ISSN :2582-9793

Representation of a Crisp Set as a Pair of Dual Fuzzy Sets

Original Research (Published On: 17-Nov-2022 )
Representation of a Crisp Set as a Pair of Dual Fuzzy Sets
DOI : 10.54364/AAIML.2022.1133

Gia Sirbiladze

Adv. Artif. Intell. Mach. Learn., 2 (4):477-500

Gia Sirbiladze : phd

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DOI: 10.54364/AAIML.2022.1133

Article History: Received on: 24-Sep-22, Accepted on: 30-Oct-22, Published on: 17-Nov-22

Corresponding Author: Gia Sirbiladze


Citation: Gia Sirbiladze (2022). Representation of a Crisp Set as a Pair of Dual Fuzzy Sets. Adv. Artif. Intell. Mach. Learn., 2 (4 ):477-500



Expert knowledge representations often fail to determine compatibility levels on all objects, and these levels are represented for a certain sampling of universe. The samplings for the fuzzy terms of the linguistic variable, whose compatibility functions are aggregated according to a certain problem, may also be different. In such a case, neither L.A. Zadeh's analysis of fuzzy sets and even the dual forms of developing today R.R. Yager’s q-rung orthopair fuzzy sets cannot provide the necessary aggregations. This fact, as a given, can be considered as a source of new types of information, in order to obtain different levels of compatibility according to Zadeh, presented throughout the universe. This source of information can be represented as a pair , A fA , where there is some crisp subset of the universe A that determines the sampling of objects from the universe, and a function A f determines the compatibility levels of the elements of that sampling. It is a notion of split fuzzy set, constructed in this article, that allows for the semantic representation and aggregation of such information. This notion is again and again based on the notion of Zadeh fuzzy set. In particular, the operation of splitting a crisp subset into dual fuzzy sets is introduced. The dual splitted fuzzy sets lattice I creates a unified environment for aggregations of expert evaluations on different samplings. For this purpose, the splitting a crisp subset indicator in dual fuzzy sets is defined. The representations of splitting indicator for union, intersection, Cartesian product, and other operations on sets are given. The lattice of split elements of the Boolean lattice of indicators I is studied, where it is proven that the lattice of splitting of all elements of former lattice form the Braeuer’s lattice I . A number of facts about the properties of this lattice are given. Splitting operation for a crisp set is defined, which is equivalent to splitting operation of its indicator. The main properties of this operation are given. The concept of a generalized degree of the universe is given, which is the lattice of the elements obtained by splitting all the subsets of the universe and, as it is proved, represents the Braeuer’s lattice. In order to study this lattice, some formulas for conditional pseudoaddition of a lattice elements are considered. Some properties of the operation of splitting sets are given. It is proved that, the ideal of split elements of I and their pseudo-additions is equivalent to a Boolean lattice I .


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