Many-valued aspects of tense an related operators
Michal Botur, Jan Paseka, Richard Smolka
TL;DR
The paper generalizes tense-operator theory to quantale-valued settings by developing three core constructions over a commutative unital quantale ${\mathbf V}$: ${\mathbf A}^{\mathbf J}$ from a ${\mathbf V}$-module and a ${\mathbf V}$-frame, ${\mathbf J} \otimes {\mathbf H}$ from a ${\mathbf V}$-frame and a ${\mathbf V}-\textnormal{F}$-semilattice, and ${\mathbf J}[{\mathbf H},{\mathbf L}]$ from a ${\mathbf V}-\textnormal{F}$-semilattice and a ${\mathbf V}$-module. These give rise to three adjoint situations $(\eta,\varepsilon)$, $(\varphi,\psi)$, and $(\nu,\mu)$ between categories of ${\mathbf V}$-modules, ${\mathbf V}$-frames, and ${\mathbf V}-\textnormal{F}$-semilattices, extending Bot's results to arbitrary commutative unital quantales. The framework unifies algebra, logic, and topology by treating fuzzy relations and tense operators via ${\mathbf V}$-frames and ${\mathbf V}$-modules, with natural transformations encoding the tense-operator interactions. A prototypical example using ${\mathbf M}_3$ illustrates the constructions and shows limitations such as non-injectivity of a key morphism, highlighting both the richness and boundaries of the theory. Overall, the work offers a rigorous, category-theoretic lens on tense-operator representations in non-classical logics and potential connections to quantum logic and locale theory.
Abstract
Our research builds upon Halmos's foundational work on functional monadic Boolean algebras and our previous work on tense operators to develop three essential constructions, including the important concepts of fuzzy sets and powerset operators. These constructions have widespread applications across contemporary mathematical disciplines, including algebra, logic, and topology. The framework we present generates four covariant and two contravariant functors, establishing three adjoint situations.