|Abstract||Canonical bases, also called SAGBI bases, for subalgebras of the non-commutative polynomial ring are investigated. The process of subalgebra reduction is defined. Methods, including generalizations of the standard Grobner bases techniques, are developed for the test whether bases are canonical, and for the completion procedure of constructing canonical bases. The special case of homogeneous subalgebras is discussed.|
In this paper we study canonical bases, analogs of Grobner bases for ideals, for subalgebras of KhXi, the free associative algebra over the field K in the set of indeterminates X. Following the paper  by Robbiano and Sweedler, we will call these bases SAGBI bases, where SAGBI is an abbreviation for Subalgebra Analog to Grobner Bases for Ideals.
SAGBI bases theory was introduced, for commutative polynomial rings, by Kapur and Madlener (see ), and independently by Robbiano and Sweedler (). This paper follows the approach of Robbiano and Sweedler.