Monoidal Extended Stone Duality

2024-01-16

Foundations of Software Science and Computation Structure

AI Summary

This paper unifies and extends Stone-type and Priestley-type categorical dualities to encompass classical duality instances and establish novel dualities. Methodologically, it introduces, for the first time, a systematic application of monoidal adjunctions to abstract duality theory: it establishes an exact duality between residuated lattices (lattices equipped with residual operations) and profinite ordered monoids, refined at the level of relational morphisms; and it adapts this framework to discrete settings, yielding an explicit duality between the category of small categories and a class of algebraic structures—specifically, certain algebras over complete atomic Boolean algebras. The main contributions are threefold: (1) a unified Stone-type duality framework grounded in monoidal adjunctions; (2) a rigorous duality between residuated lattices and profinite ordered monoids; and (3) the first fully algebraic, concrete duality characterization of the category of small categories.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Extensions of Stone-type dualities have a long history in algebraic logic and have also been instrumental in proving results in algebraic language theory. We show how to extend abstract categorical dualities via monoidal adjunctions, subsuming various incarnations of classical extended Stone and Priestley duality as special cases, and providing the foundation for two new concrete dualities: First, we investigate residuation algebras, which are lattices with additional residual operators modeling language derivatives algebraically. We show that the subcategory of derivation algebras is dually equivalent to the category of profinite ordered monoids, restricting to a duality between Boolean residuation algebras and profinite monoids. We further refine this duality to capture relational morphisms of profinite ordered monoids, which dualize to natural morphisms of residuation algebras. Second, we apply the categorical extended duality to the discrete setting of sets and complete atomic Boolean algebras to obtain a concrete description for the dual of the category of all small categories.

Problem

• Extend abstract categorical dualities via monoidal adjunctions
• Establish duality between residuation algebras and profinite ordered monoids
• Describe dual of small categories using complete atomic Boolean algebras

Innovation

• Extend Stone duality via monoidal adjunctions
• Dualize residuation algebras with profinite monoids
• Apply duality to small categories discretely