Reduction operators and completion of rewriting systems

Abstract

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 notions of confluence and of Church–Rosser property. We show that these notions are equivalent. We give an algebraic formulation of completion and show that such a completion exists using the lattice structure. We interpret the confluence for reduction operators in terms of Gröbner bases. Finally, we introduce generalised reduction operators relative to non totally ordered sets.