Abstract
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 Gröbner–Shirshov bases based on bracketed polynomials. As an application, we give a basis of the free unitary averaging algebra on a nonempty set.
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