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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| Format: | Article |
| Language: | English |
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Wiley
1996-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171296001056 |
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| author | Arturo A. L. Sangalli |
| author_facet | Arturo A. L. Sangalli |
| author_sort | Arturo A. L. Sangalli |
| collection | DOAJ |
| description | 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). |
| format | Article |
| id | doaj-art-b2121f8cc188416da494446019dce017 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1996-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-b2121f8cc188416da494446019dce0172025-02-03T05:54:20ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251996-01-0119475976610.1155/S0161171296001056Lattices of fuzzy objectsArturo A. L. Sangalli0Department of Mathematics, Champlain Regional College, P.O. Box 5003, Lennoxville, Quebec JIM 2A1, CanadaThe 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).http://dx.doi.org/10.1155/S0161171296001056Complete latticesfuzzificationfuzzy subalgebras. |
| spellingShingle | Arturo A. L. Sangalli Lattices of fuzzy objects International Journal of Mathematics and Mathematical Sciences Complete lattices fuzzification fuzzy subalgebras. |
| title | Lattices of fuzzy objects |
| title_full | Lattices of fuzzy objects |
| title_fullStr | Lattices of fuzzy objects |
| title_full_unstemmed | Lattices of fuzzy objects |
| title_short | Lattices of fuzzy objects |
| title_sort | lattices of fuzzy objects |
| topic | Complete lattices fuzzification fuzzy subalgebras. |
| url | http://dx.doi.org/10.1155/S0161171296001056 |
| work_keys_str_mv | AT arturoalsangalli latticesoffuzzyobjects |