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)

Now I see. But I think, these "directions" (left and right) exist only on paper (or screen etc), but not in the mathematical universe. Notations are a matter of metamathematics. In mathematics, we are given two partially ordered sets; that is all. Some maps preserve order, some invert. But, yes, we may ask, what happens if we endow the same set with the opposite order (thus constructing another partially ordered set). Boris (talk) 11:26, 9 April 2016 (UTC)

Indeed. I cannot think of a way to distinguish [ilmath]\le[/ilmath] from [ilmath]\ge[/ilmath] (on say [ilmath]\mathbb{R} [/ilmath] without a notion of "think of [ilmath]>[/ilmath] like a crocodile, it wants to eat the bigger number (eg [ilmath]5>4[/ilmath])" (this is how we are taught to remember [ilmath]>[/ilmath] and [ilmath]<[/ilmath] in school by the way) and of course as they are duals....

This is what I'll look into. I have heard the word "isotonic" before, but I wont make a definition without references as I am not confident (where as I know monotonic. I am confident enough to leave finding a reference for later). I shall add/work-towards [ilmath]\sigma[/ilmath]-ideal today. Alec (talk) 11:45, 9 April 2016 (UTC)