| ### 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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