## Representations with fuzzy darts.(English)Zbl 0926.94044

The problem raised in this study is concerned with fuzzy darts. Fuzzy darts are developed based on the notion of fuzzy level sets. A fuzzy level set over $${\mathbf X}$$ is defined as a structure $$\{(r,A_r) \mid r \in [0,1], A_r\subset {\mathbf X}\}$$. That is $$(r,A_r)$$ associates with an elementary fuzzy set in $${\mathbf X}$$ of support $$A_r$$ and height “$$r$$”. A fuzzy number can be represented using level fuzzy sets, namely $$[f,g]= \{(r,A_r) \mid$$ for $$r\in (0,1)]\}$$, meaning that $$A_r= [f(r), g(r)]$$ for $$r\in (0,1]$$. Subsequently, a fuzzy dart is a fuzzy number $$[f,g]$$ such that: (i) $$f$$ and $$g$$ are real valued linear functions on $$[0,1]$$, (ii) either $$f$$ or $$g$$ is a constant valued function on $$[0,1]$$, (iii) $$f\leq g$$ on $$[0,1]$$, (iv) $$f(1)= g(1)$$.
By partitioning the unit interval into “$$n$$” layers of fuzzy strata, fuzzy information can be organized into $$n$$-tuples of fuzzy darts. The main properties of fuzzy darts are revealed. A number of detailed examples are also discussed.

### MSC:

 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)

### Keywords:

fuzzy numbers; vector spaces; fuzzy darts; fuzzy level sets
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### References:

 [1] Dubois, D.; Prade, H., Operations on fuzzy numbers, Internat. J. Systems Sci., 9 (1978) · Zbl 0383.94045 [2] Goetschel, R.; Voxman, W., Topological properties of fuzzy numbers, Fuzzy Sets and Systems, 9, 87-99 (1983) · Zbl 0521.54001 [3] Goetschel, R.; Voxman, W., Elementary fuzzy calculus, Fuzzy Sets and Systems, 18, 31-43 (1986) · Zbl 0626.26014
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