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({\mathrm{\lambda }}_{P-{М}_{1}}, {\mathrm{\lambda }}_{N-{М}_{1}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{1}}+\iota {\mathrm{\lambda }}_{IP-{М}_{1}}, {\mathrm{\lambda }}_{RN-{М}_{1}}+\iota {\mathrm{\lambda }}_{IN-{М}_{1}}\right) $ and BCFRI $ {М}_{2} = \left({\mathrm{\lambda }}_{P-{М}_{2}}, {\mathrm{\lambda }}_{N-{М}_{2}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{2}}+\iota {\mathrm{\lambda }}_{IP-{М}_{2}}, {\mathrm{\lambda }}_{RN-{М}_{2}}+\iota {\mathrm{\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({\mathrm{\lambda }}_{P-{М}_{1}}, {\mathrm{\lambda }}_{N-{М}_{1}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{1}}+\iota {\mathrm{\lambda }}_{IP-{М}_{1}}, {\mathrm{\lambda }}_{RN-{М}_{1}}+\iota {\mathrm{\lambda }}_{IN-{М}_{1}}\right) $ and for each BCFRI $ {М}_{2} = \left({\mathrm{\lambda }}_{P-{М}_{2}}, {\mathrm{\lambda }}_{N-{М}_{2}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{2}}+\iota {\mathrm{\lambda }}_{IP-{М}_{2}}, {\mathrm{\lambda }}_{RN-{М}_{2}}+\iota {\mathrm{\lambda }}_{IN-{М}_{2}}\right) $ over $ Ş $ if and only if a semigroup $ Ş $ is regular and intra-regular.
Citation: Ubaid Ur Rehman, Tahir Mahmood, Muhammad Naeem. Bipolar complex fuzzy semigroups[J]. AIMS Mathematics, 2023, 8(2): 3997-4021. doi: 10.3934/math.2023200
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({\mathrm{\lambda }}_{P-{М}_{1}}, {\mathrm{\lambda }}_{N-{М}_{1}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{1}}+\iota {\mathrm{\lambda }}_{IP-{М}_{1}}, {\mathrm{\lambda }}_{RN-{М}_{1}}+\iota {\mathrm{\lambda }}_{IN-{М}_{1}}\right) $ and BCFRI $ {М}_{2} = \left({\mathrm{\lambda }}_{P-{М}_{2}}, {\mathrm{\lambda }}_{N-{М}_{2}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{2}}+\iota {\mathrm{\lambda }}_{IP-{М}_{2}}, {\mathrm{\lambda }}_{RN-{М}_{2}}+\iota {\mathrm{\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({\mathrm{\lambda }}_{P-{М}_{1}}, {\mathrm{\lambda }}_{N-{М}_{1}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{1}}+\iota {\mathrm{\lambda }}_{IP-{М}_{1}}, {\mathrm{\lambda }}_{RN-{М}_{1}}+\iota {\mathrm{\lambda }}_{IN-{М}_{1}}\right) $ and for each BCFRI $ {М}_{2} = \left({\mathrm{\lambda }}_{P-{М}_{2}}, {\mathrm{\lambda }}_{N-{М}_{2}}\right) = \left({\mathrm{\lambda }}_{RP-{М}_{2}}+\iota {\mathrm{\lambda }}_{IP-{М}_{2}}, {\mathrm{\lambda }}_{RN-{М}_{2}}+\iota {\mathrm{\lambda }}_{IN-{М}_{2}}\right) $ over $ Ş $ if and only if a semigroup $ Ş $ is regular and intra-regular.
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Ubaid Ur Rehman, Tahir Mahmood, Muhammad Naeem. Bipolar complex fuzzy semigroups[J]. AIMS Mathematics, 2023, 8(2): 3997-4021. doi: 10.3934/math.2023200
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