Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let equivalent to be the equivalence relation defined on A(m) such that (x(1),..., x(m)) equivalent to (y(1),..., y(m)) if there exists a permutation sigma on {1,..., m} such that y(sigma(i)) = x(i) for all i. Let A((m)) denote the set of all equivalence classes determined by equivalent to. Two elements X and Y in A((m)) are said to be adjacent if (x(1),..., x(m-1), a) is an element of X and (x(1),..., x(m-1), b) is an element of Y for some x(1),..., x(m) (1) is an element of A and some distinct elements a, b is an element of A. We study the structure of functions from A((m)) to B((n)) that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.