×

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

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.”

MSC:

03G27 Abstract algebraic logic
06B15 Representation theory of lattices
06B23 Complete lattices, completions
06D20 Heyting algebras (lattice-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Balbes, R.; Dwinger, Ph., Distributive lattices, (1974), University of Missouri Press Missouri · Zbl 0321.06012
[2] Babyonyshev, S., Fully Fregean logics, Reports on mathematical logic, 37, (2003) · Zbl 1058.03016
[3] Dunn, J.M.; Gehrke, M.; Palmigiano, A., Canonical extensions and relational completeness of some substructural logics, Journal of symbolic logic, 70, 3, 713-740, (2005) · Zbl 1101.03021
[4] Font, J.M.; Jansana, R., A general algebraic semantics for sentential logics, (), vi+135 pp · Zbl 0865.03054
[5] Font, J.M.; Jansana, R.; Pigozzi, D., A survey of abstract algebraic logic. abstract algebraic logic, part II (Barcelona, 1997), Studia logica, 74, 1-2, 13-97, (2003) · Zbl 1057.03058
[6] Gehrke, M.; Jónsson, B., Bounded distributive lattices with operators, Mathematica japonica, 40, 2, 207-215, (1994) · Zbl 0855.06009
[7] Gehrke, M.; Jónsson, B., Monotone bounded distributive lattice expansions, Mathematica japonica, 52, 2, 197-213, (2000) · Zbl 0972.06005
[8] Gehrke, M.; Jónsson, B., Bounded distributive lattice expansions, Mathematica scandinavica, 94, 1, 13-45, (2004) · Zbl 1077.06008
[9] Gehrke, M.; Harding, J., Bounded lattice expansions, Journal of algebra, 238, 1, 345-371, (2001) · Zbl 0988.06003
[10] Gehrke, M.; Nagahashi, H.; Venema, Y., A sahlqvist theorem for distributive modal logic, Annals of pure and applied logic, 131, 1-3, 65-102, (2005) · Zbl 1077.03009
[11] Gehrke, M.; Priestley, H.A., Canonical extensions of double quasioperator algebras: an algebraic perspective on duality for certain algebras with binary operations, Journal of pure and applied algebra, 209, 1, 269-290, (2007) · Zbl 1110.06015
[12] Gehrke, M.; Priestley, H.A., Duality for double quasioperator algebras via their canonical extensions, Studia logica, 86, 1, 31-68, (2007) · Zbl 1127.06009
[13] Gehrke, M.; Priestley, H.A., Canonical extensions and completions of posets and lattices, Reports on mathematical logic, 43, 133-152, (2008) · Zbl 1147.06005
[14] M. Gehrke, R. Jansana, A. Palmigiano, Some \(\Delta_1\)-completions of a poset and the canonical extension, Preprint (2010).
[15] Jansana, R., Selfextensional logics with a conjunction, Studia logica, 84, 1, 63-104, (2006) · Zbl 1115.03094
[16] Jansana, R., Selfextensional logics with implication, (), 65-88 · Zbl 1081.03065
[17] Jónsson, B.; Tarski, A., Boolean algebras with operators, I & II, American journal of mathematics, 73, 891-939, (1951), 74 (1952) 127-162 · Zbl 0045.31505
[18] Köhler, P.; Pigozzi, D., Varieties with equationally definable principal congruences, Algebra universalis, 11, 2, 213-219, (1980) · Zbl 0448.08005
[19] Rasiowa, H., An algebraic approach to non-classical logics, (1974), North Holland Amsterdam · Zbl 0299.02069
[20] Wójcicki, R., Dual counterparts of consequence operations, Polish Academy of sciences. institute of philosophy and sociology. bulletin of the section of logic, 2, 1, 54-57, (1973)
[21] Wójcicki, R., Theory of logical calculi, (1988), Kluwer Ac. Pub. Dordrecht · Zbl 0682.03001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.