Talk:Partially ordered set

message: I think the introduction could be made at the same time easier to understand and more concise. (I don't like 1st paragraph introductions that contain more than the strict necessary, because to modify it, you have to modify the whole page which may give problems if it grows large.)
I suggest to give the example of set theoretical inclusion and a counter example like {1} and {2} (for non-comparability). The example of political organizations is unnecessarily complicated and in some sense even wrong. (Who knows if strict inclusion does not hold, if not today then maybe somewhen in the future? Especially in politics, nothing is impossible...)
I suggest that the real life example be changed. "Is a descendant of" is not a partial ordering of people, since I am not a descendant of myself.  —Preceding unsigned comment added by 128.101.184.204 (talk) 03:44, 22 March 2009 (UTC)
thread_title:  Introduction 

message: The article is a bit light on the difference between partially and totally ordered sets. If we think of "R" as the numerical comparison "≤" and the set is the integers, then the three rules clearly apply:
* For all integers a, a ≤ a.
* For all integers a and b, if a ≤ b and b ≤ a then a = b.
* For all integers a, b, and c, if a ≤ b and b ≤ c then a≤c.
What makes the integers a totally ordered set is that there is no pair of integers for which the "≤" relationship is unspecified. For all integers a and b, at least one of these statements is true:
* a ≤ b
* b ≤ a
In a partially ordered set that is not totally ordered, there is at least one pair of members a and b for which neither aRb nor bRa is true.
Example 1, dependencies in Make. A makefile specifies the dependencies between objects (usually files) that must be observed in building a piece of software as a multiple-step process. For instance an executable foobar might be built by linking object files foo.o and bar.o. In turn, foo.o is built by compiling foo.c, and bar.o is built by compiling bar.c. Both foo.c and bar.c include a header file plugh.h. Then we could write:
* plugh.h R foo.o
* foo.c R foo.o
* plugh.h R bar.o
* bar.c R bar.o
* foo.o R foobar
* bar.o R foobar
where "R" here means "is required in order to build". A makefile contains this same information, in a slightly different syntax. This could also be drawn as a directed graph.
Notice that there is no "R" relationship between foo.c and bar.c, or between foo.o and bar.o. In the directed graph, the foo branch and the bar branch are separate. It doesn't matter whether you build foo.o first or bar.o first, you are free to build them in either order.
Example 2, import statements in the Python programming language. Source file X may "import" source file Y, so that the objects defined in Y are available in X's namespace, and can be used by code in X. This could be notated as "Y ≤ X" and it often means that the file Y was written before the file X.
In a large collection of files, many of the files may import others, and transitivity applies: if X imports Y and Y imports Z, then the objects defined in Z are accessible somewhere in X's namespace. But there may be two files U and V where neither imports the other, either directly or through a transitive chain.
:The above (light on total vs. partial) is a reasonable criticism if applied to the section on inverses.  I expanded that section a little to try to clarify the difference.  --Vaughan Pratt 08:39, 15 July 2007 (UTC)
thread_title:  Clarification 

message: I've found the Hasse diagram, which provides one way to visualize a poset.  Are there other ways to do this?  One thought that comes to mind is something akin to a hierarchical category listing with cross-references:
* grandparent 1
** parent 1
*** child 1
*** child 2
**** grandchild
** parent 2
*** child 2 (see parent 1)
*** child 3
* grandparent 2
** parent 2 (see grandparent 1)
** parent 3
Is there an article on this general topic? modify 14:08, 27 April 2007 (UTC)
replies: text: The Hasse diagram includes only the relations that cannot be inferred from other ones using the transitive law. So your inclusion diagram is not a Hasse diagram. Yes, the Hasse diagram fully describes the partial order, but not directly by its arrows. Two elements are ordered when there is a directed path between them, not just a single edge. —David Eppstein (talk) 17:17, 17 August 2018 (UTC)
thread_title:  Visualizing a poset 

message: If so, how comes that in the table, the ones counted in column "Transitive" are much less than "All"? --Army1987 14:23, 27 October 2007 (UTC)
replies: text: The relevant column is the one labeled "partial order". The one labeled "transitive" counts all transitive relations, some of which (e.g. the complete relation) are not partial orders. —David Eppstein 18:05, 27 October 2007 (UTC)
thread_title:  Aren't partial orders transitive by definitions? 

