I am trying to understand this question here that considers Sobolev spaces apparently and hence partial derivatives. What is the definition of minimality there? Is the minimality defined by cardinality or by the less-or-equal operation i.e. "x is the minimal element if it is less or equal to any other element" where the less-or-equal can also mean $\subseteq$? So simply
What does the minimality usually mean in the case of the partial derivatives and Sobolev spaces?
In the context of the question you linked to, $\alpha$ is a real number and $\alpha_3$ being minimal is to be interpreted in the usual sense for real numbers.