Lattices of fuzzy objects
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)...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1996-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171296001056 |
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| Summary: | 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). |
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| ISSN: | 0161-1712 1687-0425 |