Aggregation operations play a fundamental role in a large number of disciplines, from mathematics and natural sciences to economics and social sciences. This paper is focused on the problem of Distributivity for some special classes of aggregation operations, and quasi-linear means. Characterization of Distributivity pairs for uninorms, semi-uninorms and associative a-CAOA vs quasi-linear means is given.

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.

Due to the fact that Zadeh's quantifiers constitute the usual method to modify fuzzy propositions, the so-called family of T-power based implications was proposed. In this paper,  the four basic distributive laws  related to T-power based fuzzy implications  and fuzzy logic operations (t-norms and t-conorms) are deeply studied. This study shows that two of the four distributive laws of the T-power based implications have a unique solution, while the other two have multiple solutions.

Let $R$ be a commutative ring with identity and $\mathcal{R}(_{R}M)$ denotes the bounded lattice of radical submodules of an $R$-module $M$. We say that $M$ is a radical distributive module, if $\mathcal{R}(_{R}M)$ is a distributive lattice. It is shown that the class of radical distributive modules contains the classes of multiplication modules and finitely generated distributive modules properly. It is shown that if $M$ is a semisimple $R$-module and for any radical submodule $N$ of $M$ with direct sum complement $\tilde{N}$, the complementary operation on $\mathcal{R}(_{R}M)$ is defined by $N':=\tilde{N}+rad(0)$, then $\mathcal{R}(_{R}M)$ with this unary operation forms a Boolean algebra. In particular, if $M$ is a multiplication module over a semisimple ring $R$, then $\mathcal{R}(_{R}M)$ is a Boolean algebra, which is also a homomorphic image of $\mathcal{R}(_{R}R)$.

Conditional Distributivity of aggregation functions, which has received wide attention from the researchers, is vital for many different fields, for example, integration theory, utility theory and so on. This article is mainly devoted to dealing with the conditional Distributivity of continuous t-norms over 2-uninorms. As the first step for investigating the conditional Distributivity of 2-uninorms, we give the complete characterization of all pairs $(T,\mathcal{H})$ {fulfilling} this property. Compared to the case of Distributivity of continuous t-norms over 2-uniorms, which leads to the 2-uninorm must be idempotent, the results obtained in this paper demonstrate that conditional Distributivity and Distributivity on this topic, are not equivalent.

Recently the distributive equations involving various classes of aggregation operators have aroused widespread attention because of their importance in the theoretic and applied communities of fuzzy set theory. 2-uninorms and semi-t-operators are two special classes of aggregation operators and have been proved to be useful in many areas such as fuzzy decision making, approximate reasoning and so on. Therefore, the aim of this paper is to investigate the left (right) Distributivity of semi-t-operators over 2-uninorms. We consider five subclasses of 2-uninorms and characterize the corresponding left (right) distributive equations of semi-t-operators over 2-uninorms.

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

This paper characterizes a uninorm distributive over a noncontinuous t-(co)norm, where the uninorm is locally internal on the boundary. Firstly, when the uninorm is disjunctive, the necessary conditions (resp. the sufficient conditions) for a uninorm distributive over a noncontinuous t-norm are analyzed under the certain condition. Secondly, the Distributivity of a conjunctive uninorm over a noncontinuous t-conorm is characterized by duality. In particular, this paper is related to the open question recalled by Klement in the Linz2000 closing session, which provides the noncontinuous solutions of that question.

In this paper, we establish the GH-derivative of multiplication of fuzzy functions, the so-called product rule. For the first time, a product rule is constructed while both of the multiplied functions are assumed to be fuzzy without any restriction on the signs of multiplied functions. The rule is extracted based on the MCE-product and its property of Distributivity. Then, we propose two important applications of the fuzzy product rule: An integration by parts formula for fuzzy functions and solving a nonlinear fuzzy differential equation. Some illustrative examples are given to verify the theoretical results.