message: Order theory is almost the same as this one, shouldn't they be merged? Itaj Sherman (talk) 23:55, 26 January 2008 (UTC)
replies: text: Order theory is a major topic of mathematics, occupying an entire number in the mathematics subject classification. There are many types of orders other than partial orders. If the articles are too similar, it is more likely because the order theory article does not devote enough space to those other topics. They should not be merged. —David Eppstein (talk) 00:29, 27 January 2008 (UTC)
thread_title:  The [[Order theory]] article 

message: The newest addition to this section appears completely out of place. As far as I could tell, it does not treat the partial orders, but states some facts about general "topolical" relations. Should it be moved to "Relation (mathematics)" or just deleted due to its narrow technical nature? Arcfrk (talk) 22:01, 29 May 2008 (UTC)
thread_title:  Partial order in topological spaces 

message: I realise this is more caused by my relative inexperience with english mathemathical language, but I had to do a double take on the description of a chain which is described as "If any two elements of a poset are comparable, the poset is called a chain". In my first reading this seemed to suggest to me that any poset with two elements that are comparable is a chain, while obviously it is saying that in a chain all elements are mutually comparable. Perhaps this is a mistake that more people would make and I was wondering if it's not possible to change the wording to something that would make that clearer? Wppds (talk) 09:40, 18 May 2010 (UTC)
replies: text: It was changed to "every two". — Carl (CBM · talk) 02:09, 16 July 2010 (UTC)
thread_title:  Wording of chain/antichain 

message: The article contains the misleading statement that "algorithms for finding linear extensions of partial orders are called topological sorting." This is incomplete because top. sort algorithms take as input a graph that already has all the relations figured out. When talking about extending a partial order, it is more natural to start with an oracle that, given any two elements, says whether they are comparable or returns the larger element. The comparison itself might be hard. Therefore, before you use any topological sort, you need an algorithm like what's described in "Sorting and Selection in Posets" (Constantinos Daskalakis, Richard M. Karp, et al) to create a data structure that describes the ordering efficiently. 173.79.253.13 (talk) 19:59, 14 July 2010 (UTC)
replies: text: I suspect you're relying on a conception of poset that feels natural for a computer scientist ("Gee, I don't know this poset unless I possess the entire n×n relation explicitly, because if there are any gaps it may be computationally expensive to figure out how to fill them in"). By contrast, the mathematical definition of poset implies such complete knowledge. You might say that the mathematical perspective is more existential, while its computer-science cousin is more constructivist.—PaulTanenbaum (talk) 01:04, 22 April 2011 (UTC)
replies: text: There is very little difference between the mathematical notion of a poset and the CS notion of the transitive closure of a directed acyclic graph. But topological sorting can be applied to any DAG, regardless of whether it is transitively closed. Since many different directed acyclic graphs correspond in this way to a single partial order, the problems of transitive closure and linear extension are distinct, although they are obviously very closely related to each other. —David Eppstein (talk) 02:52, 22 April 2011 (UTC)
replies: text: Yes, I'll buy that.—PaulTanenbaum (talk) 03:09, 22 April 2011 (UTC)
thread_title:  Relation to [[topological sort]] 

message: If there's one flaw that I can see in this otherwise wonderful article, it's that both Hasse diagrams given are those of Boolean lattices (the first for B_3, the second for B_6).  Boolean lattices are so well-behaved, it may lead the lay-reader to incorrectly assume things about posets in general.  For instance, he or she might think all posets are pure (meaning every maximal chain has the same length) or that all elements at a particular height cover the same number of elements.  All in all, I think the Boolean lattices are too nice to demonstrate how complicated and asymmetric posets can be.  I'm fine with one Boolean lattice, but both images are Boolean.  Do you think we could include a figure of a non-pure poset's Hasse diagram?  I'd be more than happy to make one.  Dotdotdotatsignapostrophe (talk) 08:18, 5 August 2011 (UTC)
replies: text: You make a very good point, and I (for one) would happily take you up on your offer to add a different image. But not without offering my own brilliant idea: I'd point out that it'd be very useful to the non-technical reader if the example's ground set and partial order were both familiar enough to encourage exploration of this Hasse-diagram thing. So, for instance, I'd suggest using as the ground set a collection of members of some well-known family (The Kennedys? The Mountbatten-Windsors?) under the partial order is-descended-from. Judicious inclusion and exclusion of sundry relatives would allow you to illustrate such notions as maximals, minimals, antichains, etc. Heck, it'd even give you an excuse to work the Terminator into your example!—PaulTanenbaum (talk) 00:13, 15 September 2011 (UTC)
replies: text: The alternative is a maze of spaghetti, so my feeling is that fewer edges but a greater need for inference is a better choice. —David Eppstein (talk) 17:54, 17 August 2018 (UTC)
thread_title:  New example? 

