Monteiro's algebraic notion of maximal consistent theory for Tarskian logics

Abstract

Antonio Monteiro gave a characterization of maximal congruences in certain semisimple varieties in order to present a representation theorem for them. Under specific conditions, this theorem can be presented with the same proof for every semisimple variety considered by him. In this note, we show that this notion of maximal congruence is closely linked to Henkin's notion of maximal theories for certain families of logics from the literature of algebraic logic. To see this relation, we study the class of n-valued Hilbert algebras with supremum enriched with Moisil operators. For this class of algebras, we present a sound and complete propositional and first-order calculus. Moreover, we show how this relation works for logics from semisimple varieties studied in the Monteiro's school. Extending the scope of applications, we present soundness and completeness results for some first-order paraconsistent logics through non-deterministic matrix semantics. Despite the fact that these logics are not algebraizable with the Blok-Pigozzi's method, they display an algebraic behaviour that allows us to give a simplified presentation.