
7.6.2 Groebner bases for twosided ideals in free associative algebras
We call a total ordering on the free monoid
(where is identified with the identity element) a
monomial ordering if the following conditions hold:

is a wellordering on , that is
,

, if , then
,

, if
and , then .
Hence the notions like a leading monomial and a leading coefficient transfer to this situation.
We say that a monomial
divides monomial
, if there exist monomials
, such that
.
In other words
is a proper subword of
.
For a subset
,
define a leading ideal of to be the twosided ideal
.
Let be a fixed monomial ordering on .
We say that a subset is a (twosided) Groebner basis for the ideal with respect to , if . That is
there exists , such that
divides .
