I am wondering if there is an extreme value distribution that is closed under both the minimum and the maximum operation. 
For example, for there is a Gumbel maximum distribution closed under the maximum (provided $\beta$ is the same for both distributions). Also there is the Gumbel minimum distribution closed under minimum. However, I am interested in a distribution that is closed under both such that, given two distributions $X_1,X_2$, I can find $Y_1=\min(X_1,X_2)$ and $Y_2=\max(X_1,X_2)$, which are of the same kind of distribution. 
If it only holds in special cases (except the trivial case if iid) that would also be interesting.

I imagine that it is possible that no such kind of distribution exists, in which case my question is why this does not exist / if there is a proof that it cannot exist.

Thank you.