Primitive, Almost Primitive, Test, and Δ-Primitive Elements of Free Algebras with the Nielsen–Schreier Property

We study generalized primitive elements of free algebras of finite ranks with the Nielsen–Schreier property and their automorphic orbits. A primitive element of a free algebra is an element of some free generating set of this algebra. Almost primitive elements are not primitive elements which are primitive in any proper subalgebra. Δ-primitive elements are elements whose partial derivatives generate the same one-sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate. We prove that an endomorphism preserving an automorphic orbit 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 described. 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 algebra L is even, n = 2m, then any Δ-primitive element of L is an automorphic image of the element w = [x1, x2] + ••• + [x2m − 1, x2m], there are no Δ-primitive elements of L if n is odd, and the group of automorphisms of the algebra L acts transitively on the set of all Δ-primitive elements.