Averaging algebras, rewriting systems and Gr$\"o$bner-Shirshov bases
  In this paper, we study the averaging operator by assigning a rewriting system to it. We obtain some basic results on the kind of rewriting system we used. In particular, we obtain a sufficient and necessary condition for the confluence. We supply the relationship between rewriting systems and Grobner-Shirshov bases based on bracketed polynomials. As an application, we give a basis of the free unitary averaging algebra on a non-empty set. 
Xing Gao and Tianjie Zhang
math.RA