In this paper, we discuss the hypothesis that an ordered Γ-semigroup can be constructed on the M-left(right)-tri-basis. In order to generalize the left(right)-tri-basis using Γ-semigroups and ordered semigroups, we examined M-tri-ideals from a purely algebraic standpoint. We also present the form of the M-tri-ideal generator. We investigated the M-left(right)-tri-ideal using the ordered Γ-semigroup. In order to obtain their properties, we used M-left(right)-tri-basis. It was possible to generate a M-left(right)-tri-basis from elements and their subsets. Throughout this paper, we will present an interesting example of order ⪯mlt(⪯mrt), which is not a partial order of S. Additionally, we introduce the notion of quasi-order. As an example, we demonstrate the relationship between M-left(right)-tri-basis and partial order.

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.

A novel method is presented for solving MADM under a sine trigonometric Pythagorean neutrosophic normal interval-valued set (ST-PyNSNIVS). An identifying feature of ST-PyNSNIVS is that it is a combination of PyNSIVS, PyNSS, and IVNSS. This article proposes a novel concept of ST-PyNSNIVWA, ST-PyNSNIVWG, ST-GPyNSNIVWA, and ST-GPyNSNIVWG. In addition, we acquired a flowchart and an algorithm that interact with MADM and are called ST-PyNSNIVWA, ST-PyNSNIVWG, ST-GPyNSNIVWA, and ST-GPyNSNIVWG, respectively. In addition to Euclidean and Hamming distances, we addressed new types of two distances in the suggested models, which are future expansions of real-life instances. The sine trigonometric aggregation operations were examined using the PyNSNIV set technique. They are more straightforward and practical, and you can arrive at the best option quickly. Consequently, the conclusions of the defined models are more accurate and closely correlated with Σ . Our analysis shows that the investigated models are valid and useful by comparing them to some of the current models. As a final result of the study, some intriguing and enthralling findings are presented.

In this paper, we extend the notion of LA-semihypergroups (resp. Hv-LA-semigroups) to neutro-LA-semihypergroups (respectively, neutro-Hv-LA-semigroups). Anti-LA-semihypergroups (respectively, anti-Hv-LA-semigroups) are studied and investigated some of their properties. We show that these new concepts are di fferent from classical concepts by several examples. These are particular cases of the classical algebraic structures generalized to neutroalgebraic structures and antialgebraic structures (Smarandache, 2019). 2000 Mathematics subject classification: 20N20, 20N99.