A New Approach to Evaluate Regular Semirings in terms of Bipolar Fuzzy k-Ideals Using k-Products

Abstract

In this paper, we provide a generalized form of ideals that is k-ideals of semirings with the combination of a bipolar fuzzy set (BFS). The BFS is a generalization of fuzzy set (FS) that deals with uncertain problems in both positive and negative aspects. The main theme of this paper is to present the idea of $\left(\alpha ,\beta \right)$ -bipolar fuzzy k-subsemiring (k-BFSS), $\left(\alpha ,\beta \right)$ -bipolar fuzzy k-ideals (k-BFIs), and $\left(\alpha ,\beta \right)$ -bipolar fuzzy k-bi-ideals (k-BFbIs) in semirings by applying belongingness $\left(\in \right)$ and quasi-coincidence $\left(q\right)$ of the bipolar fuzzy (BF) point. After that, upper and lower parts of k-product of BF subsets of semirings are introduced. Lastly, the notions of k-regular and k-intraregular semirings in terms of $\left(\in ,\in \vee q\right)$ -k–BFIs and $\left(\in ,\in \vee q\right)$ -k–BFbIs are characterized.