Definition talk:Decreasing/Mapping

Birkhoff's definition of antitone in Lattice Theory looks very strange to me: strange enough I wonder if it was an error. He defines an isotone function to be an increasing function in the usual sense, but then he comes up with the following peculiar definition of antitone:

A function $\theta:P \to Q$ is antitone if and only if

$(2)\quad x \leqq y$ implies $\theta(x) \geqq \theta(y)$,

$(2')\quad \theta(x) \leqq \theta(y)$ implies $x \geqq y$.

He then defines a dual isomorphism as a bijection satisfying these conditions. --Dfeuer (talk) 22:08, 16 February 2013 (UTC)

In fact, such a function must always be an injection:

Suppose $\theta(x) = \theta(y)$. Then $\theta(x) \leqq \theta(y)$ and $\theta(y) \leqq \theta(x)$.

Then $y \leqq x$ and $x \leqq y$.

By antisymmetry: $x = y$.

Thus such a function is a dual embedding, (possibly) lacking only surjectivity. It hardly seems likely that he intended that. --Dfeuer (talk) 22:13, 16 February 2013 (UTC)

You may be right.

Let $P = \{1, 2, 3, 4\}$ and let $Q = \{1, 2, 3\}$. Let $\theta: P \to Q$ be: $\theta(1) = 3, \theta(2) = \theta(3) = 2, \theta(4) = 1$. Then $\theta$ is decreasing according to the definition given in this page, but not according to Birkhoff: $\theta(3) \le \theta(2)$ but $3 \not \le 2$.

Recommend you add an "also defined as", specifically referencing Birkhoff's definition, and bear it in mind as you progress through the work and establish what was behind his idea to define it that way. I do not have this work, although I understand it was highly regarded, and is widely cited. So if it was actually in error someone would have flagged it up by now. As per usual, I would tread carefully before declaring it a "mistake" as such, until you can find corroborative evidence to back this up (e.g. another eminent mathematician commenting so). --prime mover (talk) 22:25, 16 February 2013 (UTC)

I'm waiting for him to actually use the term, at which point I will probably be able to determine whether his use is consistent with his definition. Errors in sufficiently simple material can easily be glossed over again and again by editors for whom they look sufficiently familiar to be assumed correct. --Dfeuer (talk) 22:28, 16 February 2013 (UTC)

Hmph. He appears to use the term a grand total of three times in the entire book. I don't (yet) know enough about lattice theory to be able to determine if it's possible to determine from any of them whether his definition was what he intended. --Dfeuer (talk) 22:32, 16 February 2013 (UTC)

In that case add an "also defined as" section above and, as I say, point out who defined it as it. Once you've read the book you will then (probably) be able to put it into context appropriately. This is all part and parcel of what I have been saying all along: make sure you know your stuff before posting it up. --prime mover (talk) 22:38, 16 February 2013 (UTC)