One-Sided Noncommutative Gröbner Bases with Applications to Computing Greenʹs Relations

Standard noncommutative Gröbner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gröbner basis procedures for one-sided ideals in finitely presented noncommutative algebras over fields. The polynomials defining a K-algebra A as a quotient of a free K-algebra are combined with the polynomials defining a one-sided ideal I of A, by using a tagging notation whose essential effect is to forbid left multiplication. Standard noncommutative Gröbner basis techniques can then be applied to the mixed set of polynomials, thus calculating A/I whilst working in a free structure, avoiding the complication of computing in A. The paper concludes by showing how the results can be applied to completable presentations of semigroups and so enable calculations of Greenʹs relations.