# Compatible extensions and consistent closures: a fuzzy approach

## TL;DR

This paper explores fuzzy relation extensions and consistent closures, generalizing classical theorems from crisp relations to fuzzy contexts, providing new theoretical insights and methods.

## Contribution

It introduces fuzzy compatible extensions and consistent closures, extending classical relation theorems to fuzzy relations and broadening their theoretical framework.

## Key findings

- Generalization of Duggan's results to fuzzy relations
- Fuzzy versions of Szpilrajn, Hansson, Suzumura theorems
- Introduction of two notions of consistent closure

## Abstract

In this paper $\ast$--compatible extensions of fuzzy relations are studied, generalizing some results obtained by Duggan in case of crisp relations. From this general result are obtained as particular cases fuzzy versions of some important extension theorems for crisp relations (Szpilrajn, Hansson, Suzumura). Two notions of consistent closure of a fuzzy relation are introduced.