NEW APPROACH TOWARDS DIFFERENT TYPES OF BI-QUASI IDEALS IN b-SEMIRINGS AND ITS EXTENSION

We introduce two types of bi-quasi ideals in b-semirings. Each bi-quasi ideal generated by a single element and set is established. For any a. S, the generalized 1-right bi-quasi ideal generated by "a" is < a >(g1rbq) = {a} boolean OR [(a lozenge S-2) boolean AND (a lozenge S-2 lozenge(2)a)]. We characterize various 1-regular (2-regular) by using generalized 1-bi-quasi ideal, 1-bi-quasi ideal, weak-1-right ideal, weak-1-left ideal, right ideal, left ideal, bi-ideal, quasiideal (generalized 2-bi-quasi ideal, 2-bi-quasi ideal, weak-2-right ideal, weak-2-left ideal, right ideal, left ideal, bi-ideal, quasi-ideal). Every quasi-ideal is a bi-quasi ideal, and the reverse implication does not hold. Examples are provided to strengthen our results. The motivation of this paper is to show that the categories of quasi-ideals over b-semirings have gained prominence in algebraic structures. Although many other algebraic concepts of these valuations are worthy of a more extensive study, here we only settle down the algebraic considerations that we will need in order to investigate their 1-bi-quasi ideals and 2-bi-quasi ideals in b-semirings.