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Fuzzy ordered sets. (English) Zbl 0765.06008

The author introduces a definition of fuzzy ordered set (foset) \((P,\mu)\) and presents an example on the set of positive integers. He extends this concept to obtain a fuzzy lattice upon which he defines a (fuzzy) similarity relation as a generalization of equivalence. A few propositions of these topics are followed by a section on the fuzzy interval topology generated by fuzzy sets \(P\backslash \downarrow e\) and \(P\backslash \uparrow e\), where, for a fuzzy point \(e=x_ \lambda\), \(\downarrow e(y)=(\mu(y,x)+\lambda-1)\vee 0\) and \(\uparrow e(y)=(\mu(x,y)+\lambda-1)\vee 0\). All fuzziness is taken with respect to the unit interval.

MSC:

06A99 Ordered sets
54A40 Fuzzy topology
06B99 Lattices
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References:

[1] Bezdek, J.C.; Harris, J.D., Fuzzy partitions and relations; an axiomatic basis for clustering, Fuzzy sets and systems, 1, 111-127, (1978) · Zbl 0442.68093
[2] ErnĂ©, M., Separation axioms for interval topologies, (), 185-190 · Zbl 0398.54017
[3] Pao-Ming, Pu; Ying-Ming, Liu, Fuzzy topology I. neighborhood structure of a fuzzy point and Moore-Smith convergence, J. math. anal. appl., 76, 571-599, (1980) · Zbl 0447.54006
[4] Pao-Ming, Pu; Ying-Ming, Liu, Fuzzy topology II. product and quotient spaces, J. math anal. appl., 77, 22-37, (1980) · Zbl 0447.54007
[5] Zadeh, L.A., Similarity relations and fuzzy orderings, Inform. sci., 3, 177-200, (1971) · Zbl 0218.02058
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