The Suszko operator relative to truth‐equational logics

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 S 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 classLMod Su ( S )if and only if Alg ( S ) is a τ∞ ‐algebraic semantics for S and the Suszko operatorΩ ∼ S Fm : Th S → Co Alg ( S ) Fm preserves suprema and commutes with substitutions.