Fuzzy function as an approximate solution to a system of fuzzy relation equations

This paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations. However, we put a new light on this problem connecting it with the interpolation of a fuzzy function. We introduce a notion of fuzzy function and its representation by fuzzy relation. We demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations. First, we explain the general framework and then we prove some results related to the problem of the best approximation. In particular, we have shown that fuzzy relations introduced by Sanchez and Mamdani are the best approximations in certain approximation spaces.