Abstract
The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp and fuzzy multisets have found numerous applications but still the basic operations are based on functions min and max. It is straightforward to replace these functions in the fuzzy part of fuzzy multisets; however, it is not as easy but is feasible to do the same with the multisets and the “crisp” part of fuzzy multisets. The new mathematical structures are called triangular multisets and triangular fuzzy multisets, respectively. The aim is to facilitate the processing of certain data sets so they can be used in multi-criteria decision making and computing.
Highlights
In set theory an element either belongs to or does not belong to a set
The basic set operations between triangular fuzzy multisets are defined in the following definition
We have introduced the notion of triangular multisets and triangular multi-fuzzy sets as an obvious generalization of multisets and multi-fuzzy sets
Summary
In set theory an element either belongs to or does not belong to a set. Suppose we allow multiple copies of an element to be part of a “set”, new structures emerge. The basic set theoretic operations between fuzzy sets are defined using functions min, max. The purpose of this work is to introduce an extension, not a generalization, of both multisets and fuzzy multisets, where the union and the intersection between two such structures would be defined using functions other than min and max. If we choose not to use the unit interval but a totally ordered set (L, ≤, α, β), where α and β are the minimum and the maximum elements of L, such that α = β, we define discrete t-norms and t-conorms (see [17,18] for an overview and [19] for a discussion about the usefulness of these functions in the definition of a version of fuzzy numbers). It is easy to see that lcmext(m, n) is a triangular conorm on (N+∞,
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