Canonical extensions for congruential logics with the deduction theorem

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart Alg S 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 Alg S 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.