Discrete dualities for some algebras from rough sets

TL;DR

The work surveys discrete dualities between algebraic classes and relational frames inspired by rough-set theory, focusing on two main construction paths: monadic algebras from equivalence-based approximations and lattice-based algebras from rough sets, including regular double Stone and De Morgan algebras. It develops canonical-structure and complex-algebra representations, establishing bidirectional embeddings and representation theorems that link algebras with their canonical frames and vice versa. Key contributions include precise dualities for monadic, sufficiency, regular double Stone, De Morgan, and rough relation algebras, along with the notion of rough-set frames and the corresponding algebraic structures. The results provide a unified framework for characterizing approximation spaces and offer a foundation for extending discrete dualities to broader information systems with varying degrees of determinism.

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 Jónsson and Tarski, Kripke, and van Benthem. In this section we recall discrete dualities for various types of algebras arising from rough sets.

Theorems & Definitions (16)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 4.1