Parametrized Gröbner–Shirshov bases

Abstract

ABSTRACTWe consider the problem of describing Gröbner–Shirshov bases for free associative algebras in finite terms. To this end we consider parametrized elements of an algebra and give methods for working with them which under favorable conditions lead to a basis given by finitely many patterns. On the negative side we show that in general there can be no algorithm. We relate our study to the problem of verifying that a given set of words in certain groups yields Bokut’ normal forms (or groups with a standard basis).