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:
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Find an order theory book, also I think that huge category theory PDF (Harold Simmons) has it
Contents
[hide]Definition
A map, f:X→Y between two posets, (X,⊑) and (Y,⪯) is monotonic or monotone if:
- ∀a,b∈X[a⊑b⟹f(a)⪯f(b)], or in words:
- It preserves the ordering.
For a sequence
Recall that a sequence, (An)∞n=1⊆X (for some poset, (X,⊑)) can be considered as a mapping:
- A:N→X given by A:n↦An
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 ≤ is to induce a unique ≥ - 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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