Automorphic orbits in free non-associative algebras

We consider finitely generated free non-associative algebras, free commutat ive non-associative algebras, and free anti-commutative non-associative alg ebras. We study orbits of elements of these algebras under the action of au tomorphism groups. Using free differential calculus we obtain matrix criter ia for a system of elements to have given rank (or to be primitive). It giv es us a possibility to construct fast algorithm to recognize primitive syst ems of elements. We show that if an endomorphism of a free algebra preserve s the automorphic orbit of a nonzero element, then it is an automorphism of this algebra. In particular, endomorphisms preserving primitivity of eleme nts are automorphisms. (C) 2001 Academic Press.