Flats - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Matroids

 Flats
 return the flats of a matroid

 Calling Sequence Flats(M)

Parameters

 M -

Description

 • The flats of a matroid are those subsets which are maximal with respect to their rank. Given a matroid, the Flats command returns the flats of that matroid as a list of sets.
 • If the flats of this matroid haven't been computed before, they are computed by this command and stored for any future computations that use the flats.

Examples

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

Find the sets of indices whose corresponding columns form a flat in the column space 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)
 > $\mathrm{Flats}\left(M\right)$
 $\left[{\varnothing }{,}\left\{{1}\right\}{,}\left\{{2}\right\}{,}\left\{{3}\right\}{,}\left\{{4}\right\}{,}\left\{{5}\right\}{,}\left\{{6}\right\}{,}\left\{{1}{,}{2}\right\}{,}\left\{{1}{,}{3}\right\}{,}\left\{{2}{,}{3}\right\}{,}\left\{{1}{,}{4}\right\}{,}\left\{{2}{,}{4}\right\}{,}\left\{{3}{,}{4}\right\}{,}\left\{{1}{,}{5}\right\}{,}\left\{{2}{,}{5}\right\}{,}\left\{{3}{,}{5}\right\}{,}\left\{{4}{,}{5}\right\}{,}\left\{{1}{,}{6}\right\}{,}\left\{{2}{,}{6}\right\}{,}\left\{{3}{,}{6}\right\}{,}\left\{{4}{,}{6}\right\}{,}\left\{{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}\right\}{,}\left\{{1}{,}{3}{,}{4}\right\}{,}\left\{{2}{,}{3}{,}{4}\right\}{,}\left\{{1}{,}{3}{,}{5}\right\}{,}\left\{{2}{,}{3}{,}{5}\right\}{,}\left\{{3}{,}{4}{,}{5}\right\}{,}\left\{{1}{,}{2}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{6}\right\}{,}\left\{{1}{,}{4}{,}{6}\right\}{,}\left\{{2}{,}{4}{,}{6}\right\}{,}\left\{{3}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{5}{,}{6}\right\}{,}\left\{{3}{,}{5}{,}{6}\right\}{,}\left\{{4}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{4}{,}{5}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{4}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{5}{,}{6}\right\}{,}\left\{{3}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}\right\}{,}\left\{{1}{,}{2}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}\right\}\right]$ (3)

References

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