## Representability in interval-valued fuzzy set theory.(English)Zbl 1144.03033

Authors’ abstract: “Interval-valued fuzzy set theory is an increasingly popular extension of fuzzy set theory where traditional $$[0,1]$$-valued membership degrees are replaced by intervals in $$[0,1]$$ that approximate the (unknown) membership degrees. To construct suitable graded logical connectives in this extended setting, it is both natural and appropriate to “reuse” ingredients from classical fuzzy set theory. In this paper, we compare different ways of representing operations on interval-valued fuzzy sets by corresponding operations on fuzzy sets, study their intuitive semantics, and relate them to an existing, purely order-theoretical approach. Our approach reveals, amongst others, that subtle differences in the representation method can have a major impact on the properties satisfied by the generated operations, and that contrary to popular perception, interval-valued fuzzy set theory hardly corresponds to a mere twofold application of fuzzy set theory. In this way, by making the mathematical machinery behind the interval-valued fuzzy set model fully transparent, we aim to foster new avenues for its exploitation by offering application developers a much more powerful and elaborate mathematical toolbox than existed before.”

### MSC:

 3e+72 Theory of fuzzy sets, etc.
Full Text:

### References:

 [1] DOI: 10.1016/S0019-9958(65)90241-X · Zbl 0139.24606 [2] DOI: 10.1016/0022-247X(67)90189-8 · Zbl 0145.24404 [3] DOI: 10.1080/11663081.1999.10510967 · Zbl 0993.03079 [4] DOI: 10.1142/S0218488598000434 · Zbl 1087.68639 [5] Arieli O., IEEE Transactions on Fuzzy Systems [6] DOI: 10.1002/(SICI)1098-111X(199610)11:10<751::AID-INT3>3.0.CO;2-Y · Zbl 0865.04006 [7] Mendel J. M., Uncertain rule-based fuzzy logic systems (2001) · Zbl 0978.03019 [8] DOI: 10.1016/0165-0114(89)90205-4 · Zbl 0674.03017 [9] DOI: 10.1016/S0165-0114(96)00135-2 · Zbl 0922.03074 [10] DOI: 10.1016/S0888-613X(03)00072-0 · Zbl 1075.68089 [11] DOI: 10.1109/TFUZZ.2003.822678 · Zbl 05452531 [12] DOI: 10.1007/978-94-011-5300-3 [13] DOI: 10.1016/0165-0114(89)90215-7 · Zbl 0685.03037 [14] DOI: 10.1142/S021848850300248X · Zbl 1074.03022 [15] DOI: 10.1016/j.fss.2005.02.002 · Zbl 1090.03024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.