Talk:Monotonicity

There are now two articles on monotonicity, the other one being monotonic function. Monotonic function is more extensive and contains the general definitions. It might therefore be good to include the "monotonicity" article into the structure of "monotonic function".

I also do not know about the definitions given here: where do we actually deal with "commensurate difference functions" and "equal monotonicity" (which is clearly different from monotonicity in the general sense)? Are there examples? Is this generally accepted terminology (reference books?)? The given notion of "convergence" also needs some explanation/motivation: Where is it applied? How does it compare to all the other concepts of convergence in mathematics?

There is also a problem with redirects. We cannot redirect "monotone function" to "monotonicity" if there is some "monotonic function".

--Markus Krötzsch 13:31, 21 Mar 2004 (UTC)

Fixed the redirects and links to this page. Actually all links to this page really wanted the monotonicity from order theory as given in monotonic function or the calculus version (also given there). No articles seem to refer to the actual content of this page here... --Markus Krötzsch 14:17, 21 Mar 2004 (UTC)

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Thanks for the correction, Frank W ~@) R 05:05 Jan 14, 2003 (UTC).

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This article needs:

1. a layperson-frienldy intro
2. a section of monotonic sequences, which is probalby the number one referrer to the page

-- Tarquin 17:03 Jan 14, 2003 (UTC)

This page makes little sense

This page makes little sense to me.

1. "Monotonicity" should point to monotonic function (in fact one could make an argument that "monotonic function" should be renamed to "monotonicity", and "monotonic function" should redirect to it.)
2. It introduces terminology "equal monotonicity" and "opposite monotonicity", which I've never heard used before. The usual terms are "monotone" (or "order-preserving) and "antitone" (or "order-reversing") respectively.
3. It uses an odd definition for monotonicity: x < y -> f(x) ≤ f(y) instead of the usual (and equivalent): x ≤ y -> f(x) ≤ f(y).
4. It defines a generalized notion of convergence for functions between posets. However I've never seen this definition before and I wonder where this definition has been used before.

I propose the following:

1. Merge with Monotonic function whatever needs merging from this article. (I've already merged, and generaslized, the bit about constant maps — the only part which seems relevant to me.)
2. If the generaslized convergence stuff is important, then either merge it with limit (mathematics) say, or move it to it's own article perhaps.
3. Redirect this article to monotonic function.

Comments? Paul August ☎ 16:52, Mar 8, 2005 (UTC)

I don't find the "odd" definition so odd; the two definitions mentioned are obviously exactly equivalent. Also, the terms "monotone increasing" and "monotone decreasing" are commonplace, but I've seldom heard or seen "antitone". Michael Hardy 02:12, 9 Mar 2005 (UTC)

The reason probably is that "monotone decreasing" is far more common in functional analysis, while it is typically not used in order or lattice theory. Other names which are also common in the latter areas are "order-reversing" and "anti-monotone". Etymologically, "antitone" probably is derived from the latter of these, and may thus be the youngest of these terms. I like it because (a) its short and (b) it has a unique spelling (computer searches may care if someone searches "order reversing" instead of "order-reversing") (c) there is no other notion named like this. --Markus Krötzsch 14:44, 14 Mar 2005 (UTC)

Regarding Paul's 4 objections:

1. The good article title is Monotonic function (or Monotone function, if that's more common), since this is a more concrete idea (than the abstract property of monotonicity).
2. This terminology (using the noun "monotony" rather than "monotonicity", BTW) seems to have been introduced to Wikipedia by Fwappler. Normally I'd say, ask the original editor where they got the terms, so that we can correctly cite their usage. In Fwappler's case, however ... well, let me just say that in the decade that I've known him online, I've seen him use more nonstandard terminology than standard (and not just in mathematics). Since he hasn't been active lately, I wouldn't bother trying very hard to document those terms.
3. I agree with Michael that this definition is not particularly odd. Still, it's inferior to the version that uses ≤ in the antecedent because (if for no other reason) you need ≤ for constructivist analysis. (As a part-time constructivist, at least I think that this is important ^_^.)
4. This is the one thing of real value, IMO, in the current article. IIRC, you can find this documented (albeit not in quite the same form) in Handbook of Analysis and its Foundations, by Eric Schechter.