message: Lead reads:
:A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.
This is the definition of a strict poset. But the article goes on to define an (unqualified) partial order as a non-strict partial order. I'm not sure how to edit the lead's definition so that it is both correct and simple. Ideas? --Macrakis (talk) 20:29, 16 September 2016 (UTC)
thread_title: Strictness in lead

message: MOS:FORMULA says:
:"Large scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting boldly from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. Proposed changes should generally be discussed on the talk page of the article before implementation."
No discussion of this sort happened with this article, and it appears to me that the previous version should be restored unless a consensus for a change here is developed. — Carl (CBM · talk) 11:46, 18 December 2017 (UTC)
replies: text: I agree. Too many editors are changing formulas from math to html or vice versa. It's pointless churn, it messies our edit histories and watchlists for no good reason, and until we convince the wikimedia developers to give us working math support it's a waste of effort. For now I think we'd be better off maintaining a hiatus on format changes. —David Eppstein (talk) 18:22, 18 December 2017 (UTC)
thread_title:  LaTeX vs. HTML 

message: The article says very little about the ordering of ordinal numbers, and when it does, it is in too explanative/introductory phrases that makes it hard to descipher if they represent a tehnical part of the concept.
The only appearance of some sort of "ordinal1 < ordinal2" before the "Von Neumann definition" section is:
* "they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on".
It is hard to tell if this is a part of the definition, a consequence of the ordinal generation definition, or just an intuition point.
Particularly, when explaining the the ordinal as the equivalence class of sets with same order type, I can't tell how the order of these equivalence classes appear.
Things are better in the "Von Neumann definition" section, since there "element of" is explicitly the order relation -- even though I consider working with "subset of" would be shorter and simpler.
PS: I bet for most readers the punchline is how ω appears and the introductions seems just to be ad-hoc explanations of equivalence classes and orders. Usually the first phrase in Wikipedia articles uniquely targets the concept more accurately than just saying "generalization of natural numbers"188.27.126.226 (talk) 20:04, 20 October 2018 (UTC)
replies: text: I guess, your critics is meant for Ordinal number, not Partially ordered set? If yes, you better re-post it at the talk page there. - Jochen Burghardt (talk) 13:13, 21 October 2018 (UTC)
thread_title:  Lack of definition 

message: I have trouble remembering which conditions are on which kind of order. In the name 'pARTial order' the three letters ART remind me that the relation must be antisymmetric, reflexive, and transitive. Acorrector (talk) 10:08, 19 September 2019 (UTC)
replies: text: I like this memory aid, and I think it could be helpful for people reading the article; memorising/remembering axioms can be difficult, especially in this case where there are axioms for strict partial orders and non-strict partial orders with similar names (e.g. "antisymmetric" and "asymmetric" - although these both begin with "a" so perhaps this isn't the best example). I would support this memory aid being added to the article if other people agree. Joel Brennan (talk) 21:19, 11 December 2020 (UTC)
thread_title:  p-ART-ially ordered - A-ntisymmetric, R-eflexive, and T-ransitive 

message: Below, isn't the highlighted "a \leq a" a typo? Shouldn't it be "a = a"?
A non-strict partial order \leq may be converted to a strict partial order by removing all relationships of the form ; that is, the strict partial order is the set < \; := \ \leq\  \setminus \  \Delta_P, where \Delta_P := \{ (p, p) : p \in P \} is the identity relation on P \times P and \;\setminus\; denotes set subtraction. BrianH123 (talk) 20:09, 2 September 2021 (UTC)
replies: text: No, it's ok as is. We are modifying the ≤ relation, not the = relation, so it is the a≤a pairs of the ≤ relation that should be removed. —David Eppstein (talk) 20:32, 2 September 2021 (UTC)

text: Well, technically a \leq a is short for "(a,a)\in\leq". So a \leq a is isn't a relationship at all, it's a formula. It's somewhat of an abuse of notation. (a,a) is the relationship (tuple element). I've corrected this. --Mathnerd314159 (talk) 19:02, 1 January 2022 (UTC)
thread_title:  Error in "Correspondence of strict and non-strict partial order relations"? 

message: Who is ordering?  Give an example. 2409:4043:2E8E:C8FC:0:0:8089:220D (talk) 17:19, 9 February 2022 (UTC)
replies: text: "Ordering" is used as a noun throughout the article, so your question doesn't make sense. - Jochen Burghardt (talk) 09:38, 10 February 2022 (UTC)
thread_title:  Who is ordering?  Give an example. 

