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 (omega, eta) -cut, negative (rho, sigma) -cut, positive and ((omega, eta), (rho, sigma)) -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 semisimple, 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 S and for each BCFLI M-1 = (lambda(P)-(M1), lambda(N)-(M1)) = (lambda(RP)-(M1) +tau lambda(IP)-(M1,) lambda R-N.-(M1) +tau lambda(IN)-(M1)) and BCFRI M-2= (lambda(P)-(M2), lambda(N)-(M2)) = (lambda(RP)-(M2) +tau lambda(IP)-(M2,) lambda R-N.-(M2) +tau lambda(IN)-(M2)) over (sic)M-1 boolean AND M-2 = M-1 circle dot M-2 if and only if S is a regular semigroup. At last, we introduce regular, intra-regular semigroups and show thatM(1) boolean AND M-2 <= M-1 circle dot M-2 for each BCFLI M-2 = (lambda(P)-(M2), -lambda(N)-M-2) = (lambda(P)-(M2).-.1 +lambda(P)-(M2) -.1,-.1 +lambda(P)-(M2)-.1) and for each BCFRI.2 = (lambda(P)-(M2)-lambda(P)-(M2)-lambda(P)-(M2)2) = (lambda(P)-(M2)-.2 +lambda(P)-(M2)-.2,lambda(P)-(M2)-.2 +lambda(P)-(M2)-.2) over S if and only if a semigroup S is regular and intra-regular.