Averaging algebras, rewriting systems and Gr$\"o$bner-Shirshov bases
Xing Gao, Tianjie Zhang
TL;DR
This paper investigates averaging operators through rewriting systems, establishing confluence conditions, linking to Grobner-Shirshov bases, and providing a basis for free unitary averaging algebras.
Contribution
It introduces a rewriting system approach to averaging operators, connects it with Grobner-Shirshov bases, and constructs a basis for free averaging algebras.
Findings
Established necessary and sufficient conditions for rewriting system confluence.
Linked rewriting systems with Grobner-Shirshov bases for bracketed polynomials.
Provided a basis for the free unitary averaging algebra.
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 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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology