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Quantales for Fuzzy Sets and Relations of Higher Types

Author

Listed:
  • 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
    Retired.)

  • Magnus Löfstrand

    (School of Science and Technology, Örebro University, SE-701 82 Örebro, Sweden)

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”.

Suggested Citation

  • Patrik Eklund & Jari Kortelainen & Magnus Löfstrand, 2025. "Quantales for Fuzzy Sets and Relations of Higher Types," Mathematics, MDPI, vol. 13(13), pages 1-26, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:13:p:2159-:d:1692557
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