### 7.4.2 Groebner bases in G-algebras

We follow the notations, used in the SINGULAR Manual (e.g. in Standard bases).

For a -algebra , we denote by the left submodule of a free module , generated by elements .

Let be a fixed monomial well-ordering on the -algebra with the Poincar@'e-Birkhoff-Witt (PBW) basis . For a given free module with the basis , denotes also a fixed module ordering on the set of monomials .

### Definition

For a set , define to be the -vector space, spanned on the leading monomials of elements of , .

We call the span of leading monomials of .

Let be a left -submodule. A finite set is called a left Groebner basis of if and only if , that is for any there exists a satisfying , i.e., if , then with .

Remark: In general non-commutative algorithms are working with global well-orderings only (see PLURAL, Monomial orderings and Term orderings), unless we deal with graded commutative algebras via Graded commutative algebras (SCA).

A Groebner basis is called minimal (or reduced) if and if for all . Note, that any Groebner basis can be made minimal by deleting successively those with for some .

For and we say that is completely reduced with respect to if no monomial of is contained in .

### Left Normal Form

A map , is called a (left) normal form on if for any and any left Groebner basis the following holds:

(i) ,

(ii) if then does not divide for all ,

(iii) .

is called a left normal form of with respect to (note that such a map is not unique).

Remark: As we have already mentioned in the definitions `ideal` and `module` (see PLURAL), by `NF` (or `reduce`) PLURAL understands a left normal form. Note, that `rightNF` from nctools_lib allows to compute a right normal form.

### Left ideal membership (plural)

For a left Groebner basis of the following holds: if and only if the left normal form .

For computing a left Groebner basis `G` of `I`, use std (plural).

For computing a left normal form of `f` with respect to `G`, use reduce (plural).

### Right ideal membership (plural)

The right ideal membership is analogous to the left one:

for computing a right Groebner basis `G` of `I`, use rightstd (letterplace) from nctools_lib,

for computing a right normal form of `f` with respect to `G`, use rightNF from nctools_lib.

### Two-sided ideal membership (plural)

Let be a two-sided ideal and be a two-sided Groebner basis of .

Then if and only if the left normal form .

For computing a two-sided Groebner basis `T` of `J`, use twostd (plural),

for computing a normal form of `f` with respect to `T`, use reduce (plural).

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