# Algebraic Semantics of Proto-Transitive Rough Sets

arXiv: https://arxiv.org/abs/1410.0572 · HTML: https://mlsanity.com/paper/1410.0572/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2014.

## Built on by (most cited first, 3 of 3)

- [Algebraic Models for Qualified Aggregation in General Rough Sets, and Reasoning Bias Discovery](https://mlsanity.com/paper/2309.03217/) (2023)
- [Granular Generalized Variable Precision Rough Sets and Rational Approximations](https://mlsanity.com/paper/2205.14365/) (2022)
- [Granular Directed Rough Sets, Concept Organization and Soft Clustering](https://mlsanity.com/paper/2208.06623/) (2022)

## Abstract (arXiv)

Rough sets over generalized transitive relations like proto-transitive ones had been initiated by the present author in the year 2012. Subsequently, approximation of proto-transitive relations by other relations was investigated and the relation with rough approximations was developed towards constructing semantics that can handle fragments of structure. It was also proved that difference of approximations induced by some approximate relations need not induce rough structures. In this research we develop different semantics of proto transitive rough sets (PRAX) after characterizing the structure of rough objects and also develop a theory of dependence for general rough sets and use it to internalize the Nelson-algebra based approximate semantics developed earlier. The theory of rough dependence initiated later by the present author is extended in the process. This monograph is reasonably self-contained and includes proofs and extensions of representation of objects that were not part of earlier papers.