Definition talk:Initial Segment

Definition does not need to require that $\prec$ form a poset. Initial segments are very important to foundational relations and the axiom of regularity. See Axiom of Foundation (Strong Form). --Andrew Salmon (talk) 07:13, 22 August 2012 (UTC)

We've had this conversation before.

In the context from which your source work approaches it, maybe not. In the context from which my source work approaches it, it does.

If you need to add the definition of initial segment that you require in order to progress your own thread, then add the definition as an "also defined as", but I would contend that the object you are defining is a different object which happens to have the same name. --prime mover (talk) 09:03, 22 August 2012 (UTC)

IMO adding stuff under 'Also defined as' is not the way to go when it constitutes a generalisation, or a properly different approach. In such cases, a page with multiple definitions needs to be crafted, or wholly different pages altogether, linked via a disambiguation. I vote for the latter option in this situation (and maybe also for the other page that recently got such a 'A d a' section - I forgot the name). Of course, the two are interlinked in that one is an instance of the other, but such happens all the time. --Lord_Farin (talk) 09:16, 22 August 2012 (UTC)

Let me add to that the suggestion that the poset def be transcluded onto the general page, like on Definition:Small Class, or a construct like Definition:Open Set. --Lord_Farin (talk) 09:20, 22 August 2012 (UTC)

Yes that would probably work. As long as we don't lose the fact that the "conventional understanding" is that $\prec$ is an ordering but that the more modern abstract approach lifts this requirement and merely stipulates that it be foundational. That the traditional $\preceq$ is a foundational relation with transitivity union with the diagonal relation can be deduced from there. --prime mover (talk) 13:53, 22 August 2012 (UTC)

It need not be foundational. $<$ for Real numbers is not foundational. It can just be an abstract definition for any relation $\mathcal R \subseteq A \times A$. --Andrew Salmon (talk) 17:44, 22 August 2012 (UTC)

So what constraints are necessary on this relation for this to be valid? --prime mover (talk) 18:59, 22 August 2012 (UTC)

Actually, nothing is necessary. --Andrew Salmon (talk) 23:40, 22 August 2012 (UTC)

Naming

Definition:Strict Upper Closure vs. Definition:Initial Segment. Need I say more? --Dfeuer (talk) 17:57, 6 February 2013 (UTC)

The terminology "initial segment" is usual in especially well-order theory. That definition preceded the advent of the other sorts of upper and lower closures. IOW, it grew so hysterically. While rather unfortunate, please note that there is a redirect in place from Definition:Strict Lower Closure so that reference needn't suffer. --Lord_Farin (talk) 18:19, 6 February 2013 (UTC)

There are other inconsistencies going on. Initial segment is defined in terms of strict orders, while the other in terms of weak orders. It's kind of a mess. --Dfeuer (talk) 18:22, 6 February 2013 (UTC)

I disagree with you on that one. I contend that both definitions are perfectly clear and that no confusion is likely to arise. Nor is it messy; a "strict order" is not really mentioned anywhere (only symbolically, but after the definition in terms of the "weak" order is given). 't Is really only the name. Unfortunate, but not really relevant because there's a redirect. --Lord_Farin (talk) 18:28, 6 February 2013 (UTC)

I'm sorry, I confused matters a bit. The strict ordering bit was in Order Isomorphism Preserves Initial Segments. --Dfeuer (talk) 18:34, 6 February 2013 (UTC)

Grrr

The definition of "initial segment" used in pages about well-founded relations is not the one it links to, or that we have anywhere on the site, as far as I know. It applies to arbitrary relations. --Dfeuer (talk) 21:07, 25 February 2013 (UTC)

A brief search suggests that "initial segment" is sometimes used to mean something akin to "lower set" in the context of more general relations. --Dfeuer (talk) 21:16, 25 February 2013 (UTC)

See this page. --Dfeuer (talk) 21:20, 25 February 2013 (UTC)

That use seems obscure and ad hoc. Left-closed would be a more sensible term, also more in line with our current scheme. — Lord_Farin (talk) 21:27, 25 February 2013 (UTC)

We currently have Definition:Transitive with Respect to a Relation and Definition:Closed Relation, both of which I would like to burn in a fire. --Dfeuer (talk) 22:07, 25 February 2013 (UTC)

No deletion of material, as you know. However, both fit under the Takeuti/Zaring remark I made on your talk. Please just ignore them. — Lord_Farin (talk) 22:10, 25 February 2013 (UTC)

I'm okay with "left closed" corresponding to "lower set" and "right closed" corresponding to "upper set" as long as that's not going to break some standard terminology. --Dfeuer (talk) 22:16, 25 February 2013 (UTC)

I don't know. It seems most generalisations of these concepts aren't used very often in any case. No point in inventing things ourselves when we won't use them anyway. Suggest to leave the remark as-is until we hit it from another direction. — Lord_Farin (talk) 22:20, 25 February 2013 (UTC)

I'm pretty sure we'll hit it whenever we get around to dealing with foundational relations properly, but I guess it can wait till then. --Dfeuer (talk) 22:33, 25 February 2013 (UTC)