A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

A Hesitant Intuitionistic Fuzzy Set Approach to Study Ideals of Semirings

Debabrata Mandal
Copyright: © 2021 |Pages: 17
DOI: 10.4018/IJFSA.2021070101
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Abstract

The classical set theory was extended by the theory of fuzzy set and its several generalizations, for example, intuitionistic fuzzy set, interval valued fuzzy set, cubic set, hesitant fuzzy set, soft set, neutrosophic set, etc. In this paper, the author has combined the concepts of intuitionistic fuzzy set and hesitant fuzzy set to study the ideal theory of semirings. After the introduction and the priliminary of the paper, in Section 3, the author has defined hesitant intuitionistic fuzzy ideals and studied several properities of it using the basic operations intersection, homomorphism and cartesian product. In Section 4, the author has also defined hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals of a semiring and used these to find some characterizations of regular semiring. In that section, the author also has discussed some inter-relations between hesitant intuitionistic fuzzy ideals, hesitant intuitionistic fuzzy bi-ideals and hesitant intuitionistic fuzzy quasi-ideals, and obtained some of their related properties.
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2. Preliminaries

In this section, we have reviewed to some definitions which will be required to develop the paper.

2.1. Definition (Golan, 1999)

A semiring is a nonempty set S on which operations addition and multiplication have been defined such that the following conditions are satisfied:

  • 1.

    (S,+) is a commutative monoid with identity 0.

  • 2.

    (S,.) is a monoid with identity IJFSA.2021070101.m01.

  • 3.

    Multiplication distributes over addition from either side.

  • 4.

    0s=0=s0 for all sIJFSA.2021070101.m02S.

  • 5.

    IJFSA.2021070101.m03

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