# Notation for maximum over an arbitrary total order

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.

• The $\max$ operator generally takes a set (rather than a function) as an argument. Specify the set sufficiently, and there should be no ambiguity. For example $$\max\{ v \mid v \in (V, \ge^A)\},$$ where $(V,\ge^A)$ is the ordered set you are considering. I think that $\max (V,\ge^A)$ would also be unambiguous, or even $\max V$, assuming that there are no other total orders running around. – Xander Henderson Aug 23 '19 at 21:01
• @Xander Thanks, that's helpful! I hadn't seen the $(V, \geq^A)$ notation for ordered sets before. That's definitely a lot nicer than the multiple super- and subscripts. (I'd consider this question answered by this comment - not sure what to do about that since I'm new here!) – Kasenberg Aug 25 '19 at 1:06

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 $$\ge^A$$.
• 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.