Lattices of fuzzy objects

Arturo A. L. Sangalli

International Journal of Mathematics and Mathematical Sciences, 1996, vol. 19, 1-8

Abstract:

The collection of fuzzy subsets of a set X forms a complete lattice that extends the complete lattice 𝒫 ( X ) of crisp subsets of X . In this paper, we interpret this extension as a special case of the fuzzification of an arbitrary complete lattice A . We show how to construct a complete lattice F ( A , L ) the L -fuzzificatio of A , where L is the valuation lattice that extends A while preserving all suprema and infima. The fuzzy objects in F ( A , L ) may be interpreted as the sup-preserving maps from A to the dual of L . In particular, each complete lattice coincides with its 2 -fuzzification, where 2 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 set X may be seen as the fuzzy subsets of X with respect to the valuation lattice 𝒫 ( X ) .

Date: 1996
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:970953

DOI: 10.1155/S0161171296001056

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