The Suszko operator relative to truth‐equational logics

Abstract

This note presents some new results from [1] about the Suszko operator and truth‐equational logics, following the works of Czelakowski [11] and Raftery [17]. It is proved that the Suszko operator relative to a truth‐equational logic scriptS preserves suprema and commutes with endomorphisms. Together with injectivity, proved by Raftery in [17], the Suszko operator relative to a truth‐equational logic is a structural representation, as defined in [15]. Furthermore, if Alg(S) is a quasivariety, then the Suszko operator relative to a truth‐equational logic is continuous. Finally, it is proved that truth is equationally definable in the class sans-serifLModSufalse(scriptSfalse) if and only if Alg(S) is a τ-0.16em-0.16em∞‐algebraic semantics for scriptS and the Suszko operator boldΩ∼scriptSbold-italicFm:ThscriptS→CoAlg(S)Fm preserves suprema and commutes with substitutions.