@article{Rehman2023, 
author = {Ubaid Ur Rehman and Tahir Mahmood and Muhammad Naeem},
title = {Bipolar complex fuzzy semigroups},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {2},
pages = {3997-4021},
keywords = {bipolar complex fuzzy set, bipolar complex fuzzy sub-semigroups, bipolar complex fuzzy left (right) ideals},
url = {https://www.sciopen.com/article/10.3934/math.2023200},
doi = {10.3934/math.2023200},
abstract = {The notion of the bipolar complex fuzzy set (BCFS) is a fundamental notion to be considered for tackling tricky and intricate information. Here, in this study, we want to expand the notion of BCFS by giving a general algebraic structure for tackling bipolar complex fuzzy (BCF) data by fusing the conception of BCFS and semigroup. Firstly, we investigate the bipolar complex fuzzy (BCF) sub-semigroups, BCF left ideal (BCFLI), BCF right ideal (BCFRI), BCF two-sided ideal (BCFTSI) over semigroups. We also introduce bipolar complex characteristic function, positive        (          ω      ,      η        )  -cut, negative        (          ϱ      ,      σ        )  -cut, positive and        (                  (                  ω          ,          η                )            ,              (                  ϱ          ,          σ                )              )  -cut. Further, we study the algebraic structure of semigroups by employing the most significant concept of BCF set theory. Also, we investigate numerous classes of semigroups such as right regular, left regular, intra-regular, and semi-simple, by the features of the bipolar complex fuzzy ideals. After that, these classes are interpreted concerning BCF left ideals, BCF right ideals, and BCF two-sided ideals. Thus, in this analysis, we portray that for a semigroup        Ş   and for each BCFLI                      М                    1        =      (                                        λ                                    P          −                                                    М                                                    1                                          ,                                    λ                                    N          −                                                    М                                                    1                                            )    =      (                                        λ                                    R          P          −                                                    М                                                    1                                          +      ι                                    λ                                    I          P          −                                                    М                                                    1                                          ,                                    λ                                    R          N          −                                                    М                                                    1                                          +      ι                                    λ                                    I          N          −                                                    М                                                    1                                            )   and BCFRI                      М                    2        =      (                                        λ                                    P          −                                                    М                                                    2                                          ,                                    λ                                    N          −                                                    М                                                    2                                            )    =      (                                        λ                                    R          P          −                                                    М                                                    2                                          +      ι                                    λ                                    I          P          −                                                    М                                                    2                                          ,                                    λ                                    R          N          −                                                    М                                                    2                                          +      ι                                    λ                                    I          N          −                                                    М                                                    2                                            )   over        Ş  ,                      М                    1        ∩                    М                    2        =                    М                    1        ⊚                    М                    2       if and only if        Ş   is a regular semigroup. At last, we introduce regular, intra-regular semigroups and show that                      М                    1        ∩                    М                    2        ≼                    М                    1        ⊚                    М                    2       for each BCFLI                      М                    1        =      (                                        λ                                    P          −                                                    М                                                    1                                          ,                                    λ                                    N          −                                                    М                                                    1                                            )    =      (                                        λ                                    R          P          −                                                    М                                                    1                                          +      ι                                    λ                                    I          P          −                                                    М                                                    1                                          ,                                    λ                                    R          N          −                                                    М                                                    1                                          +      ι                                    λ                                    I          N          −                                                    М                                                    1                                            )   and for each BCFRI                      М                    2        =      (                                        λ                                    P          −                                                    М                                                    2                                          ,                                    λ                                    N          −                                                    М                                                    2                                            )    =      (                                        λ                                    R          P          −                                                    М                                                    2                                          +      ι                                    λ                                    I          P          −                                                    М                                                    2                                          ,                                    λ                                    R          N          −                                                    М                                                    2                                          +      ι                                    λ                                    I          N          −                                                    М                                                    2                                            )   over        Ş   if and only if a semigroup        Ş   is regular and intra-regular.}
}