Partial orderings for hesitant fuzzy sets

New partial orderings =o=o, =p=p and =H=H are defined, studied and compared on the set HH 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=H is repeating their elements in order to obtain two series with the same length. If lcm(n,m)lcm(n,m) is the least common multiple of n and m we can repeat every element of A lcm(n,m)/mlcm(n,m)/m times and every element of B lcm(n,m)/nlcm(n,m)/n times to obtain such series and compare them (Definition 2.2). (H,=H)(H,=H) is a bounded partially ordered set but not a lattice. Nevertheless, it will be shown that some interesting subsets of (H,=H)(H,=H) have a lattice structure. Moreover in the set BB of finite bags or multisets (i.e. allowing repetition of objects) of the unit interval a preorder =B=B can be defined in a similar way as =H=H in HH and considering the quotient set View the MathML sourceB¿=B/~ of BB by the equivalence relation ~ defined by A~BA~B when A=BBA=BB and B=BAB=BA, View the MathML source(B¿,=B) is a lattice and (H,=H)(H,=H) can be naturally embedded into it.Peer ReviewedPostprint (author's final draft

Construction of aggregation operators with noble reinforcement

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements. <br /

Generalized and Customizable Sets in R

We present data structures and algorithms for sets and some generalizations thereof (fuzzy sets, multisets, and fuzzy multisets) available for R through the sets package. Fuzzy (multi-)sets are based on dynamically bound fuzzy logic families. Further extensions include user-definable iterators and matching functions.

First-order Nilpotent Minimum Logics: first steps

Following the lines of the analysis done in [BPZ07, BCF07] for first-order G\"odel logics, we present an analogous investigation for Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra. We establish a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, we analyze axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The introduction section has been rewritten, and many modifications have been done to improve the readability; moreover, numerous references have been added. Concerning the technical side, some proofs has been shortened or made more clear, but the mathematical content is substantially the same of the previous versio

Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft