Recently, Fang and Hu introduced the definition of semi-t-operators on bounded lattices. In this study, we propose several construction methods to obtain such a semi-t-operator from a given semi-t-conorm and semi-t-norm on bounded lattices. Furthermore, we discuss the presence of idempotent semi-t-operators on bounded lattices, and show several different methods for construction of idempotent semi-t-operators on bounded lattices with additional constraints.

As a generalization of nullnorms, semi-t-operators are interesting both in theory and practical applications. In this paper, we investigate some properties of idempotent semi-t-operators on bounded. Furthermore, we propose two construction methods of idempotent semi-t-operators on bounded lattices containing only two different elements which are incomparable with $a$ but comparable with $b$. These methods are generalization of several known ones in the literature.

The study of operators satisfying Bishop's property (b) is of significant interest and is currently being done by a number of mathematicians around the world. Recently Uchiyama and Tanahashi [Oper. Matrices 4 (2009), 517–524] showed that a paranormal operator has Bishop's property (b). In this paper we introduce a new class of operators which we call the class of kquasi- paranormal operators. An operator T is said to be a k-quasi-paranormal operator if it satisfies ||Tk+1x||2£||Tk+2x|||Tkx|| for all xÎH where k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi-class A operators. We prove basic properties and give a structure theorem of k-quasi-paranormal operators. We also show that Bishop’s property (b) holds for this class of operators. Finally, we prove that if E is the Riesz idempotent for a nonzero isolated point l0 of the spectrum of a k-quasi-paranormal operator T, then E is self-adjoint if and only if the null space of T-l0, ker(T-l0)  ker(T*-`l0).

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

The cross-migrativity has been investigated for families of certain aggregation operators, such as t-norms, t-subnorms and uninorms. In this paper, we aim to study the cross-migrativity property for semi-t-operators, which are generalizations of t-operators by omitting commutativity. Specifically, we give all solutions of the cross-migrativity equation for all possible combinations of semi-t-operators. Moreover, it is shown that if a semi-t-operator F is alpha-cross-migrative over another semi-t-operator G, then G must be a semi-nullnorm except one case. Finally, it is pointed out that the cross-migrativity property between two semi-t-operators is always determined by their underlying operators except a few cases.

‎Let $R$ be a commutative ring with identity‎, ‎$S$ be a multiplicatively closed subset of $R$‎, ‎and $M$ be an $R$-module‎. A submodule $N$ of $M$ is said to be \emph{idempotent} if $N=(N:_RM)^2M$‎. ‎Also‎, ‎$M$ is said to be \emph{fully idempotent} if every submodule of $M$ is idempotent‎. The aim of this paper is to introduce the notion of fully $S$-idempotent modules as a generalization of fully idempotent modules and investigate some properties of this class of modules‎.

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.