Regular ternary semirings in terms of bipolar fuzzy ideals
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Regular ternary semirings in terms of bipolar fuzzy ideals Shahida Bashir1 · Rabia Mazhar1 · Hasnain Abbas1 · Muhammad Shabir2 Received: 29 August 2019 / Revised: 16 June 2020 / Accepted: 29 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The central objective of this paper is to introduce (α, β)-bipolar fuzzy ideals (left, lateral and right) and bi-ideals in ternary semirings by applying the definitions of belongingness (∈) and quasi-coincidence (q) of a bipolar fuzzy point with a bipolar fuzzy set. In this work, upper parts and lower parts of bipolar fuzzy set in ternary semirings are also discussed. Regular and intra-regular ternary semirings in terms of (∈, ∈ ∨q)-bipolar fuzzy (left, lateral and right) ideals and (∈, ∈ ∨q)-bipolar fuzzy bi-ideals are characterized. Keywords Ternary semiring · (α, β)-Bipolar fuzzy subsemirings · (α, β)-Bipolar fuzzy ideals · Regular and intra-regular ternary semiring Mathematics Subject Classification 20N10 · 03E72 · 16Y60
1 Introduction Theory of bipolar fuzzy sets deals with the uncertain and complex problems, both in positive and negative aspects. For example, a house near to the commercial area of a city is both good (it is convenient) and bad (it is pollutant). Similarly, 50 dollars are much more for poor people while on the same circumstances rich man does not give any value to this amount. In the same way, sweetness and sourness of food, effects and side effects of medicines, good and bad human behaviors, thinness and thickness of fluid all are two-sided aspects of a situation.
Communicated by Anibal Tavares de Azevedo.
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Shahida Bashir [email protected]; [email protected] Rabia Mazhar [email protected] Hasnain Abbas [email protected] Muhammad Shabir [email protected]
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Department of Mathematics, University of Gujrat, Gujrat 50700, Pakistan
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Department of Mathematics, Quaid-I-Azam University Islamabad, Islamabad 44000, Pakistan 0123456789().: V,-vol
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For more applications of bipolar fuzzy set theory, see (Hayat et al. 2017; Jan et al. 2019; Mahmood and Ejaz 2015; Mahmood and Hayat 2015; Mahmood and Munir 2013; Ullah et al. 2018). Bipolar fuzzy set theory is an extension of fuzzy set theory. Zhang (1994) introduced the bipolar fuzzy set theory in which the function is mapped to interval [− 1, 1] instead of [0, 1]. Lee (2000) gave the concept of bipolar fuzzy ideals. The idea of finite state machine on bipolar fuzzy theory is given by Jun and Kavikumar in (2011). In engineering, environmental science and medical science we may face uncertainty and complexity in the data or information. In classical mathematics, all the mathematical formulae and methods are exact, so we cannot deal with the problems having uncertainty. After a long effort, many tools are created to handle such problems. Zadeh (1965) was first who gave the idea of fuzzy set theory in 1965 to handle such complex problems. This concept is applied on logics, measure theory,
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