Find in Library

In practical applications, the basic fuzzy set is used via symmetric uncertainty variables. In the research field, it is comparatively rare to discuss two-fold uncertainty due to its complication. To deal with the multi-polar uncertainty in real life problems, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-polar (multi-polar) fuzzy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PF) sets are put forward. The main objective of this paper is to explore the idea of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PF sets, which is a generalization of bipolar fuzzy (BPF) sets, in ternary semirings. The major aspects and novel distinctions of this work are that it builds any multi-person, multi-period, multi-criteria, and complex hierarchical problems. The main focus of this study is to confine generalization of some important results of BPF sets to the results of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PF sets. In this research, the notions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-polar fuzzy ternary subsemiring (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PFSS), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-polar fuzzy ideal (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PFI), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-polar fuzzy generalized bi-ideal (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PFGBI), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-polar fuzzy bi-ideal (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PFBI), and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-polar fuzzy quasi-ideal (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PFQI) in ternary semirings are introduced. Moreover, this paper deals with several important properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>m</mi></semantics></math></inline-formula>-PFIs and characterizes regular and intra-regular ternary semiring in terms of these ideals.