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Associativity of triangular norms characterized by the geometry of their level sets
PetríK M., Sarkoci P. Fuzzy Sets and Systems202 100-109,2012.Type:Article
Date Reviewed: Sep 24 2012

The concepts of triangular norms (t-norm), triangular conorms (t-conorm), and complementation operations all play crucial roles in generalizing the standard operations of intersection, union, and complementation on crisp sets to the framework of fuzzy sets. This allows for the definition of more sophisticated types of aggregation operators in a series of new areas such as neurofuzzy and fuzzy control theory. This paper was inspired by the Reidemeister closure condition that characterizes the algebraic structures of groups.

The authors propose counterpart concepts to characterize the associativity of t-norms. Following a brief introduction and a section on the motivation of the work, the basic notions are introduced in the third section of the paper, including conjunctors (nondecreasing binary operations with neutral element 1), equivalence, and strongly and weakly aligned rectangles with respect to a given conjunctor.

Next, the paper presents a series of interesting characterizations for general t-norms, continuous t-norms, and continuous Archimedean t-norms. The characterization provided by Theorem 4.2 for a general conjunctor is expressed in terms of two binary relations--being strongly aligned and equivalent--defined on the set of a (1,1)-local rectangle. According to this result, a general conjunctor is associative if and only if the set of the strongly aligned relation is a subset of the equivalence relation. A similar characterization is given by Theorem 5.1 for continuous conjunctors. To obtain a similar characterization for continuous Archimedean t-norms, the authors use the concept of positive cancellativity, the binary operation F being positively cancellative if the equality F (x, y)= F (x, z) implies y=z whenever both (x, y), (x, z) belong to the support of F. The characterization provided by Theorem 6.4 refers to the class of continuous positively cancellative conjunctors in terms of the binary relations of being weakly aligned and equivalent, defined on the set of rectangles included in the support of the conjunctor. According to this result, a continuous positively cancellative conjunctor is associative if and only if the relation of being weakly aligned is a subset of the equivalence relation. In the final part of the paper, the authors formulate a series of open problems.

The results supplied in the paper are interesting, and enable insight into the relation between the associativity property of t-norms and the geometry of their level sets. Moreover, they could inspire researchers in the field of computational intelligence and related areas to further generalize and extend this research.

Reviewer:  L. State Review #: CR140558 (1302-0150)
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Geometric (I.5.1 ... )
 
 
Fuzzy Set (I.5.1 ... )
 
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