Difference between revisions of "Monotonic"
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(Created page with "{{Stub page|I made this just to make it blue}} {{Requires references|Find an order theory book, also I think that huge category theory PDF (Harold Simmons) has it}} ==Definiti...") |
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# How can we have monotonically decreasing things? Via the dual partial ordering of course! To have {{M|\le}} is to induce a unique {{M|\ge}} - these are distinct orderings. | # How can we have monotonically decreasing things? Via the dual partial ordering of course! To have {{M|\le}} is to induce a unique {{M|\ge}} - these are distinct orderings. | ||
| + | # Not sure, but probably some call this isotonic, while monotonic is either increasing or decreasing. | ||
# Unite with [[monotonic set function]] | # Unite with [[monotonic set function]] | ||
==References== | ==References== | ||
Latest revision as of 04:50, 9 April 2016
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I made this just to make it blue
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This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
The message provided is:
Find an order theory book, also I think that huge category theory PDF (Harold Simmons) has it
Definition
A map, [ilmath]f:X\rightarrow Y[/ilmath] between two posets, [ilmath](X,\sqsubseteq)[/ilmath] and [ilmath](Y,\preceq)[/ilmath] is monotonic or monotone if:
- [ilmath]\forall a,b\in X[a\sqsubseteq b\implies f(a)\preceq f(b)][/ilmath], or in words:
- It preserves the ordering.
For a sequence
Recall that a sequence, [ilmath] ({ A_n })_{ n = 1 }^{ \infty }\subseteq X [/ilmath] (for some poset, [ilmath](X,\sqsubseteq)[/ilmath]) can be considered as a mapping:
- [ilmath]A:\mathbb{N}\rightarrow X[/ilmath] given by [ilmath]A:n\mapsto A_n[/ilmath]
We can now apply the above definition directly.
Work needed
TODO: These
- How can we have monotonically decreasing things? Via the dual partial ordering of course! To have [ilmath]\le[/ilmath] is to induce a unique [ilmath]\ge[/ilmath] - these are distinct orderings.
- Not sure, but probably some call this isotonic, while monotonic is either increasing or decreasing.
- Unite with monotonic set function
References
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