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Relational correspondences for L-fuzzy rough approximations defined on De Morgan Heyting algebras

Jouni Järvinen, Michiro Kondo

TL;DR

This work addresses how to characterize and relate fuzzy rough approximations defined on complete De Morgan Heyting algebras to underlying fuzzy relations. It introduces a general, single-axiom framework that yields correspondences between $L$-fuzzy rough operators and their defining $L$-relations, including compositions of reflexive, transitive, mediate, Euclidean, adjoint, functional, alliance, and serial properties. A core contribution is a general existence-uniqueness theorem that derives a unique $L$-relation $R$ from a finite composition of operators, via the rule $\mathsf{U}(I_y)(x) = R(x,y)$, and a series of precise axiomatic characterizations for reflexive, symmetric, transitive, mediate, Euclidean, adjoint, functional, and serial relations. The paper also solves an open problem from Pang2019 by providing unified single-axiom criteria for a broad class of compositions, and it clarifies the landscape of serial and symmetric cases, offering new insights into how to recover relations from approximation operators in this non-classical fuzzy setting.

Abstract

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras by using only a single axiom.

Relational correspondences for L-fuzzy rough approximations defined on De Morgan Heyting algebras

TL;DR

This work addresses how to characterize and relate fuzzy rough approximations defined on complete De Morgan Heyting algebras to underlying fuzzy relations. It introduces a general, single-axiom framework that yields correspondences between -fuzzy rough operators and their defining -relations, including compositions of reflexive, transitive, mediate, Euclidean, adjoint, functional, alliance, and serial properties. A core contribution is a general existence-uniqueness theorem that derives a unique -relation from a finite composition of operators, via the rule , and a series of precise axiomatic characterizations for reflexive, symmetric, transitive, mediate, Euclidean, adjoint, functional, and serial relations. The paper also solves an open problem from Pang2019 by providing unified single-axiom criteria for a broad class of compositions, and it clarifies the landscape of serial and symmetric cases, offering new insights into how to recover relations from approximation operators in this non-classical fuzzy setting.

Abstract

We consider fuzzy rough sets defined on De Morgan Heyting algebras. We present a theorem that can be used to obtain several correspondence results between fuzzy rough sets and fuzzy relations defining them. We characterize fuzzy rough approximation operators corresponding to compositions of reflexive, transitive, mediate, Euclidean and adjoint fuzzy relations defined on De Morgan Heyting algebras by using only a single axiom.
Paper Structure (12 sections, 31 theorems, 77 equations, 1 figure)

This paper contains 12 sections, 31 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binary relations and rough approximation operators
  4. Approximation operators in De Morgan Heyting algebras
  5. Correspondence results
  6. A general result
  7. Reflexive and symmetric relations
  8. Transitive and mediate relations
  9. Euclidean and adjoint relations
  10. Functional and alliance relations
  11. Serial relations
  12. From L-approximations to L-relations

Key Result

Proposition 2.3

Let $U$ be a set, $L$ a complete De Morgan Heyting algebra and $R$ an $L$-relation on $U$. For $\{A_i\}_{i \in I} \subseteq \mathcal{F}_L(U)$, $A \in \mathcal{F}_L(U)$ and $x \in U$, the following assertions hold:

Figures (1)

  • Figure 1: Positive alliance relation $\rho$. Elements of $U = \{1,2,3,4\}$ are represented with circles, and if an element $x$ is $\rho$-related to an element $y$, there is an arrow from the circle representing $x$ to the circle representing $y$.

Theorems & Definitions (77)

  • Example 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • ...and 67 more