Talk:Monotonic

Isotonic

For one relation to be isotonic BUT NOT the dual you would need a way to separate them. Isotonic is a word I've read though. But "isotonic: monotonic but where the relations are visually facing the same way" is not how I want to define it! Will look into later Alec (talk) 08:44, 9 April 2016 (UTC)

Oops, I fail to get your hint. Quite unclear, what do you mean? Boris (talk) 09:01, 9 April 2016 (UTC)

Another option: "order preserving" versus "order inverting". Not sure whether it is in use. Boris (talk) 09:06, 9 April 2016 (UTC)

While I get what you mean by "order preserving" and "order reversing" I cannot come up with a definition. Suppose we have:

• Isotonic: $\forall a,b\in X[a\mathcal{R} b\implies f(a)\mathcal{S}f(b)]$

This only works if $f$ is "order preserving" itself. Suppose $\mathcal{R}$ and $\mathcal{S}$ are $\le$ and $f:\mathbb{R} \rightarrow\mathbb{R}$, if we define $f:x\mapsto -x$ this is no longer isotonic.

BUT! If we define $\mathcal{S}$ as $\ge$ it is now "isotonic".

If both $\mathcal{ R }$ and $\mathcal{ S }$ are "to the right" (eg $\le$) this works as expected, as if they're both to the left (eg $\ge$) then it's actually the same thing.

• That is: $\forall a,b\in X[a\le b\implies f(a)\le f(b)]\leftrightarrow\forall a,b\in X[a\ge b\implies f(a)\ge f(b)]$ where $\ge$ is the dual of whatever $\le$ is.

However I cannot define "to the left" (as these are dual concepts, I don't expect to be able to UNLESS there is some "natural order preserving map", $f$, then the above definition works)

Do you see what I mean? Alec (talk) 10:33, 9 April 2016 (UTC)