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\(L\)-fuzzy sets and codes. (English) Zbl 0782.94012

Summary: A decomposition of an \(L\)-valued finite fuzzy set (\(L\) is a lattice) gives a family of characteristic functions, which can be considered as a binary block-code. Using a previous theorem of synthesis for fuzzy sets, we give conditions under which an arbitrary block-code corresponds to an \(L\)-valued fuzzy set. An explicit description of the Hamming distance, as well as of any code distance is also given, all in lattice-theoretic terms. Finally, we give necessary and sufficient conditions under which a linear code corresponds to an \(L\)-valued fuzzy set. It turns out that in such case the lattice \(L\) has to be Boolean.

MSC:

94B05 Linear codes (general theory)
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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References:

[1] Birkhoff, G.; Bartree, T. C., Modern Applied Algebra (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0215.31302
[2] Šešelja, B.; Vojvodić, G., Fuzzy sets on \(S\) as closure operators on \(P(S)\), Rev. Res. Fac. Sci. Univ. Novi Sad, 14, 1, 117-127 (1984) · Zbl 0577.04002
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