Ambiguous representations as fuzzy relations between sets

TL;DR

This paper introduces fuzzy and crisp ambiguous representations between closed subsets of compact Hausdorff spaces, exploring their lattice structures and categorical properties.

Contribution

It defines and analyzes the structure of ambiguous representations as lattices and categories, extending the concept to fuzzy relations.

Findings

The set of ambiguous representations forms a lattice and a compact Hausdorff Lawson upper semilattice.

Categories of ambiguous and fuzzy ambiguous representations are constructed and studied.

The paper generalizes the notion of representations between closed sets using fuzzy logic.

Abstract

Crisp and $L$-fuzzy ambiguous representations of closed subsets of one space by closed subsets of another space are introduced. It is shown that, for each pair of compact Hausdorff spaces, the set of (crisp or $L$-fuzzy) ambiguous representations is a lattice and a compact Hausdorff Lawson upper semilattice. The categories of ambiguous and $L$-ambiguous representations are defined and investigated.