A Note on Categories about Rough Sets

TL;DR

This paper explores categorical structures related to rough sets, establishing properties and relationships among categories of approximation spaces, rough closure and interior spaces, and information systems using functorial approaches.

Contribution

It introduces categorical frameworks for rough sets and information systems, proving intrinsic properties and linking different categories via functors and subcategories.

Findings

Proves an intrinsic property of the category of approximation spaces.

Establishes relationships between categories of rough closure and interior spaces.

Defines a subcategory of information systems with non-expensive homomorphisms.

Abstract

Using the concepts of category and functor, we provide some insights and prove an intrinsic property of the category ${\bf AprS}$ of approximation spaces and relation-preserving functions, the category ${\bf RCls}$ of rough closure spaces and continuous functions, and the category ${\bf RInt}$ of rough interior spaces and continuous functions. Furthermore, we define the category ${\bf IS}$ of information systems and O-A-D homomorphisms, and establish the relationship between the category ${\bf IS}$ and the category ${\bf AprS}$ by considering a subcategory ${\bf NeIS}$ of ${\bf IS}$ whose objects are information systems and whose arrows are non-expensive O-A-D homomorphisms with surjective attribute mappings.