Difference between revisions of "Talk:Monotonic"
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:: This only works if {{M|f}} is "order preserving" itself. Suppose {{M|\mathcal{R} }} and {{M|\mathcal{S} }} are {{M|\le}} and {{M|f:\mathbb{R} \rightarrow\mathbb{R} }}, if we define {{M|f:x\mapsto -x}} this is no longer isotonic. | :: This only works if {{M|f}} is "order preserving" itself. Suppose {{M|\mathcal{R} }} and {{M|\mathcal{S} }} are {{M|\le}} and {{M|f:\mathbb{R} \rightarrow\mathbb{R} }}, if we define {{M|f:x\mapsto -x}} this is no longer isotonic. | ||
::: BUT! If we define {{M|\mathcal{S} }} as {{M|\ge}} it is now "isotonic". | ::: BUT! If we define {{M|\mathcal{S} }} as {{M|\ge}} it is now "isotonic". | ||
| − | :: If both {{mathcal|R}} and {{mathcal|S}} are "to the right" (eg {{M|\le}}) this works as expected, as if they're ''both'' to the left (eg {{M|\ge}}) then it's actually the same thing. | + | :: If both {{mathcal|R}} and {{mathcal|S}} are "to the right" (eg {{M|\le}}) this works as expected, as if they're ''both'' to the left (eg {{M|\ge}}) then it's actually the same thing. (But I cannot define "to the left" formally, this is what I will investigate.) |
::* That is: {{M|1=\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 {{M|\ge}} is the dual of whatever {{M|\le}} is. | ::* That is: {{M|1=\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 {{M|\ge}} is the dual of whatever {{M|\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", {{M|f}}, then the above definition works) | :: 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", {{M|f}}, then the above definition works) | ||
:: Do you see what I mean? [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 10:33, 9 April 2016 (UTC) | :: Do you see what I mean? [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 10:33, 9 April 2016 (UTC) | ||
Revision as of 10:34, 9 April 2016
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: ∀a,b∈X[aRb⟹f(a)Sf(b)]
- This only works if f is "order preserving" itself. Suppose R and S are ≤ and f:R→R, if we define f:x↦−x this is no longer isotonic.
- BUT! If we define S as ≥ it is now "isotonic".
- If both R and S are "to the right" (eg ≤) this works as expected, as if they're both to the left (eg ≥) then it's actually the same thing. (But I cannot define "to the left" formally, this is what I will investigate.)
- That is: ∀a,b∈X[a≤b⟹f(a)≤f(b)]↔∀a,b∈X[a≥b⟹f(a)≥f(b)] where ≥ is the dual of whatever ≤ 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)
- While I get what you mean by "order preserving" and "order reversing" I cannot come up with a definition. Suppose we have:
