Definition talk:Order Embedding/Definition 4

I'm thinking maybe this would be clearer if we didn't subscript the restriction, since the subscript in this case is so huge. --Dfeuer (talk) 07:32, 14 March 2013 (UTC)

Ehhhh... I'll give the set a name. Forget it. --Dfeuer (talk) 07:33, 14 March 2013 (UTC)

Ehhhh .... I can't give it a name so easily in the theorem statement. Question back on table. --Dfeuer (talk) 07:33, 14 March 2013 (UTC)

IMO: just drop the restriction symbol and everything after it, it doesn't add to the understanding. --prime mover (talk) 07:36, 14 March 2013 (UTC)

I just realized I can give it a name if I delay introducing the word order embedding. As for your suggestion ... hmmm ... I certainly see the site being more formal about such things on some pages and less formal on others, and I'm not sure why. --Dfeuer (talk) 07:40, 14 March 2013 (UTC)

Whatever. It's your page. Your difficulty perhaps highlights my emphasis on planning out what you want to do first.

I reiterate my plea for you to source your pages from published works. Indeed, it may be possible to define these objects as equivalent definitions, but unless you can find a source work which specifically uses one of these as the definition it uses for this object, then merely stating the result as an equivalence is adequate.

We have had this discussion before, in the context of something completely different, where one of the contributors established an equivalence to a truly obscure and complicated condition and tried to use it as a definition. Now nobody in their right mind would have actually used it as a definition because it was useless for this purpose.

I wonder whether the same applies here. The initial definition as stated is completely simple and straightforward: it's a mapping from $S$ to $T$ which (weakly) preserves order in both directions. It can subsequently be proved that such a mapping is also strictly order-preserving both ways, and has to be an injection. Equally similarly, this definition is that it's an order isomorphism between $S$ and its image in $T$. While it's an elementary consequence of the (simple) definition, is it necessarily worth documenting as an actual definition? And the acid test for the latter question: is it used anywhere in the literature as a definition? If it is, provide a link to the source, however "dull" setting up such a source work is (it only needs to be done once, and there are surely a limited number of books in your source library). If not, scrap it as a definition and merely include it as a result. --prime mover (talk) 07:55, 14 March 2013 (UTC)

The third definition expresses an essential element of the concept of embedding. The second definition is the debatable one in the context of orderings. However, a text dealing primarily with strict orderings is likely to use that one, and I don't really think we want a whole separate layer for the notion of order isomorphism of ordered set vs. ordered isomorphism of strictly ordered set. --Dfeuer (talk) 07:59, 14 March 2013 (UTC)

"is likely to use that one" - So find one and cite it. If you can't, then you can't use it. And despite the obvious and essential nature of the concept of embedding, if def 3 is not used as a definition, don't use it as one. When you've written and published your own textbook on the subject, then maybe you can cite that, but until then it is a suboptimal move to include equivalences as definitions "just because you can" when in fact all it does is clutter up the definition space with confusing detail. If you specifically need to invoke the fact stated in def 3, then link to that page where it is stated as an equivalence. --prime mover (talk) 08:14, 14 March 2013 (UTC)