Categorical Abstract Algebraic Logic: Equivalence of Closure Systems

In their famous “Memoirs” monograph, Blok and Pigozzi defined algebraizable deductive systems as those whose consequence relation is equivalent to the algebraic consequence relation associated with a quasivariety of universal algebras. In characterizing this property, they showed that it is equivalent with the existence of an isomorphism between the lattices of theories of the two consequence relations that commutes with inverse substitutions. Thus emerged the prototypical and paradigmatic result relating an equivalence between two consequence relations established by means of syntactic translations and the isomorphism between corresponding lattices of theories. This result was subsequently generalized in various directions. Blok and Pigozzi themselves extended it to cover equivalences between k-deductive systems. Rebagliato and Verdú and, later, also Pynko and Raftery, considered equivalences between consequence relations on associative sequents. The author showed that it holds for equivalences between two term π-institutions. Blok and Jónsson considered equivalences between structural closure operations on regular M -sets. Gil-Férez lifted the author’s results to the case of multi-term π-institutions. Finally, Galatos and Tsinakis considered the case of equivalences between closure operators on A-modules and provided an exact characterization of those that are induced by syntactic translations. In this paper, we contribute to this line of research by further abstracting the results of Galatos and Tsinakis to the case of consequence systems on Sign-module systems, which are set-valued functors SEN : Sign → Set on complete residuated categories Sign.