A.2 Groebner bases
Let be a fixed monomial well-ordering on the -algebra .
Definition
For a set , define
to be the monoid, generated by the leading exponents
of elements of , that is
. We call the monoid of leading exponents.
There exist
,
such that
.
We define a set of leading monomials of be
.
A finite set is called Groebner basis of if and
only if , that is for any
there exists a
satisfying
.
Remark: In the non-commutative case we are working with well
ordering only. (See PLURAL conventions)
A Groebner basis is called minimal if and if
lm
for all .
Note, that any Groebner basis can be made minimal by deleting successively those
with
for some
.
For and we say that is reduced with
respect to if no monomial of is contained in .
Normal Form
A map
, is called a normal
form on if for any and any Groebner basis the following
holds:
(i)
then does not divide
for all .
(ii)
is called a normal form of with
respect to (note that such a map is not unique).
Remark:
With respect to the definitions of ideal and module (see
PLURAL conventions ) PLURAL works with left normal form only.
- Ideal membership:
-
For a Groebner basis
of the following holds:
if and only if
.
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