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Gehrke, Mai; Jansana, Ramon; Palmigiano, Alessandra

Canonical extensions for congruential logics with the deduction theorem. (English) Zbl 1238.03051

Ann. Pure Appl. Logic 161, No. 12, 1502-1519 (2010).
Authors’ abstract: “We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart Alg\(\mathcal{S}\) of any finitary and congruential logic \(\mathcal{S}\). 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 Alg\(\mathcal{S}\) are based on lattices. As a case study on logics purely based on implication, we prove that the variety of Hilbert and Tarski algebras are canonical in this new sense.”
Reviewer: Michiro Kondo (Inzai)

Cited in 1 Review
Cited in 8 Documents

MSC:

03G27 Abstract algebraic logic
06B15 Representation theory of lattices
06B23 Complete lattices, completions
06D20 Heyting algebras (lattice-theoretic aspects)

Keywords:

abstract algebraic logic; canonical extensions; deduction theorem; Hilbert algebras; Tarski algebras
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\textit{M. Gehrke} et al., Ann. Pure Appl. Logic 161, No. 12, 1502--1519 (2010; Zbl 1238.03051)
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