Novelty for Different Prime Partial Bi-Ideals in Non-Commutative Partial Rings and Its Extension

In computer programming languages, partial additive semantics are used. Since partial functions under disjoint-domain sums and functional composition do not constitute a field, linear algebra cannot be applied. A partial ring can be viewed as an algebraic structure that can process natural partial orderings, infinite partial additions, and binary multiplications. In this paper, we introduce the notions of a one-prime partial bi-ideal, a two-prime partial bi-ideal, and a three-prime partial bi-ideal, as well as their extensions to partial rings, in addition to some characteristics of various prime partial bi-ideals. In this paper, we demonstrate that two-prime partial bi-ideal is a generalization of a one-prime partial bi-ideal, and three-prime partial bi-ideal is a generalization of a two-prime partial bi-ideal and a one-prime partial bi-ideal. A discussion of the mpb1,(m(pb2),m(pb3)) systems is presented. In general, the m(pb2) system is a generalization of the m(pb1) system, while the m(pb3) system is a generalization of both m(pb2) and m(pb1) systems. If f is a prime bi-ideal of (sic), then Phi is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if (sic)\Phi is a m(pb1) system (m(pb2) system, m(pb3) system) of (sic). If Theta is a prime bi-ideal in the complete partial ring (sic) and Delta is an mpb3 system of (sic) with Theta boolean AND Delta = phi, then there exists a three-prime partial bi-ideal Phi of (sic), such that Theta subset of Phi with Phi boolean AND Delta=phi. These are necessary and sufficient conditions for partial bi-ideal Theta to be a three-prime partial bi-ideal of (sic). It is shown that partial bi-ideal Theta is a three-prime partial bi-ideal of (sic) if and only if H-Theta is a prime partial ideal of (sic). If Theta is a one-prime partial bi-ideal (two-prime partial bi-ideal) in (sic), then H-Theta is a prime partial ideal of (sic). It is guaranteed that a three-prime partial bi-ideal f with a prime bi-ideal T does not meet the mpb3 system. In order to strengthen our results, examples are provided.