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Interval valued intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups

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Abstract

The notion of interval valued intuitionistic fuzzy sets was introduced by Atanassov and Gargov as a generalization of the notion of intuitionistic fuzzy sets and interval valued fuzzy sets. In this paper, we initiate a study on interval valued intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups. We introduce the notions of interval valued intuitionistic fuzzy left (right, two sided) \(\Gamma \)-hyperideal, interval valued intuitionistic fuzzy bi-\(\Gamma \)-hyperideal and interval valued intuitionistic fuzzy (1,2) \(\Gamma \)-hyperideal in a \(\Gamma \)-semihypergroup and some properties of them are obtained. We use the interval valued intuitionistic fuzzy left, right, two-sided and bi-\(\Gamma \)-hyperideals to characterize some classes of \(\Gamma \)-semihypergroups. We also introduce the notion of an interval valued intuitionistic fuzzy \(M\) (resp. \(N\))-hypersystem of a \(\Gamma \)-semihypergroup and some properties of them are investigated.

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Abbreviations

\(\mathcal {P}^{*}(S)=\mathcal {P}(S)\backslash \{\emptyset \}\) :

The set of all non-empty subsets of \(S\)

\(\Gamma \) :

The set of all hyperoperations

\(D\) [0,1]:

The collections of all closed subinterval of \( [0,1]\)

\(\widetilde{\lambda }\) :

An interval valued fuzzy set

i.v fuzzy set:

Interval valued fuzzy set

\(A=\langle \widetilde{\mu }_{A},\) \(\widetilde{\gamma }_{A}\rangle \) :

Notion for interval valued intuitionistic fuzzy set

i.v intuitionistic fuzzy set:

Interval valued intuitionistic fuzzy set

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Acknowledgments

We wish to express our heart thanks to Pakistan Science Foundation. This research work was supported by a grant of Pakistan Science Foundation, Islamabad, Pakistan.

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Authors and Affiliations

  1. Department of Mathematics, Quaid-i-Azam University, Islamabad, 45320, Paksitan

    Saleem Abdullah

  2. Department of Mathematics, King Khalid University, Abha, Saudi Arabia

    Muhammad Aslam

  3. Department of Mathematics and Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, Albania

    Kostaq Hila

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  1. Saleem Abdullah
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  2. Muhammad Aslam
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  3. Kostaq Hila
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Correspondence to Saleem Abdullah.

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Abdullah, S., Aslam, M. & Hila, K. Interval valued intuitionistic fuzzy sets in \(\Gamma \)-semihypergroups. Int. J. Mach. Learn. & Cyber. 7, 217–228 (2016). https://doi.org/10.1007/s13042-014-0250-4

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