Bipolar complex fuzzy semigroups

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 $\left(\omega ,\eta \right)$-cut, negative $\left(\varrho ,\sigma \right)$-cut, positive and $\left(\left(\omega ,\eta \right),\left(\varrho ,\sigma \right)\right)$-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 ${М}_1=\left({\lambda }_{P-{М}_1},{\lambda }_{N-{М}_1}\right)=\left({\lambda }_{RP-{М}_1}+\iota {\lambda }_{IP-{М}_1},{\lambda }_{RN-{М}_1}+\iota {\lambda }_{IN-{М}_1}\right)$ and BCFRI ${М}_2=\left({\lambda }_{P-{М}_2},{\lambda }_{N-{М}_2}\right)=\left({\lambda }_{RP-{М}_2}+\iota {\lambda }_{IP-{М}_2},{\lambda }_{RN-{М}_2}+\iota {\lambda }_{IN-{М}_2}\right)$ over $Ş$, ${М}_1\cap {М}_2={М}_1⊚{М}_2$ if and only if $Ş$ is a regular semigroup. At last, we introduce regular, intra-regular semigroups and show that ${М}_1\cap {М}_2\preccurlyeq {М}_1⊚{М}_2$ for each BCFLI ${М}_1=\left({\lambda }_{P-{М}_1},{\lambda }_{N-{М}_1}\right)=\left({\lambda }_{RP-{М}_1}+\iota {\lambda }_{IP-{М}_1},{\lambda }_{RN-{М}_1}+\iota {\lambda }_{IN-{М}_1}\right)$ and for each BCFRI ${М}_2=\left({\lambda }_{P-{М}_2},{\lambda }_{N-{М}_2}\right)=\left({\lambda }_{RP-{М}_2}+\iota {\lambda }_{IP-{М}_2},{\lambda }_{RN-{М}_2}+\iota {\lambda }_{IN-{М}_2}\right)$ over $Ş$ if and only if a semigroup $Ş$ is regular and intra-regular.