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Two matroids are isomorphic if there is a relabeling of their ground sets so that they are equal.

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

$\mathrm{A1}≔\mathrm{Matrix}\left(\left[\left[1\,1\,0\,0\right]\,\left[-1\,0\,1\,0\right]\,\left[0\,-1\,-1\,0\right]\right]\right)$

$\mathrm{A1}≔\left[\begin{array}{cccc}1& 1& 0& 0\\ \mathrm{-1}& 0& 1& 0\\ 0& \mathrm{-1}& \mathrm{-1}& 0\end{array}\right]$

$\mathrm{A2}≔\mathrm{Matrix}\left(\left[\left[0\,4\,0\,2\right]\,\left[0\,-6\,3\,0\right]\,\left[0\,0\,-4\,-4\right]\right]\right)$

$\mathrm{A2}≔\left[\begin{array}{cccc}0& 4& 0& 2\\ 0& \mathrm{-6}& 3& 0\\ 0& 0& \mathrm{-4}& \mathrm{-4}\end{array}\right]$

$\mathrm{AreIsomorphic}\left(\mathrm{Matroid}\left(\mathrm{A1}\right)\,\mathrm{Matroid}\left(\mathrm{A2}\right)\right)$

$\mathrm{true}$

$\mathrm{A3}≔\mathrm{Matrix}\left(\left[\left[1\,1\,0\,0\right]\,\left[0\,0\,1\,1\right]\,\left[1\,1\,0\,1\right]\right]\right)$

$\mathrm{A3}≔\left[\begin{array}{cccc}1& 1& 0& 0\\ 0& 0& 1& 1\\ 1& 1& 0& 1\end{array}\right]$

$J≔⟨x+y\,z+w\,x+y+w⟩$

$J≔⟨x+y\,z+w\,x+y+w⟩$

$\mathrm{AreIsomorphic}\left(\mathrm{Matroid}\left(\mathrm{A3}\right)\,\mathrm{Matroid}\left(J\right)\right)$

$\mathrm{false}$

$\mathrm{AreIsomorphic}\left(\mathrm{Dual}\left(\mathrm{Matroid}\left(\mathrm{A3}\right)\right)\,\mathrm{Matroid}\left(J\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{Matroid}\left(J\right)\,\mathrm{Matroid}\left(\mathrm{A1}\right)\right)$

$\mathrm{false}$

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