Fuzzy accompanied approximation space under fuzzy relation

If there is a fuzzy relation $\widetilde{R}$ between two spaces $U$ and $V$, the fuzzy approximations in both spaces based on $\widetilde{R}$ are widely studied, and they basically reflect only the influence from one space to another. In this paper, on each space of $U$ and $V$, two new fuzzy relations are derived from $\widetilde{R}$, a positive low-value relation and a conservative high-value relation, to reflect the interaction and feedback between the two spaces. So, the fuzzy approximations based on them can reflect the combination of the action and the reaction from one space to another. Therefore, two spaces $U$ and $V$ are closely accompanied, and $(U, V, \widetilde{R})$ is a whole, so it is called a fuzzy accompanied approximation space (FAAS). In an FAAS, the properties of the fuzzy approximation models on each space are studied, the relationships between fuzzy approximation models of two spaces are researched, and examples to show how the approximation operator models in the FAAS to solve practical problems from multiple perspectives are also illustrated. More importantly, when the fuzzy relation $\widetilde{R}$ is a binary relation $R$ or the two spaces are the same, the special cases of FAAS are investigated and some important new models and new results are obtained, which add new ideas and methods to the current research.