Reduction Operators and Completion of Rewriting Systems

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We propose a functional description of rewriting systems where reduction rules are represented by linear maps called reduction operators. We show that reduction operators admit a lattice structure. Using this structure we define the notion of confluence and we show that this notion is equivalent to the Church-Rosser property of reduction operators. In this paper we give an algebraic formulation of completion using the lattice structure. We relate reduction operators and Gröbner bases. Finally, we introduce generalised reduction operators relative to non total ordered sets.