Lattices of fuzzy objects. (1996)

• Record Type: Journal Article
• Authors: Sangalli, Arturo A. L.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 19:Number 4(1996)
• Page Start: 759
• Page End: 766
• Publication Date: 1996
• Subjects: Complete lattices -- fuzzification -- fuzzy subalgebras; Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171296001056
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : The collection of fuzzy subsets of a setX forms a complete lattice that extends the complete lattice𝒫 ( X ) of crisp subsets ofX . In this paper, we interpret this extension as a special case of the “fuzzification” of an arbitrary complete latticeA . We show how to construct a complete latticeF ( A, L ) –theL -fuzzificatio ofA, whereL is the valuation lattice– that extendsA while preserving all suprema and infima. The “fuzzy” objects inF ( A, L ) may be interpreted as the sup-preserving maps fromA to the dual ofL . In particular, each complete lattice coincides with its2 -fuzzification, where2 is the twoelement lattice. Some familiar fuzzifications (fuzzy subgroups, fuzzy subalgebras, fuzzy topologies, etc.) are special cases of our construction. Finally, we show that the binary relations on a setX may be seen as the fuzzy subsets ofX with respect to the valuation lattice𝒫 ( X ) .