A new completeness proof at the propositional level for mbC and some of its extensions

Abstract Antonio Monteiro developed several techniques for the study of algebraic systems. One of the most important is the characterization of congruences through deductive systems for certain semisimple varieties of algebras. This characterization allowed for the development of a Representation Theorem, which generalizes the corresponding one for Boolean algebras. In [31, p. 18], the notion of Systèmes deductifs liés à “$a$”, where $a$ is an element of a given algebra, can be found. In the case that the lattice is a Boolean algebra, this notion of deductive system characterizes the maximal congruences. Figallo–Orellano and Slagter [24] and Slagter [37] presented a technique to perform completeness proofs in various classes of algebras, relating linked deductive systems with maximal consistent theories, using the notion of consistent theory given by H. Rasiowa in [36] and the notion of the maximal consistent theory given by Wókcicki in [40]. This new technique allows for completeness proofs that link the aforementioned relationship and the first isomorphism theorem. In [2], Avron presented the first Nmatrices for $\textbf{mbC}$. Then, in [7], Swap structures were introduced as a generalization of these Nmatrices. Subsequently, in [10], the first Representation Theorem for Swap structures was established, showing that Avron’s 5-valued matrix for $\textbf{mbC}$ behaves as a subdirectly irreducible algebra.