Scale-valued sets: a minimal framework for generalized set models

TL;DR

This paper introduces scale-valued sets (SV-sets), a unified framework for various generalized set models using lattice-valued functions, enabling richer representations and applications.

Contribution

It defines SV-sets as a minimal, flexible framework encompassing many existing set models, and explores their structure and applications.

Findings

SV-sets include ordinary, fuzzy, soft, and other set types as special cases.

SV-sets provide a natural topological and algebraic structure for complete chains and groups.

Applications demonstrate how SV-sets retain graded suitability and supporting evidence.

Abstract

Many generalized set models have the same basic form: they assign a value to each object, and the main difference lies in the kind of values that are allowed. This paper studies that common form through scale-valued sets (SV-sets), defined as maps $U\times E\to\Sigma$, where $U$ is a universe, $E$ is a parameter set, and $\Sigma$ is a bounded De Morgan lattice. With a suitable choice of scale, SV-sets include ordinary sets, fuzzy sets, soft sets, bounded multisets, intuitionistic fuzzy sets, $L$-fuzzy sets, and Type-2 fuzzy sets. We study the basic structure of SV-sets. The relation between SV-sets and lattice-valued interval soft sets is also discussed. For complete chains, the SV setting gives a natural topological construction, and for groups, it gives an algebraic structure through SV-subgroups. The applications show how graded suitability and supporting evidence can be kept together in a single model, whereas one-coordinate reductions lose information.