Primitive, almost primitive, test, and Delta-primitive elements of free algebras with the Nielsen-Schreier property

We study generalized prirmitive elements of free algebras of finite ranks w ith the Nielsen-Schreier property and their automorphic orbits. A primitive element of a free algebra is an element of same free generating set of thi s algebra. Almost primitive elements are not primitive elements which are p rimitive in any proper subalgebra. Delta-primitive elements are elements wh ose partial derivatives generate the same one-sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free genera tors generate. We prove that an endomorphism preserving an automorphic orbi t of a nonzero element of a free algebra of rank two is an automorphism. An algorithm to determine test elements of free algebras of rank two is descr ibed. A series of almost primitive elements is constructed and new examples of test elements are given. We prove that if the rank n of the free Lie al gebra L is even, n = 2m, then any Delta-primitive element of L is an automo rphic image of the element w = [x(1), x(2)] +...+ [x(2m-1), x(2m)], there a re no Delta-primitive elements of L if n is odd, and the group of automorph isms of the algebra L acts transitively on the set of all Delta-primitive e lements. (C) 2000 Academic Press.