| ### 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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