Details:
Title  Monomial orderings, rewriting systems, and Gröbner bases for the commutator ideal of a free alegbra  Author(s)  Susan M. Hermiller, Xenia H. Kramer, Reinhard C. Laubenbacher  Type  Article in Journal  Abstract  In this paper we consider a free associative algebra on three generators over an arbitrary field K. Given a term ordering on the commutative polynomial ring on three variables over K, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.
The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such, more general, monomial orderings in noncommutative Gröbner basis theory.  ISSN  07477171  Copyright  Academic Press 
URL 
dx.doi.org/10.1006/jsco.1998.0245 
Language  English  Journal  Journal of Symbolic Computation  Volume  27  Number  2  Pages  133141  Publisher  Academic Press, Inc.  Address  Duluth, MN, USA  Year  1999  Month  February  Translation 
No  Refereed 
No 
