operations

Fuzzy soft set is a general mathematical tool to deal with uncertain and fuzziness information. But inadequacy lies in which cannot express various uncertainty in a more natural and accurate manner, due to the reason of adopting the fuzzy membership which restricts to [0, 1]. As a result, it merely concerns non-negative fuzzy belongingness of elements, so as to be incapable of capturing the conflict information that supporting, neutral and opposing scenarios regarding belongingness of elements in a set to taken into account. To overcome the limitation, this paper extends fuzzy soft set to bilateral fuzzy soft set (BFSS), which considers both non-negative and negative sides for fuzzy membership and normalizes the range to [−1, 1]. Then defines some basic operations on BFSS, as well as a detail study of relevant properties. In order to eliminate redundant parameters and draw the essential part of BFSS, the parameters reduction on BFSS is also investigated, and the significance of parameters is introduced accordingly. On this basis, an assessment approach for multi-criteria multi-person decision making under conflict information, together with the algorithm for implementation is presented. Finally, a typical numerical example regarding voting demonstrates the validity and feasibility of the proposed method.

An operator is a special symbol for performing a specific function. Several operators like modal operators, topological operators, level operators, etc. have been defined over intuitionistic fuzzy sets. At the same time, so many operations were introduced and studied. The key objective of this paper is to study those operations over intuitionistic fuzzy sets and to investigate their properties. Some new results are obtained and proved.

The structure of an α(β,β) -topological ring is richer in comparison with the structure of an α(β,β) -topological group. The theory of α(β,β) -topological rings has many common features with the theory of α(β,β) -topological groups. Formally, the theory of α(β,β) -topological abelian groups is included in the theory of α(β,β) -topological rings. The purpose of this paper is to introduce and study the concepts of α(β,β) -topological rings and α(β,γ) -topological R-modules. we show how they may be introduced by specifying the neighborhoods of zero, and present some basic constructions. We provide fundamental concepts and basic results on α(β,β) -topological rings and α(β,γ) -topological Rmodules.

The line graph of the graph $Gamma$ denoted by $L(Gamma)$ is a graph with a vertex set consists of the sets of edges of $Gamma$ and two vertices are adjacent in $L(Gamma)$ if they are incident in $Gamma$. In this article, we discuss and determine the effect of operations on the line graphs of simple graphs.