Regarding Paul's 3 proposals:

1. This is the obvious right thing to do.
2. Depending on room, this could be a section in Monotonic function or an article linked from Monotonic function; in either case, there should also be a link from Limit (mathematics).
3. Yes, I think that Monotonic function is the best title.

-- Toby Bartels 01:11, 2005 Mar 11 (UTC)

Thanks Michael Hardy and [User:Toby Bartels|Toby Bartels]] for your comments.

Based on the comments so far, what I would do is simply make this page a redirect to monotonicity, and since Toby thinks that the generalized convergence bit is worth keeping, find a home for it. The only remaining question for me then is where should it go?:

• Monotonic function?
• Limit (mathematics)?
• Its own page? what title?

Paul August ☎ 23:15, Mar 11, 2005 (UTC)

I think that Order convergence would be the right title for its own page -- I've seen that phrase before. The material could probably be placed on any of those three pages, as long as each of them linked to the material. Perhaps the cleanest solution is to rename this page Order convergence, remove the redundant material, and create appropriate links. -- Toby Bartels 23:36, 2005 Mar 12 (UTC)

Removing the redundant content from this article leaves the following:

(Beginning of quoted text)

The notion of monotonicity allows one to express the principal instances of convergence (to a limit):

Given that a commensurate difference relation is defined between the members of S; that is, such that for any four (not necessarily distinct) members g, h, j, and k of S, either g − h ≤ j − k, or g − h ≥ j − k, and given that M from T to S is a map of equal monotonicity, then the values M(s) are called converging (to an upper limit), as the argument s increases, if either:

• the set T has a last and largest member (which M maps explicitly to the corresponding limit value l in set S); or
• for each member m of T, there exists a member <n > m such that for any two further members x > y with y > n, M(n) − M(m) ≥ M(x) − M(y).

As far as the set of all values M(s) does therefore have an upper bound (either within set S, or besides), and as far as every set which is bounded (from above) does have a least upper bound l, the values M(s) are called converging to the upper limit l as the argument s increases.

Similarly one may consider convergence of the values M(s) to a lower limit, as the argument s decreases; as well as convergence involving maps of opposite monotonicity.

(End of quoted text)

If I were going to try to turn the above into its own article with the title of "Order convergence" as suggested by Toby above, I would create something like the following:

(beginning of new text)

In mathematics, order convergence is a generalized notion of convergence defined for partially ordered sets.

Suppose that T and S are partially ordered sets, and f : T → S, is a order-preserving function, that is x ≤ y implies that f(x) ≤ f(y), and that a commensurate difference relation is defined between the members of S; that is, for any four (not necessarily distinct) members g, h, j, and k of S, either g − h ≤ j − k, or g − h ≥ j − k, then the values f(x) are said to converge (to an upper limit), as the argument x increases, if either:

• the set T has a last and largest member (which f maps explicitly to the corresponding limit value L in S); or
• for each member m of T, there exists a member n > m such that for any two further members x > y with y > n, f(n) − f(m) ≥ f(x) − f(y).

As far as the set of all values f(x) does therefore have an upper bound (either within set S, or besides), and as far as every set which is bounded (from above) does have a least upper bound L, the values f(x) are called converging to the upper limit L as the argument x increases.

Similarly one may consider convergence of the values f(x) to a lower limit, as the argument x decreases; as well as convergence involving order-reversing maps.

(end of new text)

I would also spend more time reworking the above (since in my view it could be worded better). However I'm flying blind here, as I'm unfamiliar with this concept. Googling for "order convergence", I do see the term used, but I can't find a definition, and none of my books mentions it or anything like it. Consequently I'm not comfortable making such a change.

What I would like to do is make this article a redirect to Monotonic function — and let other, more competent editors, either create a new article from the above or rename (and make appropriate changes to) this article and create a new article titled "Monotonicity" which redirects to Monotonic function.

Paul August ☎ 14:39, Mar 13, 2005 (UTC)

I completely agree with the objections and the proposed solution. Someone be bold please. --Markus Krötzsch 14:44, 14 Mar 2005 (UTC)

I have now made this page a redirect to Monotonic function. Paul August ☎ 20:46, Mar 14, 2005 (UTC)