Talk:Preorder

Reflexive vs Irreflexive

I put a flag on this article because evidently there are two ways to define quasi order, reflexive or irreflexive. Can someone clarify?

[[1]]  —Preceding unsigned comment added by 130.70.11.93 (talk) 14:16, 18 October 2009 (UTC) 

The current definition of a "preorder" arises more naturally as a generalization of the concept of partial orders but, like any such situation, the most important thing is consistency. Jwuthe2 (talk) 10:51, 3 December 2009 (UTC)

Types of preorders

Is there a commonly-used name for a complete preorder (sequence A000670 in OEIS) within set theory? (Not a total order, mind you -- it need not be antisymmetric.) I've heard the term "weak order", but that's from the same field that uses "linear order" for total orders, so I wanted some clarification if anyone knows of something else. CRGreathouse (t | c) 20:02, 30 August 2006 (UTC)

Total preorders, of course. Thanks, Hurkyl! CRGreathouse (t | c) 04:55, 31 August 2006 (UTC)

Table

The table which shows the number of preorders needs some explanation. What precisely does the value n (given for n=1,2,3,4) denote? —Preceding unsigned comment added by 66.108.155.231 (talk) 02:19, August 28, 2007 (UTC)

That's the number of elements to be preordered. If there's just one element there's only one preorder (a <~ a). If there are two elements there are four preorders: (a <~ a, b <~ b, a <~ b), (a <~ a, b <~ b, b <~ a), (a <~ a, b <~ b, a <~ b, b <~ a), (a <~ a, b <~ b). CRGreathouse (t | c) 16:11, 4 October 2007 (UTC)

Formal definition

I was confused by the next-to-final statement in the formal definition "A preorder which is preserved in all contexts is called a precongruence." Upon reading it I was like "huh ?!" All what contexts ? There didn't seem to be any context (pun) for the notion of "contexts". Netrapt (talk) 16:43, 23 November 2008 (UTC)

I agree with Netrapt. I too find that statement confusing, because in fact the definition is not formal. I did a Google search of precongruence but could not decipher from what I read the intended meaning of the sentences about precongruences. Could someone who knows what a precongruence is correct this section? Undsoweiter (talk) 02:21, 20 February 2010 (UTC)

Digging some more, it turns out that Repton added the bit about precongruence, and later an anonymous(?) user with IP 141.76.75.213 added the bit about symmetry implying congruence. Clearly these folks knew the context for their statements, but I wish they'd left us with explicit details. The relevant edit dates are May 26, 2006 and July 28, 2006, respectively. Undsoweiter (talk) 02:31, 20 February 2010 (UTC)

Suggestions for clarifications

1. It would be helpful to understand why the "pre" in "preorder". Does it come before in some sense? Is there a "postorder"? For that matter, why the "quasi"? Might that be because the ordering described here allows items to be equal, hence not ordered? Yet that seems to be covered by "partial order". (Apparently there's also the possibility of elements simply not being related by the relation in question, not sure how that fits here.)

2. What do ~ and $\lesssim$ mean? The section Constructions appears to be trying to explain that, but seems to have contradictions, at one point describing ~ as equivalence, and elsewhere as complement. Similar odd comparisons of < and $\lesssim$.

Possible sources of vagueness: am I correct in assuming that a and b are individual members of the set S? Does ~ have a fixed meaning, or can one define it in alternative ways?

3. How does one verbally speak ~ and $\lesssim$?

Gwideman (talk) 14:15, 15 December 2009 (UTC)

It's called "preorder" because it's nearly/'just before' an order. $\lesssim$ is a generic symbol for some [pre]order; its particular meaning can vary (and often, like here, no particular meaning is intended). $\lesssim$ can be pronounced "precedes" if there's a need to say it. CRGreathouse (t | c) 00:37, 21 February 2010 (UTC)

Table, again

Currently the table is titled: "Number of n-element binary relations of different types". However, shouldn't it be something more like: "Number of preorders for different binary relations and different set sizes"? (I'm assuming n is set size) Gwideman (talk) 14:22, 15 December 2009 (UTC)

No, preorders are only one column in that chart. CRGreathouse (t | c) 19:17, 15 December 2009 (UTC)

Oops, yes of course. But I'm still stumped by the title. What is an "n-element binary relation". Doesn't it really mean "Number of binary relations (of different types) for sets having n elements". Ie: "n-elements" pertains to the sets not the relations, right? Gwideman (talk) 20:27, 15 December 2009 (UTC)

I think that those phrases mean the same thing. I find "n-element binary relation" clearer. CRGreathouse (t | c) 17:16, 16 December 2009 (UTC)