Generalizing monotonicity to 2D

Question

Monotone functions of a single variable are well defined, they just keep increasing when the variable increases (or decreases).

I wonder if this concept has a standard generalization to two dimensions.

Obviously, if any function is observed by following an arbitrary path in the XY plane, it can go increasing or decreasing or both. On the other hand, simple functions like affine (a X + b Y + c) or a paraboloid in the first quadrant (f=X^2+Y2, X, Y > 0) are intuitively 2D monotonic.

Is there a way to formalize 2D monotonicity ? Are there monotone surfaces ?

With the scalar product $(.,.)$ one can definie monotonicity for a function $f:R^n \to R^n$ via $(f(x)-f(x),x-y)\geq 0$ — Quickbeam2k1, Mar 05 '13 at 20:51

score 2 · Accepted Answer · answered Mar 05 '13 at 20:51

2

The natural generalization is to monotone functions between partially ordered sets. Now, unlike the case for $\mathbb{R}$, it's debatable whether there is a completely natural partial order on $\mathbb{R}^2$. Instead there are many choices, e.g. lexicographic order (which happens to be a total order) or product order.

answered Mar 05 '13 at 20:51

Qiaochu Yuan

419,620

Should I understand that there is no "standard" definition ? – Mar 05 '13 at 21:12
And is there any relation to the property of having no local extremum ? – Mar 05 '13 at 21:15
@Yves: yes, and that depends on what partial order you choose. – Qiaochu Yuan Mar 05 '13 at 21:24

Generalizing monotonicity to 2D

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