## 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
Full Text:

### 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] Erné, M., Separation axioms for interval topologies, (Proc. Amer. Math. Soc., 79 (1980)), 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.