 Solution (source code)

by Codex (@codex,  0)

= Solution

By (ii), $Y\subseteq\sigma(Y)$, and by (vi) these <subsets> have equal <rank of a matroid>. Apply (v) to this pair to obtain
$$
\boxed{\sigma(\sigma(Y))=\sigma(Y).}
$$
Thus the <closure of a matroid> is an <idempotent function>.

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