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,(mpb2,mpb3) systems is presented. In general, the mpb2 system is a generalization of the mpb1 system, while the mpb3 system is a generalization of both mpb2 and mpb1 systems. If Φ is a prime bi-ideal of ℧, then Φ is a one-prime partial bi-ideal (two-prime partial bi-ideal, three-prime partial bi-ideal) if and only if ℧\Φ is a mpb1 system (mpb2 system, mpb3 system) of ℧. If Θ is a prime bi-ideal in the complete partial ring ℧ and Δ is an mpb3 system of ℧ with Θ∩Δ=ϕ, then there exists a three-prime partial bi-ideal Φ of ℧, such that Θ⊆Φ with Φ∩Δ=ϕ. These are necessary and sufficient conditions for partial bi-ideal Θ to be a three-prime partial bi-ideal of ℧. It is shown that partial bi-ideal Θ is a three-prime partial bi-ideal of ℧ if and only if HΘ is a prime partial ideal of ℧. If Θ is a one-prime partial bi-ideal (two-prime partial bi-ideal) in ℧, then HΘ is a prime partial ideal of ℧. It is guaranteed that a three-prime partial bi-ideal Φ with a prime bi-ideal Θ does not meet the mpb3 system. In order to strengthen our results, examples are provided.