2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103835We introduce the notion of Groebner 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 Groebner 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 Groebner basis is infinite and our Groebner S-basis is finite. We obtain also explicit minimal Groebner bases of these ideals.15 pagesRings and Algebras16S15 (Primary). 16R10; 16S30; 16W50 (Secondary)text