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: [ilmath]\forall a,b\in X[a\mathcal{R} b\implies f(a)\mathcal{S}f(b)][/ilmath]

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

BUT! If we define [ilmath]\mathcal{S} [/ilmath] as [ilmath]\ge[/ilmath] it is now "isotonic".

If both [ilmath]\mathcal{ R } [/ilmath] and [ilmath]\mathcal{ S } [/ilmath] are "to the right" (eg [ilmath]\le[/ilmath]) this works as expected, as if they're both to the left (eg [ilmath]\ge[/ilmath]) then it's actually the same thing. (But I cannot define "to the left" formally, this is what I will investigate.)

• That is: [ilmath]\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)][/ilmath] where [ilmath]\ge[/ilmath] is the dual of whatever [ilmath]\le[/ilmath] 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", [ilmath]f[/ilmath], then the above definition works)

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

PS: We can however say "[ilmath]f[/ilmath] is isotonic with respect to (two partial orders)" what I'm saying here is we first need to determine if both relations "are facing the same direction") Alec (talk) 10:47, 9 April 2016 (UTC)