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)