Suppose we have some functions ${f}_{1}(x),{f}_{2}(x),\dots ,{f}_{n}(x)$ with $x\in {\mathbb{Z}}^{n}$.

We can denote the subset ${X}_{1}$ of ${\mathbb{Z}}^{n}$ that maximizes ${f}_{1}(x)$ as:

${X}_{1}=\underset{x\in {\mathbb{Z}}^{n}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{1}(x)$

Now, suppose there is a kind of "priority" in which I also want to maximize ${f}_{2}$, as long as I keep maximizing ${f}_{1}$. This could be represented as:

${X}_{2}=\underset{x\in {X}_{1}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{2}(x)$

The same for ${f}_{3}$:

${X}_{3}=\underset{x\in {X}_{2}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{3}(x)$

So on and so forth...

Is there some, more concise, notation to represent this "maximization priority"?

We can denote the subset ${X}_{1}$ of ${\mathbb{Z}}^{n}$ that maximizes ${f}_{1}(x)$ as:

${X}_{1}=\underset{x\in {\mathbb{Z}}^{n}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{1}(x)$

Now, suppose there is a kind of "priority" in which I also want to maximize ${f}_{2}$, as long as I keep maximizing ${f}_{1}$. This could be represented as:

${X}_{2}=\underset{x\in {X}_{1}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{2}(x)$

The same for ${f}_{3}$:

${X}_{3}=\underset{x\in {X}_{2}}{\mathrm{a}\mathrm{r}\mathrm{g}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\mathrm{a}\mathrm{x}}\phantom{\rule{thinmathspace}{0ex}}{f}_{3}(x)$

So on and so forth...

Is there some, more concise, notation to represent this "maximization priority"?