Fuzzy algebraic structure

Summary

A fuzzy algebraic structure replaces a substructure of an algebra by a fuzzy subset. A fuzzy subset gives each element a degree of membership between $0$ and $1$. The fuzzy subset must respect the operations: the result of an operation belongs at least as much as the least member among its arguments. The first case studied was the fuzzy subgroup of a group, introduced by Rosenfeld in 1971. Fuzzy subrings, fuzzy ideals and fuzzy subalgebras of general algebras followed. A fuzzy subalgebra is determined by its level sets, which form a nested family of ordinary subalgebras, one for each membership level.