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Canonical extensions for congruential logics with the deduction theorem |
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| Authors: | Mai Gehrke Alessandra Palmigiano |
| Affiliation: | a Radboud Universiteit Nijmegen, FNWI, IMAPP, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands b Dept. Lògica, Història i Filosofia, de la Ciència., Universitat de Barcelona, Montalegre, 6, 08001 Barcelona, Spain c FNWI, ILLC, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE Amsterdam, The Netherlands |
| Abstract: | We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart  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  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. |
| Keywords: | 03G27 06B23 06B15 06D20 |
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