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.

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

$A≔\mathrm{Matrix}\left(\left[\left[1\,1\,1\,0\,0\,0\right]\,\left[0\,0\,0\,1\,1\,1\right]\,\left[1\,0\,0\,1\,0\,0\right]\,\left[0\,1\,0\,0\,1\,0\right]\,\left[0\,0\,1\,0\,0\,2\right]\right]\right)$

$A≔\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 2\end{array}\right]$

$M≔\mathrm{Matroid}\left(A\right)$

$M≔⟨\begin{array}{c}\mathrm{th\ⅇ\; lin\ⅇar\; matroi\ⅆ\; whos\ⅇ\; groun\ⅆ\; s\ⅇt\; is\; th\ⅇ\; s\ⅇt\; of\; column\; v\ⅇctors\; of\; th\ⅇ\; matrix:}\\ \left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\\ 1& 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 2\end{array}\right]\end{array}⟩$

$\mathrm{Flats}\left(M\right)$

$\left[\varnothing \,\left\{1\right\}\,\left\{2\right\}\,\left\{3\right\}\,\left\{4\right\}\,\left\{5\right\}\,\left\{6\right\}\,\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{2\,3\right\}\,\left\{1\,4\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\,\left\{1\,5\right\}\,\left\{2\,5\right\}\,\left\{3\,5\right\}\,\left\{4\,5\right\}\,\left\{1\,6\right\}\,\left\{2\,6\right\}\,\left\{3\,6\right\}\,\left\{4\,6\right\}\,\left\{5\,6\right\}\,\left\{1\,2\,3\right\}\,\left\{1\,3\,4\right\}\,\left\{2\,3\,4\right\}\,\left\{1\,3\,5\right\}\,\left\{2\,3\,5\right\}\,\left\{3\,4\,5\right\}\,\left\{1\,2\,6\right\}\,\left\{1\,3\,6\right\}\,\left\{2\,3\,6\right\}\,\left\{1\,4\,6\right\}\,\left\{2\,4\,6\right\}\,\left\{3\,4\,6\right\}\,\left\{1\,5\,6\right\}\,\left\{2\,5\,6\right\}\,\left\{3\,5\,6\right\}\,\left\{4\,5\,6\right\}\,\left\{1\,2\,4\,5\right\}\,\left\{1\,2\,3\,6\right\}\,\left\{1\,3\,4\,6\right\}\,\left\{2\,3\,4\,6\right\}\,\left\{1\,3\,5\,6\right\}\,\left\{2\,3\,5\,6\right\}\,\left\{3\,4\,5\,6\right\}\,\left\{1\,2\,3\,4\,5\right\}\,\left\{1\,2\,4\,5\,6\right\}\,\left\{1\,2\,3\,4\,5\,6\right\}\right]$

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