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LEE MONOIDS ARE NONFINITELY BASED WHILE THE SETS OF THEIR ISOTERMS ARE FINITELY BASED

Part of: Semigroups Varieties

Published online by Cambridge University Press:  28 March 2018

OLGA SAPIR Open the ORCID record for OLGA SAPIR [Opens in a new window]
OLGA SAPIR*
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA email olga.sapir@gmail.com
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Abstract

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We establish a new sufficient condition under which a monoid is nonfinitely based and apply this condition to Lee monoids $L_{\ell }^{1}$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ with the relation $0=abab\cdots \,$ (length $\ell$). We show that every monoid $M$ which generates a variety containing $L_{5}^{1}$ and is contained in the variety generated by $L_{\ell }^{1}$ for some $\ell \geq 5$ is nonfinitely based. We establish this result by analysing $\unicode[STIX]{x1D70F}$-terms for $M$, where $\unicode[STIX]{x1D70F}$ is a certain nontrivial congruence on the free semigroup. We also show that if $\unicode[STIX]{x1D70F}$ is the trivial congruence on the free semigroup and $\ell \leq 5$, then the $\unicode[STIX]{x1D70F}$-terms (isoterms) for $L_{\ell }^{1}$ carry no information about the nonfinite basis property of $L_{\ell }^{1}$.

Keywords

Lee monoidsidentityfinite basis problemnonfinitely basedvarietyisoterm

MSC classification

Primary: 20M07: Varieties and pseudovarieties of semigroups
Secondary: 08B05: Equational logic, Mal′cev (Mal′tsev) conditions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 422 - 434
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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