Gröbner bases of ideals invariant under endomorphisms

We introduce the notion of Gröbner S-basis of an ideal of the free associative algebra K〈X〉 over a field K invariant under the action of a semigroup S of endomorphisms of the algebra. We calculate the Gröbner S-bases of the ideal corresponding to the universal enveloping algebra of the free nilpotent of class 2 Lie algebra and of the T-ideal generated by the polynomial identity [x, y, z] = 0, with respect to suitable semigroups S. In the latter case, if |X | > 2, the ordinary Gröbner basis is infinite and our Gröbner S-basis is finite. We obtain also explicit minimal Gröbner bases of these ideals.