Notation for maximum over an arbitrary total order

Question

I feel like this is a question to which I should already know the answer, but my Google searches so far are coming up empty.

Is there a common notation for specifying the maximum/minimum over an arbitrary total order? For example, let's define $\geq^\mathbf{A}$ over $n$-dimensional vectors by $\mathbf{v} \geq^\mathbf{A} \mathbf{v}'$ iff $\mathbf{A}(\mathbf{v}-\mathbf{v'}) \geq^{lex} \mathbf{0}$.

If I then want to find the maximal vector according to $\geq^\mathbf{A}$ over some set $V$, is there some canonical/common notation for specifying this element?

I've was thinking of using $\max^{\geq^\mathbf{A}}_{\mathbf{v} \in V} \mathbf{v}$ or $\max_{\substack{\geq^\mathbf{A} \\ \mathbf{v} \in V}} \mathbf{v}$, but if there's a canonical way of specifying this, I'd prefer to use that.

Answer 1

In general, the $\max$ operator takes a set as its argument. For example, the maximum value attained by a function $f$ is the maximum of the set of values attained by $f$ over its domain. If $f : \mathbb{R} \to \mathbb{R}$, then this is denoted $$ \max \{ f(x) \mid x\in\mathbb{R} \}. $$ A common shorthand for this is to write $$ \max_{x\in\mathbb{R}} f(x) \qquad\text{or}\qquad \max f(x), $$ where the second notation might be used if the domain is understood or otherwise unambiguous. Additionally, if a set has some extra structure, we can use a tuple to represent the set-plus-structure. For example, a metric space consists of a set $X$ and a metric $d$, which can be written as $(X,d)$. Hence it is reasonable to write $$ (V, \ge^A) $$ for a totally ordered set $V$, where the order is given by $\ge^A$.

Combining these ideas, there are several natural notations which seem reasonable in the context of the question:

• A rather pedantic notation might be $$ \max\left\{ v \ \middle|\ v \in \left(V,\ge^A\right) \right\}. $$
• Less pedantically, but retaining all of the important data, we could write $$ \max \left(V, \ge^A\right) \qquad\text{or}\qquad \max_{v \in (V,\ge^A)} v. $$ Personally, I think that the first notation is likely the best of all those discussed in this answer. The second notation is meant to parallel the notation for the maximum of a function over some domain, though I don't much like it (the subscript is overly complicated and small).
• Finally, if we are not worried about specifying the total order, we can just write $$ \max V. $$ This might be a very convenient notation if there are no extra total orders running around.