Equational theories of unstable involution semigroups

Edmond W. H. Lee

doi:10.3934/era.2017.24.002

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2017, Volume 24: 10-20. Doi: 10.3934/era.2017.24.002

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Equational theories of unstable involution semigroups

Edmond W. H. Lee

Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, Florida 33314, USA

The author is indebted to the referee for insightful comments and a thorough review. Results of the present article were announced in Workshop on Groups and Semigroups: on the occasion of the 60th birthday of Mikhail Volkov held at the University of Porto on June 9,2015

Received: December 07, 2016

Published: August 2017

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Abstract

It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.

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Mathematics Subject Classification: 20M05.

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