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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 189#189–algebra 190#190, we denote by 239#239 the left submodule of a free module 240#240, generated by elements 241#241.

Let 226#226 be a fixed monomial well-ordering on the 189#189-algebra 190#190 with the Poincar@'e-Birkhoff-Witt (PBW) basis 242#242. For a given free module 240#240 with the basis 243#243, 226#226 denotes also a fixed module ordering on the set of monomials 244#244.

Definition

For a set 245#245, define 246#246 to be the 50#50-vector space, spanned on the leading monomials of elements of 247#247, 248#248.

We call 246#246 the span of leading monomials of 247#247.

Let 249#249 be a left 190#190-submodule. A finite set 250#250 is called a left Groebner basis of 251#251 if and only if 252#252, that is for any 253#253 there exists a 254#254 satisfying 255#255, i.e., if 256#256, then 257#257 with 258#258.


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 259#259 is called minimal (or reduced) if 260#260 and if 261#261 for all 254#254. Note, that any Groebner basis can be made minimal by deleting successively those 149#149 with 262#262 for some 263#263.

For 264#264 and 259#259 we say that 265#265 is completely reduced with respect to 189#189 if no monomial of 265#265 is contained in 266#266.

Left Normal Form

A map 267#267, is called a (left) normal form on 240#240 if for any 264#264 and any left Groebner basis 189#189 the following holds:

(i) 268#268,

(ii) if 269#269 then 270#270 does not divide 271#271 for all 254#254,

(iii) 272#272.

273#273 is called a left normal form of 265#265 with respect to 189#189 (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 189#189 of 251#251 the following holds: 274#274 if and only if the left normal form 275#275.

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 276#276 be a two-sided ideal and 277#277 be a two-sided Groebner basis of 276#276.

Then 278#278 if and only if the left normal form 279#279.

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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