Talk:Preorder

message: I put a flag on this article because evidently there are two ways to define quasi order, reflexive or irreflexive. 
Can someone clarify?
 []  —Preceding unsigned comment added by 130.70.11.93 (talk) 14:16, 18 October 2009 (UTC)
thread_title: Reflexive vs Irreflexive

message: Is there a commonly-used name for a complete preorder  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)
replies: text: Total preorders, of course.  Thanks, Hurkyl! CRGreathouse (t | c) 04:55, 31 August 2006 (UTC)
thread_title: Types of preorders

message: 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)
thread_title: Table

message: 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)
thread_title: Formal definition

message: 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)
replies: text: 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)
replies: text: Well, I think he meant that we should modify the article itself to clarify this, rather than discussing on the talk page. These are good points; if one never had to deal with relations before, this could all seem very confusing.  I'll try to updated article to do this. linas (talk) 04:38, 20 April 2012 (UTC)
replies: text: Done. I gave a big expanded intro; I think it should be much easier now, and more intuitive.  linas (talk) 05:41, 20 April 2012 (UTC)
thread_title:  Suggestions for clarifications 

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

message: I added the diamond lemma to the "see also" list, as an important lemma that holds for many pre-orders.  Are there other lemmas/theorems that should be stated? The article hints that there are theorems from topology that require pre-orders, but its unclear if any of these can be stated as holding for pre-orders in general, (i.e. if they still say anything useful after throwing away the topology bits).  linas (talk) 04:31, 20 April 2012 (UTC)
thread_title:  theorems and lemmas? 

message: I think this paragraph is misleading, so I have removed it:
:To every preorder there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. Note that, in general, the corresponding graphs may be cyclic graphs: preorders may have cycles in them.  Imposing the condition of anti-symmetry on the preorder (to force it into becoming a partial order) is the same as imposing the no-cycles condition on a directed graph (forcing it to become a directed acyclic graph).  Imposing the condition of symmetry (to force a pre-order into being an equivalence relation) is the same as removing the direction on the edges (to create an undirected graph).  In general, a preorder may have many disconnected components.  The diamond lemma is an important result for certain kinds of preorders.
In particular, most directed graphs are not transitive, and therefore do not correspond to preorders in this way. And if one uses reachability to define a preorder from an arbitrary directed graph, then many digraphs will give rise to the same preorders. —David Eppstein (talk) 05:54, 20 April 2012 (UTC)
replies: text: OK. I also cringed when I wrote "forced" but I couldn't think of a better way of saying it.  So much for late-night editing.  The main point I wanted to get across was that preorders can contain cycles, as this is not something that would ever occur to a reader who'd never heard of a pre-order before.  I think the general notion that there's a graph that corresponds to orders can also bring a lot of intuition to new readers. (If most people are like me, it takes a while to realize that algebra, in general, need not be confined to neatly-ordered rows of symbols, but has a spatial, topological, graphical "shape", as it were. Things like feynmann diagrams, or category theory diagrams, shouldn't be just for the initiates; the general ideas could/should be more widely appreciated). I tried to be careful and say that the edges are defined by the relation, and not by other concepts, such as reachability.  I'll try re-writing later today or tomorrow, and say explicitly "only transitive graphs" and "not reachability" or maybe something like "most digraphs aren't transitive but when they are.."  I dunno.   Goal is to wordsmith a poetic introduction for the young reader.  linas (talk) 15:25, 20 April 2012 (UTC)

text: To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs may be cyclic graphs: preorders may have cycles in them.  A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph.  A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph.  In general, a preorder may have many disconnected components.  The diamond lemma is an important result for certain kinds of preorders.
Would this work?  linas (talk) 15:46, 20 April 2012 (UTC)

text: The issue for me is that the transitive-closure representation of a preorder as a directed graph (the one that has an edge for every reachable pair of distinct vertices) is uniquely determined by the preorder. The transitive-reduction representation (one that has a minimal subset of those edges preserving the same reachability relation) is not. For instance, the trivial preorder in which everything is equal to everything else can be represented by any one of (n-1)! directed cycles, or also in other ways by directing the edges of a tree of cycles. So, if you're looking for a 1-1 correspondence between preorders and graphs, the transitive closure is really the only way to go. —David Eppstein (talk) 18:22, 13 December 2016 (UTC)
thread_title:  Preorders vs directed graphs 

message: which spelling is correct? Both are used in the text. --Tillmo (talk) 10:10, 9 October 2012 (UTC)
replies: text: Both are correct.  It would be preferable not to mix them, though.  In the spirit of WP:RETAIN, what we should probably do is figure out which one was used first, and change all instances to that form. --Trovatore (talk) 23:09, 9 October 2012 (UTC)
replies: text: Actually, I take it back.  The article is entitled preorder, whereas it could have used pre-order.  Therefore we should use quasiorder for consistency, regardless of which was first. --Trovatore (talk) 23:14, 9 October 2012 (UTC)
replies: text: The trouble is that preorder is nearly always written without a hyphen, whereas quasi-order gets a hyphen more often than not (IME).—Emil J. 14:42, 11 June 2013 (UTC)
thread_title: quasi-order versus quasiorder

message: I'd be curious to see what the asymptotic form (large N) is for the OEIS sequences. 67.198.37.16 (talk) 22:04, 14 February 2016 (UTC)
thread_title:  asymptotic form of the OEIS sequences? 

message: Early on in the article, ≤ is equated to ← and a ≤ b with “b reduces to a”. I'm sure this is usage in existence, but perhaps a source would be useful? The thing is, phrase “reduces to” here gives the opposite direction to what one would get when ordering problems by hardness using reductions, for example (which yields a preorder), and the direction of the arrow is opposite of that commonly used when defining a preorder as a thin category – i.e., a ≤ b indicates the existence of a unique morphism from a to b, not the other way around, and this is written as a forward arrow. So at least there seems to be reasonable usage out there that is the opposite of what is stated, though I'm sure the usage used in the article is frome some source more directly related to order relations? Maybe in addition to a source, a brief discussion of this issue might be merited? Mlhetland (talk) 19:52, 11 November 2017 (UTC)
thread_title:  Reduction and arrow direction