The research in the field of the so called Fuzzy Mathematics can be conditionally devided into two mainstreams: the first one emphasizes on the study of different fuzzy structures (topological, algebraic, analytical, etc.) on an ordinary set $X$, while $L$-valued sets $X$ (that are sets equipped with some $L$-valued equalities $E: X\times X \to L$, or, more generally, with $L$-valued relations $R: X \times X \to L$) are the starting point for the second one. ($L$ being a lattice usually with an additionally algebraic structure). The aim of this work is to discuss the problem how an $L$-valued relation given on a set $X$ can be extended to the $L$-valued relation $\R$ on the $L$-powerset $L^X$. This problem, is important, among other for the theory of $L$-fuzzy topological spaces in the sense of [15], [16].