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A new approach of bipolar valued fuzzy set theory applied on semigroups
Author(s) -
Yiarayong Pairote
Publication year - 2021
Publication title -
international journal of intelligent systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.291
H-Index - 87
eISSN - 1098-111X
pISSN - 0884-8173
DOI - 10.1002/int.22465
Subject(s) - mathematics , ideal (ethics) , semigroup , generalization , discrete mathematics , interval (graph theory) , fuzzy logic , pure mathematics , fuzzy set , minimal ideal , set (abstract data type) , algebraic number , algebra over a field , maximal ideal , combinatorics , mathematical analysis , computer science , philosophy , epistemology , artificial intelligence , programming language
In this paper, we generalize the concept of interval‐valued bipolar fuzzy sets (IVBFSs) and define interval‐valued bipolar fuzzy subsemigroups (IVBF‐subsemigroup) and interval‐valued bipolar fuzzy left (right, two‐sided) ideals (IVBF‐left [right, two‐sided] ideals) over semigroups, which is a generalization of the concept of an bipolar valued fuzzy set (BVFS) in a semigroup. The purpose of this paper is to deal with the algebraic structure of semigroups by applying IVBFS theory. We give characterizations of different classes of (intra‐regular, left [right] regular, regular, semisimple) semigroups by the properties of their IVBF‐ideals. We also characterize these classes in terms of IVBF‐left ideals, IVBF‐right ideals, and IVBF‐two‐sided ideals. In this respect, we prove that a semigroup is regular if and only if for every IVBF‐right ideal A ˜ =μ A P ˜ , μ A N ˜and every IVBF‐left ideal B ˜ =μ B P ˜ , μ B N ˜over S , we have A ˜ ∩ B ˜ = A ˜ ⊙ B ˜ . Further, we characterize intra‐regular and regular semigroups and prove that a semigroup is intra‐regular and regular if and only if for every IVBF‐left ideal A ˜ =μ A P ˜ , μ A N ˜and every IVBF‐right ideal B ˜ =μ B P ˜ , μ B N ˜over S we have A ˜ ∩ B ˜≼A ˜ ⊙ B ˜ .