Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number $\tilde{u},$ $\tilde{u}\otimes \frac{1}{\tilde{u}}=\tilde{i}=(1,1,0,0)$ does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, $\tilde{u}$ is constructed $\frac{1}{\tilde{u}}$ so that $\tilde{u}\otimes \frac{1}{\tilde{u}}=\tilde{i}=(1,1,0,0)$. Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order $m\times m$. We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order $m\times n$ with $m \ne n$. This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention.

The Citarum River Basin (DAS Citarum) in Indonesia faces significant challenges in waste management, necessitating a circular economy-based approach to reduce land-based pollution, which is critical for achieving the sustainability goals of the blue economy in the basin. This study addresses the complexity and inherent uncertainty in decision-making processes related to this challenge by developing a novel hybrid model, namely the Weighted Fuzzy N-Soft Set combined with the COmbinative Distance-based Assessment (CODAS) method. The model synergistically integrates the weighted 10R strategies in the circular economy, obtained via the Analytical Hierarchy Process (AHP), the capability of Fuzzy N-Soft Sets to represent uncertainty granularly, and the robust ranking mechanism of CODAS. Applied to a case study covering 16 types of waste in the Citarum River Basin, the model effectively processes expert assessments that are ambiguous regarding the 10R criteria. The results indicate that single-use plastics, particularly plastic bags (HDPE), styrofoam, transparent plastic sheets (PP), and plastic cups (PP), are the top priorities for intervention, in line with the high AHP weights for upstream strategies such as Refuse (0.2664) and Rethink (0.2361). Comparative analysis with alternative models, namely Fuzzy N-Soft Set-CODAS, Weighted Fuzzy N-Soft Set with row-column sum ranking, and Weighted Fuzzy N-Soft Set-TOPSIS, confirms the superiority of the proposed hybrid model in producing ecologically rational priorities, free from purely economic value biases. Further sensitivity analysis shows that the model remains highly robust across various weighting scenarios. This study concludes that the WFN-SS-CODAS framework provides a rigorous, data-driven, and reliable decision support tool for translating circular economy principles into actionable waste management priorities, directly supporting the restoration and sustainability goals of the blue economy in river basins. The findings suggest that targeting the high-priority waste types identified by the model addresses the dominant fraction of riverine pollution, indicating the potential for significant waste volume reduction. This research was conducted to directly contribute to achieving multiple targets under SDG 6 (Clean Water and Sanitation), SDG 12 (Responsible Consumption and Production), and SDG 14 (Life Below Water).

This article introduces the concept of a Sheffer stroke BN-algebra by applying the Sheffer stroke operator | to the BN-algebra axioms and aligning it with the axioms of the Sheffer stroke groupoid. From this definition, properties of Sheffer stroke BN algebras are derived, focusing on the relationship between the axioms and the properties of the special element 0. Furthermore, the notions of Sheffer stroke BN-subalgebras, BN-ideals, and BN-homomorphisms are defined, along with normal subsets of Sheffer stroke BN-algebras, and the relationships between these concepts are explored. It is shown that every normal subset in a Sheffer stroke BN-algebra is a Sheffer stroke BN-subalgebra, but the converse is not necessarily true. This implies that every normal BN-ideal in a Sheffer stroke BN-algebra is also a Sheffer stroke BN-subalgebra. Finally, the properties of the kernel of the Sheffer stroke BN-homomorphism are investigated.