
Boris M. Schein
University of Arkansas, Fayetteville, USA
Coauthors: Olga V. Poliakova

In the seventies Munn, Preston, Scheiblich and Schein found several constructions for free inverse semigroups. In this talk an alternative construction is suggested.

The elements of a free inverse semigroup with a set X of free generators can be considered as classes of equivalent words over the alphabet Y = X \cup X^-1 modulo an appropriate congruence relation. I produce a simple algorithm that reduces every word w over Y to its canonical form [w\tilde]. These canonical forms are exactly the shortest words in their equivalence classes.

Each equivalence class may contain a few different shortest words, all of them are canonical, and an easy algorithm is given to determine whether two canonical words are equivalent. In fact, in this particular context the word ``algorithm" is a bit too much, for the decision is made instantly from the external appearance of these words. Equivalent canonical words ``almost coincide, " the only difference between them may be the order of idempotents immediately following each other as subwords. Equivalent canonical words can differ only about as much as the words xx^-1yy^-1 and yy^-1xx^-1, which are equivalent (they happen to be the only canonical words in their equivalence class).

All previously known constructions for free inverse semigroups follow easily from the new one. This is not true in the opposite direction. In particular, the known and new results about free inverse semigroups follow instantly if the new construction is used. As an example of a new result, I prove the Margolis conjecture about Munn's birooted trees.

Date received: June 1, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-41.