Duality for finitely valued algebras
Marco Abbadini, Adam Přenosil
TL;DR
The work extends Stone-like dualities to infinite dualizing objects by focusing on finitely valued $oldsymbol{L}$-algebras and their representations as algebras of finite-range $oldsymbol{L}$-valued functions, via $ ext{$ ext{FinRng}$}(X,oldsymbol{L})$. It introduces $oldsymbol{L}$-spaces and $oldsymbol{L}$-constrained spaces to realize a two-tier duality: a CD duality between finitely valued $oldsymbol{L}$-algebras and compact separated $oldsymbol{L}$-spaces, followed by the Baker–Pixley representation that recasts the spatial side as $oldsymbol{L}$-Priestley spaces. Under near-unanimity/baker–pixley hypotheses, it yields the Near Unanimity Duality (NU) that specializes to Priestley duality for bounded distributive lattices and to Cignoli–Marra/Cignoli–Dubuc–Mundici dualities for MV-algebras, while subsuming positive MV-algebras as a common generalization. The framework provides universal-algebraic formulations that extend classical Stone-type dualities beyond finite dualizing objects and clarifies when local-to-global principles suffice to recover global extensions. The results have MV-algebra consequences, unify several dualities under a single theory, and offer a versatile toolkit for analyzing finitely valued logics with infinite dualizing objects.
Abstract
The theory of natural dualities provides a well-developed framework for studying Stone-like dualities induced by an algebra $\mathbf{L}$ which acts as a dualizing object when equipped with suitable topological and relational structure. The development of this theory has, however, largely remained restricted to the case where $\mathbf{L}$ is finite. Motivated by the desire to provide a universal algebraic formulation of the existing duality of Cignoli and Marra or locally weakly finite MV-algebras and to extend it to a corresponding class of positive MV-algebras, in this paper we investigate Stone-like dualities where the algebra $\mathbf{L}$ is allowed to be infinite. This requires restricting our attention from the whole prevariety generated by $\mathbf{L}$ to the subclass of algebras representable as algebras of $\mathbf{L}$-valued functions of finite range, a distinction that does not arise in the case of finite $\mathbf{L}$. Provided some requirements on $\mathbf{L}$ are met, our main result establishes a categorical duality for this class of algebras, which covers the above cases of MV-algebras and positive MV-algebras.