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# basisElements -- extract the matrix of generators from an involutive basis or factor module basis

## Synopsis

• Usage:
B = basisElements J
B = basisElements F
• Inputs:
• Outputs:
• B,

## Description

If the argument of basisElements is an instance of the type InvolutiveBasis, then the columns of B are generators for the module spanned by the involutive basis. These columns form a Gr\"obner basis for this module.

If the argument of basisElements is an instance of the type FactorModuleBasis, then the columns of B are generators for the monomial cones in the factor module basis.

 i1 : R = QQ[x,y]; i2 : I = ideal(x^3,y^2); o2 : Ideal of R i3 : J = janetBasis I; i4 : basisElements J o4 = | y2 xy2 x3 x2y2 | 1 4 o4 : Matrix R <-- R
 i5 : R = QQ[x,y,z]; i6 : M = matrix {{x*y,x^3*z}}; 1 2 o6 : Matrix R <-- R i7 : J = janetBasis M; i8 : F = factorModuleBasis J +--+------+ o8 = |1 |{z, y}| +--+------+ |x |{z} | +--+------+ | 2| | |x |{z} | +--+------+ | 3| | |x |{x} | +--+------+ o8 : FactorModuleBasis i9 : basisElements F o9 = | 1 x x2 x3 | 1 4 o9 : Matrix R <-- R i10 : multVar F o10 = {set {z, y}, set {z}, set {z}, set {x}} o10 : List