Introduction

Reasoning and computing in finitely presented algebraic structures is wide-spread in many fields in mathematics, physics and computer science. Reduction in the sense of simplification combined with appropriate completion methods is one general technique which is often successfully applied in this context, e.g. to solve the word problem and hence to compute effectively in the structure.

One fundamental application of this technique to polynomial rings was provided by B. Buchberger (1965) in his uniform effective solution of the ideal membership problem establishing the theory of Gröbner bases. These bases can be characterized in various manners, e.g. by properties of a reduction relation associated with polynomials (confluence or all elements in the ideal reduce to zero) or by special representations for the ideal elements with respect to a Gröbner basis (standard representations).