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Gentzen-style axiomatizations in equational logic

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Abstract

The notion of a Gentzen-style axiomatization of equational theories is presented. In the standard deductive systems for equational logic axioms take the form of equations and the inference rules can be viewed as quasi-equations. In the deductive systems for quasi-equational logic the axioms, which are quasi-equations, can be viewed as sequents and the inference rules as Gentzen-style rules. It is conjectured that every finite algebra has a finite Gentzen-style axiomatization for its quasi-identities. We verify this conjecture for a class of algebras that includes all finite algebras without proper subalgebras and all finite simple algebras that are embeddable into the free algebra of their variety.

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Authors and Affiliations

  1. Department of Mathematics, Iowa State University, 50011, Ames, IA, USA

    K. Palasińska & D. Pigozzi

  2. Instytut Matematyki, Politechnika Krakowska, ul. Warszawska 24, Krakow, Poland

    K. Palasińska & D. Pigozzi

Authors
  1. K. Palasińska
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  2. D. Pigozzi
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Additional information

Dedicated to the memory of Alan Day

Supported by an Iowa State University Research Assistantship.

Supported by National Science Foundation Grant #DMS 8005870.

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Palasińska, K., Pigozzi, D. Gentzen-style axiomatizations in equational logic. Algebra Universalis 34, 128–143 (1995). https://doi.org/10.1007/BF01200495

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  • DOI: https://doi.org/10.1007/BF01200495

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