The semantic isomorphism theorem in abstract algebraic logic

One of the most interesting aspects of Blok and Pigozzi's algebraizability theory is that the notion of algebraizable logic LL can be characterised by means of Syntactic and Semantic Isomorphism Theorems. While the Syntactic Isomorphism Theorem concerns the relation between the theories of the algebraizable logic LL and those of the equational consequence relative to its equivalent algebraic semantics KK, the Semantic Isomorphism Theorem describes the interplay between the filters of LL on an arbitrary algebra A and the congruences of A   relative to KK. The pioneering insight of Blok and Jónsson, and the further generalizations by Galatos, Tsinakis, Gil-Férez and Russo, showed that the concept of algebraizability was not intrinsic to the connection between a logic and an equational consequence, thus inaugurating the abstract theory of equivalence between structural closure operators. However all these works focus only on the Syntactic Isomorphism Theorem, disregarding the semantic aspects present in the original theory. In this paper we fill this gap by introducing the notion of compositional lattice, which acts on a category of evaluational frames. In this new framework the non-linguistic flavour of the Semantic Isomorphism Theorem can be naturally recovered. In particular, we solve the problem of finding sufficient and necessary conditions for transferring a purely syntactic equivalence to the semantic level as in the Semantic Isomorphism Theorem.