On fuzzy linear representations of finite groups

Abstract

A linear representation ϱ of a finite group G in a finite-dimensional vector space V induces, through Zadeh's extension principle, a function, 9≈, from I G to into I GL( V) where GL( V) is the group of all linear automorphisms of V. If W is a fuzzy subspace of V, the group of all fuzzy linear automorphisms of W GL( W ), is a subgroup of GL( V). W is said to be stable under the action of a fuzzy subgroup A of G if 9≈( A ) is a subset of GL( W ), i.e. 9≈( A ) is zero at every f inside GL( V) and outside GL( q ). If W is stable under the action of A then its support subspace is stable under the action of the support subgroup of A in the crisp sense. Finally, we show that; if there are two stable fuzzy subspaces one of them is contained in the other, then the smaller one has a fuzzy direct summand in the bigger which is also a stable fuzzy subspace.