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Partial orderings for hesitant fuzzy sets
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DOI：10.1016/j.ijar.2017.02.008.png)
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Abstract
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New partial orderings <=(0), <=(p) and <=(H) are defined, studied and compared on the set H of finite subsets of the unit interval with special emphasis on the last one. Since comparing two sets of the same cardinality is a simple issue, the idea for comparing two sets A and B of different cardinalities n and m respectively using <=(H) is repeating their elements in order to obtain two series with the same length. If lcm(n, m) is the least common multiple of n and m we can repeat every element of A lcm(n, m)/m times and every element of B lcm(n,m)/n times to obtain such series and compare them (Definition 2.2). (H, <=(H)) is a bounded partially ordered set but not a lattice. Nevertheless, it will be shown that some interesting subsets of (H, <=(H)) have a lattice structure. Moreover in the set it of finite bags or multisets (i.e. allowing repetition of objects) of the unit interval a preorder <=(B) can be defined in a similar way as <=(H) in H and considering the quotient set (B) over bar = B/similar to of B by the equivalence relation similar to defined by A similar to B when A <=(B) B and B <=(B) A, ((B) over bar, <=(B)) is a lattice and (H, <=(H)) can be naturally embedded into it. (C) 2017 Elsevier Inc. All rights reserved.
Keywords:
Hesitant fuzzy sets
Finite subsets of the unit interval
Partial ordering
T-norm
Fuzzy conjunction
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