In algebraic semigroups when we deal with classical sets it is not necessary every \(\rho\) -ideal is \(\overline{\rho }\) -ideal. Therefore, in this work it is proved that possible for their extension in non-classical sets like fuzzy sets and bipolar fuzzy sets with their new forms in algebraic semigroups. Moreover, a new structure of bipolar fuzzy ideals is shown. The new connotations like \(\rho\) -semigroup, \(\rho\) -subsemigroup, \(\rho\) -ideal semigroup, \(\overline{\rho }\) -ideal semigroup, \(\rho\) -semigroup homomorphism are investigated. Also, some connotations of fuzzy logic, like fuzzy \(\rho\) -subsemigroup, fuzzy \(\rho\) -ideal semigroup, fuzzy \(\overline{\rho }\) -ideal semigroup, bipolar fuzzy \(\rho\) -subsemigroup, bipolar fuzzy \(\rho\) -ideal semigroup and bipolar fuzzy \(\overline{\rho }\) -ideal semigroup are found. Furthermore, some basic characterizations of our connotations are studied and discussed. In this paper, the \(\rho\) -semigroup homomorphism image, translations and the product characteristics of bipolar fuzzy \(\rho\) / \(\overline{\rho }\) -ideals semigroups are studied their applications.