message: In section Partially ordered set#Informal definition, you reverted my edit "x>y" --> "y<x". 
My point is that the section tacitly assumes that the relation is called "<", and that ">" denotes its converse. However, there are also different names for partial orders, like "|" for "is a divisor of", or just "R"; in these cases there are no "natural" symbols for the converse. The authors of the Sage reference manual (citation [1]) apparently didn't think about this issue; however, in the implementation of P.compare_elements(x,y) they most likely had to consider it, since the name P has no "natural converse symbol". (I guess, the implementation will look like if (x==y) return 0; else if (P.is_member([x,y])) return -1 else if (P.is_member([y,x])) return 1; else return None;, using C-like  rather than proper Sage syntax). 
Another point is that the paragraph tacitly assumes strict partial orders, which are not yet introduced at that point. - Jochen Burghardt (talk) 11:49, 2 March 2023 (UTC)
replies: text: In the informal definition section, I am trying to push an alternative definition of partial orders, namely defining them as a comparison operator A -> A -> {GT,LT,EQ,INCOMPARABLE}. The 4-valued comparison operator and the various p.o. relations are two different ways of defining the same thing, and either can be taken as the base notion. I think the 4-valued function definition is clearer as a mental model than the typical p.o. relation presentation (which leads to 4 interconnected relations <, leq, > , geq and confusion as to which relation actually defines the poset). It is possible to define a poset completely in terms of the 4-valued function satisfying some properties (e.g. the anti-symmetry property "compare x y = GT <-> compare y x = LT"), although I haven't found a good source presenting that so it's not in the article. But, taking that definition as the base definition, there is nothing that assumes that ">" denotes a converse, or that strict partial orders exist - those can all be defined in terms of the 4-valued function, e.g. we say x > y holds if compare x y = GT, and similarly x leq y if compare x y is either LT or EQ. Mathnerd314159 (talk) 18:00, 2 March 2023 (UTC)
replies: text: Ok, I understood your intention now; and I agree that "GT" is just an operator result, and there is no converse-relation issue.  In the article section, it didn't become clear to me that it is supposed to be an alternative definition.  Maybe it is a good idea to insert into the article section your above first sentence ("... alternative definition of partial orders, namely defining them as a comparison operator A -> A -> {GT,LT,EQ,INCOMPARABLE}"). Your approach works only for strict partial orders, doesn't it? This should also be mentioned, which, in turn, would require to move the section below Partially_ordered_set#Strict_partial_order. - Jochen Burghardt (talk) 18:18, 2 March 2023 (UTC)
replies: text: I tried moving the section, it seems a little better. I'm not really sure what you want the article to say about strict partial orders. Mathnerd314159 (talk) 00:34, 3 March 2023 (UTC)
replies: text: I like your new version. I made some minor additions in the Hasse-diagram paragraph, and a suggestion about the cmp function - fell free to revert it. - Jochen Burghardt (talk) 11:42, 3 March 2023 (UTC)
thread_title:  Converse relation symbol 

message: The definition by Wallis.2013 (requiring transitivity and antisymmetry only) is quite interesting. I wonder if there are proerties mentioned in the article that are not satisfied by all Wallis-orders. Trying to check, I read through subsection "Extrema", and found that it tacitly assumes that \leq and \geq are weak partial orders, and  <  and  >  are strict ones. I guess this assumption is made thoughout the whole article, and therefore should be mentioned explicitly, preferably in section "Correspondence of strict and non-strict partial order relations", i.e. immediately after weak and strict orders were introduced. If we were to elaborate more on Wallis-orders, we should use an own symbol for them, e.g. \precsim, or \vartriangleleft, or \curlyeqprec, or something similar. In that case, I'd suggest to insert a subsection for them right after "Strict partial orders". - Jochen Burghardt (talk) 14:33, 4 March 2023 (UTC)
replies: text: Well if you think of the partial relation as the matrix P \times P then what we're talking about is the diagonal. Irreflexive is all 0's, reflexive is all 1's, and Wallis's general definition allows any mixture of 0's and 1's. There's not really any extra structure allowed by the generality, it's just a partial order with some extra information, similar to how a partial equivalence relation is just an equivalence relation plus a larger domain. Maybe I should just move Wallis to the "alternative definitions" section so that it's clear it's an unusual definition. Mathnerd314159 (talk) 20:31, 4 March 2023 (UTC)
replies: text: I did the move. I guess it would be nice if there was a better name for them than Wallis-orders or transitive antisymmetric relations, but searching for it just shows reflexive partial orders and filtering out reflexive gives noise. Mathnerd314159 (talk) 22:04, 4 March 2023 (UTC)
thread_title:  Most general definition