Canonical extensions for congruential logics with the deduction theorem

Mai Gehrke, Ramon Jansana, Alessandra Palmigiano*

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart AlgS of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in AlgS are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.

Original languageEnglish
Pages (from-to)1502-1519
Number of pages18
JournalAnnals of Pure and Applied Logic
Volume161
Issue number12
DOIs
Publication statusPublished - 1 Sep 2010
Externally publishedYes

Keywords

  • Abstract Algebraic Logic
  • Canonical extensions
  • Deduction theorem
  • Hilbert algebras
  • Tarski algebras

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