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In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies. [ABSTRACT FROM AUTHOR]

One of the most interesting aspects of Blok and Pigozzi's algebraizability theory is that the notion of algebraizable logic L 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 L and those of the equational consequence relative to its equivalent algebraic semantics K , the Semantic Isomorphism Theorem describes the interplay between the filters of L on an arbitrary algebra A and the congruences of A relative to K . 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. [ABSTRACT FROM AUTHOR]

The recent discovery that very different types of algebras have a quantale as an injective envelope is analyzed. Multi-posets are introduced as a generic structure that admits an essential embedding into a quantale, explicitly realized as a completion. Effect algebras and their non-commutative extensions, quantum B-algebras, KL -algebras, groups, and various other types of algebras are genuine multi-posets, in the sense that they determine full subcategories. Some structures like partially ordered groups or effect algebras do not carry over to the injective envelope. We provide a criterion for extendability in terms of a generalized archimedean property. Applied to non-commutative topology, multi-posets lead to a symmetric version of quantum spaces as a class of spatial quantales. [ABSTRACT FROM AUTHOR]

A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included. Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras. [ABSTRACT FROM AUTHOR]

This short note intends to replace a minor part of [2] where, due to an oversight of the author, an incorrect construction appears. [ABSTRACT FROM AUTHOR]