Three-way decision-making provides an effective framework for addressing uncertainty, aligning closely with human cognitive decision patterns. This paper proposes a novel three-way decision model based on multi-granular support intuitionistic fuzzy rough sets, integrating n-dimensional overlap and grouping functions. The model constructs optimistic and pessimistic upper and lower approximations to optimize decision rules and introduces score and precision functions for ranking. To validate the model, a consumer decision-making algorithm was developed and applied to empirical data. The results demonstrate that the proposed model effectively narrows decision boundary regions, enhances decision-making precision, and supports decision-making in complex multi-attribute scenarios. This study not only advances rough set theory but also offers practical tools for addressing real-world uncertainty in decision-making.

The main purpose of this paper is to establish a type of quantitative model by using the contangent similarity function in the three‐valued Łukasiewicz propositional logic system Ł₃. We introduce the concepts of the cotangent similarity degree, cotangent pseudo‐distance and cotangent truth degree of the propositions, together with their basic properties in Ł₃. We investigate the relationship between the cotangent truth degree and contangent pseudo‐distance, and prove the continuity of the logical connectives and in the logical metric space. We propose a graded reduction method and three types of graded reasoning frameworks on the propositions set F(S), and provide several examples and basic properties of it.

The quasi-cancellativity of Abel Grassmann‘s groupoids (AG-groupoids) are discussed and two conjectures are partially solved. First, the following conjecture is proved to be true: every AG-3-band is quasi-cancellative. Moreover, a new notion of AG-(4,1)-band is proposed, and it is also proved that every AG-(4,1)-band is quasi-cancellative. Second, the notions of left (right) quasi-cancellative AG-groupoids and power-cancellative AG-groupoids are proposed, and the following results are obtained: for an AG*-groupoid or AG**-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative; for a power-cancellative and locally power-associative AG-groupoid, it is left quasi-cancellative if and only if it is right quasi-cancellative. Finally, a general result is proved, that for any AG-groupoid, if it is left quasi-cancellative then it is right quasi-cancellative.

By introducing the concepts of intuitionistic L-fuzzy β-covering and intuitionistic L-fuzzy β-neighborhood, we define three kinds of intuitionistic L-fuzzy β-covering rough set models. The basic properties of those intuitionistic L-fuzzy β-covering rough set models are investigated. Moreover, we define the other three kinds of intuitionistic L-fuzzy β-covering rough set models by using the former three models. Finally, we present the matrix representations of the newly defined lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices.