The Remak-Krull-Schmidt Theorem on\\ Fuzzy Groups

In this paper we study a representation of a fuzzy subgroup $\mu$ of a group $G$, as a product of indecomposable fuzzy subgroups called the components of $\mu$. This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and some of their properties to prove this theorem. We illustrate with some examples.