GroundSet - Maple Help

Matroids

 GroundSet
 return the ground set of a matroid

 Calling Sequence GroundSet(M)

Parameters

 M -

Description

 • Every matroid is defined with respect to some set $E$ called its ground set. The GroundSet command returns this ground set as a list.

Examples

 > $\mathrm{with}\left(\mathrm{Matroids}\right):$

Return the ground set of a matroid formed via the independence relationships among the columns of a matrix.

 > $A≔\mathrm{Matrix}\left(\left[\left[1,1,1,0,0,0\right],\left[0,0,0,1,1,1\right],\left[1,0,0,1,0,0\right],\left[0,1,0,0,1,0\right],\left[0,0,1,0,0,2\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{cccccc}{1}& {1}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {0}& {2}\end{array}\right]$ (1)
 > $M≔\mathrm{Matroid}\left(A\right)$
 ${M}{≔}⟨\begin{array}{c}{thⅇ linⅇar matroiⅆ whosⅇ grounⅆ sⅇt is thⅇ sⅇt of column vⅇctors of thⅇ matrix:}\\ \left[\begin{array}{cccccc}{1}& {1}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {0}& {2}\end{array}\right]\end{array}⟩$ (2)
 > $\mathrm{GroundSet}\left(M\right)$
 $\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}\right]$ (3)

Return the ground set of a matroid formed explicitly via its bases.

 > $M≔\mathrm{Matroid}\left(\left["apples","bananas","coconuts"\right],\mathrm{bases}=\left[\left\{"apples","bananas"\right\},\left\{"apples","coconuts"\right\},\left\{"bananas","coconuts"\right\}\right]\right)$
 ${M}{≔}⟨{thⅇ uniform matroiⅆ of rank 2 on 3 ⅇlⅇmⅇnts}⟩$ (4)
 > $\mathrm{GroundSet}\left(M\right)$
 $\left[{"apples"}{,}{"bananas"}{,}{"coconuts"}\right]$ (5)

References

 James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.