Symmetrization Based Completion
Reinhard Bündgen · Birkhäuser Basel eBooks · 1998
We argue that most completion procedures for finitely presented algebras can be simulated by term completion procedures based on a generalized symmetrization process. Therefore we present three different constructive definitions of symmetrization procedures that can take the rôle of the orientation step in a symmetrization based completion procedure. We investigate confluence and compatibility properties of the symmetrized rules computed by the different symmetrization procedures. Based on semi-compatibility properties we can present a generic version of the critical pair theorem that specializes to the critical pair theorems of Knuth-Bendix completion procedures and algebraic completion procedures like Buchberger’s algorithm respectively. This critical pair theorem also applies to symmetrization based completion procedures using a normalized reduction relation if the result of the symmetrization is both semi-compatible and semi-stable. We conclude our paper showing how a generic Buchberger algorithm for polynomials over arbitrary finitely presented rings can be formulated as a symmetrization based completion procedure. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.