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Representing ordered structures by fuzzy sets: An overview. (English) Zbl 1026.03039

Summary: We present a survey on representations of ordered structures by fuzzy sets. Any poset satisfying some finiteness condition, semilattice, lattice belonging to a special class, e.g., distributive, Noetherian, complete and others – can be represented by a single function, i.e., by a fuzzy set. Its domain and co-domain are particular subsets of the same structure, and consist of irreducible elements. The representation is minimal in the sense that another representation could not be obtained by replacing the domain of the former by a proper subset. By this approach, the structure itself is uniquely represented by the collection of cuts ordered dually to inclusion.

MSC:

03E72 Theory of fuzzy sets, etc.
06D72 Fuzzy lattices (soft algebras) and related topics
08A72 Fuzzy algebraic structures
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