## 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)
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### 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
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