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The concept of DioNSBSs is introduced in this study. This paper is focused on the following: The introduction is in Section 1. The preliminary definitions and results are found in Section 2. Section 3 introduces the notion of DioNSBS and its several illustrative examples.

Neutosophic vague subbisemirings (NSVSBS) are discussed here, as well as their level sets. Subbisemirings are a generalization of bisemirings, and NSVSBSs are a generalization of sub-bisemirings. A number of illustrative examples are provided to illustrate the theory for (ξ, τ )-NSVSBS over bisemiring theory. Following is an outline of the preliminary definitions and results presented in Section 2. The concept of a NSVSBS is introduced in Section 3.

In this paper, we introduce the notion of bipolar-valued neutrosophic subbisemiring (BVNSBS), level sets of BVNSBS, and bipolar valued neutrosophic normal subbisemiring (BVNNSBS) of a bisemiring. The concept of BVNSBS is a new generalization of subbisemiring over bisemirings. We discussed the theory of (ξ, τ )- BVNSBS and (ξ, τ )-BVNNSBS over bisemirings and presented several illustrative examples to demonstrate the sufficiency and validity of the proposed theorems, lemmas, and propositions.

Pythagorean neutrosophic soft set (PNSS set) is a new approach towards decision making under uncertainty. The PNSS set has much stronger abilities than the neutrosophic soft set and the Pythagorean fuzzy soft set. In this paper, we discuss aggregated operations for aggregating the PNSS decision matrix. The TOPSIS and VIKOR methods are strong approaches for multi criteria group decision making (MCGDM), which is various extensions of neutrosophic soft sets. In this approach, we propose a score function based on aggregating TOPSIS and VIKOR methods to the PNSS-positive ideal solution and the PNSS-negative ideal solution. Also, the TOPSIS and VIKOR methods provide the weights of decision-making. Afterward, a revised closeness is introduced to identify the optimal alternative.

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.

We introduce the notion of neutrosophic subbisemiring( shortly NSBS), level sets of NSBS and neutrosophic normal subbisemiring (NNSBS) of a bisemiring. The concept of neutrosophic subbisemiring is a new generalization of fuzzy subbisemiring over bisemiring. Examples are provided to illustrate our results.

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.

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.

This paper describes a generalized Diophantine fuzzy sets, which can be seen as a generalization of both Diophantine fuzzy sets and Pythagorean fuzzy sets. We define the basic properties of generalized Diophantine fuzzy set, as well as their relationships and distances. We compare Diophantine fuzzy sets with other Diophantine Pythagorean fuzzy sets to demonstrate their importance in the literature. We introduce new operators including necessity, possibility, accuracy function and score function. Furthermore, we discuss the new distance between normalized Euclidean distance and normalized Hamming distance. For a generalized Diophantine fuzzy relation, image and inverse image functions are defined. Numerous real-world applications can be found for the prevalent ideas of intuitionistic fuzzy sets, Pythagorean fuzzy sets, Diophantine fuzzy sets and q-rung orthopair fuzzy sets. Regretfully, these theories about the membership and non-membership grades have their own limits. We provide a new idea the generalized Diophantine fuzzy set that eliminates these limitations by including reference parameters. Compared to other kinds of fuzzy sets, there are more applications for generalized Diophantine fuzzy sets. We offer practical examples that show how different enhanced distances might be used in everyday situations. Additionally, to demonstrate the effectiveness of the suggested approach, flowchart based multi-criteria decision-making is provided and used to a numerical example. The outcomes are assessed for various parameter values. Furthermore, a comparative analysis developed to demonstrate the superiority of the suggested technique over current methodologies., (© 2024. The Author(s).)

This article introduces the structure of the ( t , s ) -regulated interval-valued neutrosophic soft set (abbr. ( t , s ) -INSS). The structure of ( t , s ) -INSS is shown to be capable of handling the sheer heterogeneity and complexity of real-life situations, i.e. multiple inputs with various natures (hence neutrosophic), uncertainties over the input strength (hence interval-valued), the existence of different opinions (hence soft), and the perception at different strictness levels (hence ( t , s ) -regulated). Besides, a novel distance measure for the ( t , s ) -INSS model is proposed, which is truthful to the nature of each of the three membership (truth, indeterminacy, falsity) values present in a neutrosophic system. Finally, a Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and a Viekriterijumsko Kompromisno Rangiranje (VIKOR) algorithm that works on the ( t , s ) -INSS are introduced. The design of the proposed algorithms consists of TOPSIS and VIKOR frameworks that deploy a novel distance measure truthful to its intuitive meaning. The conventional method of TOPSIS and VIKOR will be generalized for the structure of ( t , s ) -INSS. The parameters t and s in the ( t , s ) -INSS model take the role of strictness in accepting a collection of data subject to the amount of mutually contradicting information present in that collection of data. The proposed algorithm will then be subjected to rigorous testing to justify its consistency with human intuition, using numerous examples which are specifically made to tally with the various human intuitions. Both the proposed algorithms are shown to be consistent with human intuitions through all the tests that were conducted. In comparison, all other works in the previous literature failed to comply with all the tests for consistency with human intuition. The ( t , s ) -INSS model is designed to be a conclusive generalization of Pythagorean fuzzy sets, interval neutrosophic sets, and fuzzy soft sets. This combines the advantages of all the three previously established structures, as well as having user-customizable parameters t and s , thereby enabling the ( t , s ) -INSS model to handle data of an unprecedentedly heterogeneous nature. The distance measure is a significant improvement over the current disputable distance measures, which handles the three types of membership values in a neutrosophic system as independent components, as if from a Euclidean vector. Lastly, the proposed algorithms were applied to data relevant to the ongoing COVID-19 pandemic which proves indispensable for the practical implementation of artificial intelligence., Competing Interests: Conflict of interestThe authors have no relevant financial or non-financial interests to disclose., (© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023, Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.)