Is $\arg$ ever used on its own?

Question

I am familiar with the use of $\arg$ in $\arg \min$ or $\arg \max$.

Would you ever use $\arg$ to define a set of arguments where a condition is true?

Can I use these two notations interchangeably or is the second line not a canonical use of $\arg$?

$X = \{i : z_i = 1\}$

$X = \arg_i z_i = 1$

Thanks!

Answer 1

If you wrote $$ X = \arg_i z_i = 1 $$ I would not know what you mean.

What branch of math uses notations $\arg$ and $\arg \min$ and $\arg \max$ ?

Answer 2

You're right about the use of $\arg \max$ etc. in optimization and discrete maths. The coverage on wikipedia is reasonable see here, where a standard expression is $$ \arg \max_{x \in S}f(x) :=\{x \in S: f(y)\leq f(x)~for~all~y\in S\}. $$

However the notation you propose is totally unclear, I doubt it's used anywhere, you must have come up with it. Stick to $$ X=\{i:x_i=1\}, $$ instead.