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7.4.2 Groebner bases in G-algebras

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

For a 190#190–algebra 191#191, we denote by 241#241 the left submodule of a free module 242#242, generated by elements 243#243.

Let 228#228 be a fixed monomial well-ordering on the 190#190-algebra 191#191 with the Poincar@'e-Birkhoff-Witt (PBW) basis 244#244. For a given free module 242#242 with the basis 245#245, 228#228 denotes also a fixed module ordering on the set of monomials 246#246.

Definition

For a set 247#247, define 248#248 to be the 50#50-vector space, spanned on the leading monomials of elements of 249#249, 250#250.

We call 248#248 the span of leading monomials of 249#249.

Let 251#251 be a left 191#191-submodule. A finite set 252#252 is called a left Groebner basis of 253#253 if and only if 254#254, that is for any 255#255 there exists a 256#256 satisfying 257#257, i.e., if 258#258, then 259#259 with 260#260.


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 261#261 is called minimal (or reduced) if 262#262 and if 263#263 for all 256#256. Note, that any Groebner basis can be made minimal by deleting successively those 149#149 with 264#264 for some 265#265.

For 266#266 and 261#261 we say that 267#267 is completely reduced with respect to 190#190 if no monomial of 267#267 is contained in 268#268.

Left Normal Form

A map 269#269, is called a (left) normal form on 242#242 if for any 266#266 and any left Groebner basis 190#190 the following holds:

(i) 270#270,

(ii) if 271#271 then 272#272 does not divide 273#273 for all 256#256,

(iii) 274#274.

275#275 is called a left normal form of 267#267 with respect to 190#190 (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 190#190 of 253#253 the following holds: 276#276 if and only if the left normal form 277#277.

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 278#278 be a two-sided ideal and 279#279 be a two-sided Groebner basis of 278#278.

Then 280#280 if and only if the left normal form 281#281.

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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