A new Composition-Diamond lemma for associative conformal algebras

Let C(B, N) be the free associative conformal algebra generated by a set B with a bounded locality N. Let S be a subset of C(B, N). A Composition-Diamond lemma for associative conformal algebras is first established by Bokut, Fong and Ke in 2004 [L. A. Bokut, Y. Fong and W.-F. Ke, Composition-Diamond Lemma for associative conformal algebras, J. Algebra 272 (2004) 739-774] which claims that if (i) S is a Grobner-Shirshov basis in C(B, N), then (ii) the set of S-irreducible words is a linear basis of the quotient conformal algebra C(B, N| S), but not conversely. In this paper, by introducing some new definitions of normal S-words, compositions and compositions to be trivial, we give a new Composition-Diamond lemma for associative conformal algebras, which makes the conditions (i) and (ii) equivalent. We show that for each ideal I of C(B, N), I has a unique reduced Grobner-Shirshov basis. As applications, we show that Loop Virasoro Lie conformal algebra and Loop Heisenberg-Virasoro Lie conformal algebra are embeddable into their universal enveloping associative conformal algebras.