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7.6.2 Groebner bases for two-sided 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 well-ordering 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 two-sided ideal
.
Let be a fixed monomial ordering on .
We say that a subset is a (two-sided) Groebner basis for the ideal with respect to , if . That is
there exists , such that
divides .
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