Relation Formulas for Protoalgebraic Equality Free Quasivarieties; Pałasińska’s Theorem Revisited

Abstract

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are \({\mathcal{Q}}\)-relation formulas for a protoalgebraic equality free quasivariety \({\mathcal{Q}}\) . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for \({\mathcal{Q}}\) when it has definable principal \({\mathcal{Q}}\)-subrelations. This is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties.

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Correspondence to Michał M. Stronkowski.

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Nurakunov, A.M., Stronkowski, M.M. Relation Formulas for Protoalgebraic Equality Free Quasivarieties; Pałasińska’s Theorem Revisited. Stud Logica 101, 827–847 (2013). https://doi.org/10.1007/s11225-013-9499-y

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Keywords

  • Equality free quasivariety
  • Protoalgebraicity
  • Relation distributivity
  • Finite axiomatization
  • Relation formulas
  • Definable principal subrelations

2000 Mathematics Subject Classification

  • 08C15
  • 08C10
  • 03G27