There has been a growing interest in the study of the notion of -migrativity and generalizations in recent years, and it has been investigated for families of certain operators such as t-norms, t-conorms, uninorms, nullnorms. This paper is mainly devoted to investigating the migrativity equations between semi-t-operators or semi-uninorms, and Mayor's Aggregation operators. The results that we obtain are complete and different from the known ones concerning migrativity for t-norms, t-conorms, uninorms and nullnorms.

The issue of distributivity of Aggregation operators is crucial for many different areas such as the utility theory  and the integral theory. The topic of  this paper is distributivity between 2-uninorms and Mayor's Aggregation operators. The presented research is an extension and upgrade ofthe previously obtained results. The full characterization of distributive pairs of operators is given.

The focus of this paper is to investigate the modularity equation involving uninorms and Mayor's Aggregation operators.  Necessary and sufficient conditions are established for this equation. And, it finds that themodularity equation of a Mayor's Aggregation operator over a uninorm is reduced to the modularityof a commutative semi-t-norm over a uninorm and the modularity equation of a uninorm over a Mayor's Aggregation operator is reduced to the modularity equation of a uninorm over a commutative semi-t-conorm. Among them, the cases of a uninorm which is locally internal on the boundary are studied in \cite{Su_Riera_Aguilera_Torrens_2019}. In this paper, we consider whether the neutral element $e$ of the uninorm is idempotent element of the Mayor's Aggregation operator in modularity equation to get solutions in the corresponding cases.

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the arithmetic and geometric Aggregation operators with hesitant fuzzy linguistic information. Then, motivated by the idea of traditional arithmetic operation, we have developed some Aggregation operators for aggregating hesitant fuzzy linguistic information: hesitant fuzzy linguistic weighted average (HFLWA) operator, hesitant fuzzy linguistic ordered weighted average (HFLOWA) operator and hesitant fuzzy linguistic hybrid average (HFLHA) operator. Furthermore, we propose the concept of the dual hesitant fuzzy linguistic set and develop some Aggregation operators with dual hesitant fuzzy linguistic information. Then, we have utilized these operators to develop some approaches to solve the hesitant fuzzy linguistic multiple attribute decision making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Aggregation operators (AOs) have been studied by many scholars. As many AOs are proposed, there is still lacking approach to classify the categories of AO, and to select the appropriate AO within the AO candidates. In this research, each AO can be regarded as a cognitive style or individual difference. A Cognitive Style and Aggregation Operator (CSAO) model is pro- posed to analyze the mapping relationship between the Aggregation operators and the cognitive styles represented by the decision attitudes. Four algorithms are proposed for CSAO: CSAO-1, CSAO-2 and two selection strategies on the basis of CSAO-1 and CSAO-2. The numerical examples illustrate how the choice of the Aggregation operators on the basis of the decision attitudes can be determined by the selection strategies of CSAO-1 and CSAO-2. The CSAO model can be applied to decision making systems with the selection problems of the appropriate Aggregation operators with consideration of the cognitive styles of the decision makers.

Recently, Zhao et al. \cite{Zhao-2021-25} characterized the distributivity equations of null-uninorms with continuous and Archimedean underlying operators over overlap or grouping functions. Moreover, Liu et al. \cite{Liu-2020-25} studied the distributive laws of continuous t-norms over overlap functions. In this paper, we proceed with the distributivity characterization of idempotent null-uninorms over overlap or grouping functions. In order to do that, we introduce a class of weak overlap and grouping functions with weak coefficients, and obtain the full characterizations of overlap and grouping functions by considering the different values of underlying uninorms' associated functions of idempotent null-uninorms on the interval endpoints and comparing them with the weak coefficients. Obviously, idempotent null-uninorms generalize idempotent uninorms. Thus, the obtained results also generalize the distributivity of idempotent uninorms proposed as future work in

In this paper, in order to apply the concept of IVI-octahedron sets to MCDGM problems, we define some Aggregation operators via IVI-octahedron sets and obtain some their properties. We define some Aggregation operators via IVI-octahedron sets and obtain some their properties. We present a MCGDM method with linguistic variables in IVI-octahedron set environment. Finally, we give a numerical examples for MCGDM problems.

In this article, we introduce the interval-valued intuitionistic fuzzy set (\textbf{IVIFS}), which are generalized forms of intuitionistic fuzzy set (\textbf{IFS}) and fuzzy set, this is because in intuitionistic fuzzy sets the non-membership function also applies to evaluations, and these sets are useful for modelling ambiguous concepts that abound in real problems. Here we try to look for new methods for more practical solutions in optimization problems for various sciences such as computer science, mathematics, engineering, medicine, psychology, climate and etc. First, with the introduction of t-norm Frank, an action we construct some Frank Aggregation operators on interval-valued intuitionistic fuzzy numbers (\textbf{IVIFN}s), including the Frank weighted averaging operator, Frank-ordered weighted averaging operator, Frank hybrid weighted averaging operator, Frank geometric weighted averaging operator, Frank geometric-ordered weighted averaging operator, and Frank geometric hybrid weighted averaging operator. Also, examine some of the characteristics of these operators. In the following, we introduce two multiple attribute group decision-making methods (\textbf{MAGDM}) based on such operators. Finally, we provide illustrative examples of these methods.

Based on a new hesitant fuzzy partial ordering proposed by Garmendia et al. [7], in this paper a fuzzy disjunction D on the set H of finite and nonempty subsets of the unit interval and a t-conorm S on the set  B of equivalence class on the set of finite bags of unit interval based on this partial ordering are introduced respectively. Then, hesitant fuzzy negations Nn on H and  n on  B are proposed. Particularly, their De Morgan’ s laws are investigated with respect to binary operations C and D on H, as well as T and S on  B respectively, where C is a commutative fuzzy conjunction on (H;  H) and T is a t-norm on ( B;  B). Finally, the new hesitant fuzzy Aggregation operators are presented on H and  B and their more general forms are given. Moreover, the validity of the Aggregation operations is illustrated by a numerical example on decision making.