Gröbner bases of ideals invariant under endomorphisms

We introduce the notion of GröbnerS-basis of an ideal of the free associative algebra K X over a field K invariant under the action of a semigroupS of endomorphisms of the algebra. We calculate the GröbnerS-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 semigroupsS. In the latter case, if X>2, the ordinary Gröbner basis is infinite and our GröbnerS-basis is finite. We obtain also explicit minimal Gröbner bases of these ideals.