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ON IDENTITIES OF REES QUOTIENTS OF FREE INVERSE SEMIGROUPS DEFINED BY POSITIVE RELATORS

Part of: Semigroups

Published online by Cambridge University Press:  10 March 2025

D. EASDOWN Open the ORCID record for D. EASDOWN [Opens in a new window]  and
L. M. SHNEERSON
D. EASDOWN*
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia
L. M. SHNEERSON
Affiliation:
Hunter College, City University of New York, 695 Park Avenue, New York, NY 10065, USA e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We give a complete description of Rees quotients of free inverse semigroups given by positive relators that satisfy nontrivial identities, including identities in signature with involution. They are finitely presented in the class of all inverse semigroups. Those that satisfy a nontrivial semigroup identity have polynomial growth and can be given by an irredundant presentation with at most four relators. Those that satisfy a nontrivial identity in signature with involution, but which do not satisfy a nontrivial semigroup identity, have exponential growth and fall within two infinite families of finite presentations with two generators. The first family involves an unbounded number of relators and the other involves presentations with at most four relators of unbounded length. We give a new sufficient condition for which a finite set X of reduced words over an alphabet $A\cup A^{-1}$ freely generates a free inverse subsemigroup of $FI_A$ and use it in our proofs.

Keywords

free inverse semigroupRees quotientinverse semigroup presentationgrowthidentities

MSC classification

Primary: 20M18: Inverse semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by James East

The authors were supported by the Research Foundation of CUNY (Grant TRADA-54-217).

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