I want to find a family of continuous distributions $H$, where if $X$ and $Y$ are random variables from $H$, $aX+b$, $max(X,Y)$, $min(X,Y)$ also have distributions from $H$. The distributions of $H$ are be parameterized by a finite number of parameters.
What I tried so far:
I tried to solve the functional equation $$G\left(\frac{x-\mu_1}{\sigma_1},\gamma_1\right )\cdot G\left(\frac{x-\mu_2}{\sigma_2},\gamma_2\right )=G\left(\frac{x-\mu_{max}(\mu_1,\sigma_1,\gamma_1,\mu_2,\sigma_2,\gamma_2)}{\sigma_{max}(\mu_1,\sigma_1,\gamma_1,\mu_2,\sigma_2,\gamma_2))},\gamma_{max}(\mu_1,\sigma_1,\gamma_1,\mu_2,\sigma_2,\gamma_2)\right )$$ to find the CDF for the 3 parameter case.(Then maybe derive a distribution that is also min-invariant.) But this doesn't appear to have an easy solution. I'm not sure what assumptions should I make to simplify this.