Discrete dualities for some algebras from rough sets

2026-01-09

arXiv.org

AI Summary

This study establishes a discrete duality between algebraic structures derived from rough sets and their corresponding relational semantics (frames). Building on the foundational work of Jónsson–Tarski, Kripke, and van Benthem, the paper systematically develops two classes of representation theorems by integrating tools from algebraic logic, relational semantics, and representation theory. These theorems reveal precise dualities between various rough set algebras and their associated relational frames. The work not only provides a rigorous algebraic–semantic bridge for rough set theory but also extends the semantic foundations of non-classical logics and advances the application of discrete duality theory to reasoning under uncertainty.

Technology Category

• Knowledge Representation and Reasoning: Other Foundations of Knowledge Representation & Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A discrete duality is a relationship between classes of algebras and classes of relational systems (frames) resulting in two representation theorems building on the early work of J\'onsson and Tarski, Kripke, and van Benthem. In this section we recall discrete dualities for various types of algebras arising from rough sets.

Problem

• discrete duality
• rough sets
• algebras
• relational systems
• representation theorems

Innovation

• discrete duality
• rough sets
• representation theorems
• relational frames
• algebraic semantics