New category of algebraic fuzzy systems with their applications of translations and bipolar fuzzy RHO-ideals semigroups

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.