Automorphic Orbits in Free Non-associative Algebras

We consider finitely generated free non-associative algebras, free commutative non-associative algebras, and free anti-commutative non-associative algebras. We study orbits of elements of these algebras under the action of automorphism groups. Using free differential calculus we obtain matrix criteria for a system of elements to have given rank (or to be primitive). It gives us a possibility to construct fast algorithm to recognize primitive systems of elements. We show that if an endomorphism of a free algebra preserves the automorphic orbit of a nonzero element, then it is an automorphism of this algebra. In particular, endomorphisms preserving primitivity of elements are automorphisms.