Quantales for Fuzzy Sets and Relations of Higher Types

Patrik Eklund (), Jari Kortelainen and Magnus Löfstrand
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Patrik Eklund: Department of Computing Science, Umeå University, SE-901 87 Umeå, Sweden
Jari Kortelainen: Department of Information Technology, South-Eastern Finland University of Applied Sciences, FI-50100 Mikkeli, Finland
Magnus Löfstrand: School of Science and Technology, Örebro University, SE-701 82 Örebro, Sweden

Mathematics, 2025, vol. 13, issue 13, 1-26

Abstract: In this paper, we open up more possibilities to define higher types of fuzzy sets in a mixed way. In doing so, we show that there are essentially two alternative definitions for fuzzy sets of higher types, one of which is widely adopted in the literature, and one where the unit interval, or subsets thereof, is used to represent membership values. The other alternative definition opens up new perspectives for the use of fuzzy sets of higher types, and it promotes the use of other algebraic structures of sets of membership values, where quantales are seen as particularly useful, also in applications. The paper also underlines the importance of making distinctions between “computing with fuzzy” and “fuzzy computing” and understanding the difference between “logic with fuzzy” and “fuzzy logic”.

Keywords: fuzzy set of higher type; fuzzy term; quantale; design structure (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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