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Lattices of fuzzy objects

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  • Arturo A. L. Sangalli

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 ) .

Suggested Citation

  • Arturo A. L. Sangalli, 1996. "Lattices of fuzzy objects," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 19, pages 1-8, January.
    • Handle: RePEc:hin:jijmms:970953
    DOI: 10.1155/S0161171296001056
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