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 class LModSu(S) if and only if Alg(S) is a tau infinity-algebraic semantics for S and the Suszko operator omega similar to SFm:ThS -> CoAlg(S)Fm preserves suprema and commutes with substitutions.