Gröbner Bases Theory and The Diamond Lemma

[摘要] Commutative Gröbner bases theory is well known and widely used. In this thesis, we will discuss thoroughly its generalization tononcommutative polynomial ring k<X> which is also an associative free algebra. We introduce some results on monomial orders due to John Lawrence and the author. We show that a noncommutative monomial order is a well order while a one-sided noncommutative monomialorder may not be. Then we discuss the generalization of polynomial reductions, S-polynomials and the characterizations ofnoncommutative Gröbner bases. Some results due to Mora are also discussed, such as the generalized Buchberger;;s algorithm and the solvability of ideal membership problem for homogeneous ideals. At last, we introduce Newman;;s diamond lemma and Bergman;;s diamond lemma and show their relations with Gröbner bases theory.

[发布机构] University of Waterloo

[效力级别] Gröbner Bases

[关键词] Mathematics;Gröbner Bases;Diamond Lemma