TY - JOUR
T1 - Canonical extensions for congruential logics with the deduction theorem
AU - Gehrke, Mai
AU - Jansana, Ramon
AU - Palmigiano, Alessandra
PY - 2010/9/1
Y1 - 2010/9/1
N2 - 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.
AB - 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.
KW - Abstract Algebraic Logic
KW - Canonical extensions
KW - Deduction theorem
KW - Hilbert algebras
KW - Tarski algebras
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U2 - 10.1016/j.apal.2010.05.003
DO - 10.1016/j.apal.2010.05.003
M3 - Article
AN - SCOPUS:77955553910
SN - 0168-0072
VL - 161
SP - 1502
EP - 1519
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 12
ER -