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7.4.2 Groebner bases in G-algebras

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

Definition


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).

Left Normal Form


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

Left ideal membership (plural)

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)

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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