Vector Ambiguity and Freeness Problems in SL (2, Z)

We study the vector ambiguity problem and the vector freeness problem in SL (2, Z). Given a finitely generated n x n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y = Mx, where M ∈ S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL (2, Z), which is the set of 2 x 2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups.