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Equational Theories for Classes of Finite Semigroups

Authors:	V. Yu. Popov

Affiliation:	(1) Marshala Zhukova 11-6, Ekaterinburg, 620077

Abstract:	It is proved that there exists an infinite sequence of finitely based semigroup varieties $mathfrak{A}_{text{1}} subset mathfrak{B}_{text{1}} subset mathfrak{A}_{text{2}} subset mathfrak{B}_{text{2}} subset ...$ such that, for all i, an equational theory for $mathfrak{A}_i$ and for the class $mathfrak{A}_i cap mathfrak{F}$ of all finite semigroups in $mathfrak{A}_i$ is undecidable while an equational theory for $mathfrak{B}_i$ and for the class $mathfrak{B}_i cap mathfrak{F}$ of all finite semigroups in $mathfrak{B}_i$ is decidable. An infinite sequence of finitely based semigroup varieties $mathfrak{A}_{text{1}} supset mathfrak{B}_i supset mathfrak{A}_{text{2}} supset mathfrak{B}_{text{2}} supset ...$ is constructed so that, for all i, an equational theory for $mathfrak{B}_i$ and for the class $mathfrak{B}_i cap mathfrak{F}$ of all finite semigroups in $mathfrak{B}_i$ is decidable whicle an equational theory for $mathfrak{A}_i$ and for the class $mathfrak{A}_i cap mathfrak{F}$ of all finite semigroups in $mathfrak{A}_i$ is not.

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