Why I cannot type $x_\min$ directly?

Question

I found that I can type such as $x_\alpha$ directly without the bracket (like $x_{\alpha}$), this just like I can type $\frac12$ instead of $\frac{1}{2}), but why I cannot omit the bracket when I type $x_\min$? I shouldn't omit the bracket or there will be an error? I just want to know the reason of this condition.