DUALITIES AND REPRESENTATIONS FOR MANY-VALUED LOGICS IN THE HIERARCHY OF WEAK NILPOTENT MINIMUM.

In this thesis we study particular subclasses of WNM algebras. The variety of WNM algebras forms the algebraic semantics of the WNM logic, a propositional many-valued logic that generalizes some well-known case in the setting of triangular norms logics. WNM logic lies in the hierarchy of schematic extensions of MTL, which is proven to be the logic of all left-continuous triangular norms and their residua. In this work, I have extensively studied two extensions of WNM logic, namely RDP logic and NMG logic, from the point of view of algebraic and categorical logic. We develop spectral dualities between the varieties of algebras corresponding to RDP logic and NMG logic, and suitable defined combinatorial categories. Categorical dualities allow to give algorithmic construction of products in the dual categories obtaining computable descriptions of coproducts (which are notoriously hard to compute working only in the algebraic side) for the corresponding finite algebras. As a byproduct, representation theorems for finite algebras and free finitely generated algebras in the considered varieties are obtained. This latter characterization is especially useful to provide explicit construction of a number of objects relevant from the point of view of the logical interpretation of the varieties of algebras: normal forms, strongest deductive interpolants and most general unifiers.