
7.4.2 Groebner bases in GalgebrasWe follow the notations, used in the SINGULAR Manual (e.g. in Standard bases). For a algebra , we denote by the left submodule of a free module , generated by elements . Let be a fixed monomial wellordering on the algebra with the Poincar@'eBirkhoffWitt (PBW) basis . For a given free module with the basis , denotes also a fixed module ordering on the set of monomials . DefinitionFor a set , define to be the vector space, spanned on the leading monomials of elements of , . We call the span of leading monomials of .Let be a left submodule. A finite set is called a left Groebner basis of if and only if , that is for any there exists a satisfying , i.e., if , then with .
A Groebner basis is called minimal (or reduced) if and if for all . Note, that any Groebner basis can be made minimal by deleting successively those with for some . For and we say that is completely reduced with respect to if no monomial of is contained in . Left Normal FormA map , is called a (left) normal form on if for any and any left Groebner basis the following holds: (i) , (ii) if then does not divide for all , (iii) . is called a left normal form of with respect to (note that such a map is not unique).
Left ideal membershipFor a left Groebner basis of the following holds: if and only if the left normal form . 