Talk:Binary relation

message: This section seems full of non-standard definitions, you might want to actually provide a solid reference for there first use, or who proposed them.  — Preceding unsigned comment added by Differentiablef (talk • contribs) 03:59, 30 July 2011 (UTC)
thread_title:  Special types of binary relations 

message: I concur. Some authors are trying to use Wikipedia, not as reference, but as a marketing tool for their preferred terminology. In particular, neither "serial relation" nor "connex relation" is common in the literature, outside, perhaps of a limited subgroup within computer science. 24.4.62.241 (talk) 22:59, 12 March 2019 (UTC)
thread_title: Not encyclopedia material

message: I have seen another definition of binary relation as an ordered triple R=(X,Y,G), where G is a subset of X × Y and is called the graph of R.  This definition is better then the definition here as it avoid the confusion while talking about the function and its graph.  Any comment?  --Wshun
:I have seen another definition of a graph - G(N, T), where N and T are set of nodes and connections/relations/associations/links/etc binding em correcpondingly. So, the notation G(R) used here for graph definition is merely unclear. Javalenok
:So, in my opinion, the steps of constructing the graph G=(V,E) for a relation must be given. The set of vertices is the union N = (X v Y) which are connected by directed edges when xRy; that is, E={(x,y)|xRy}.
In the current definition it says:
"The statement (x,y) ∈ R is read "x is R-related to y"".
Shouldn't it be like that:
"The statement (x,y) ∈ G is read "x is R-related to y""?
I mean (x,y) is an element of the Cartesian product X × Y, R refers to the triple (X,Y,G) so "(x,y) ∈ R" doesn't make sense. On the other hand "(x,y) ∈ G" does, as that's what G defines: all the ordered pairs (x,y) of X × Y such that x is R-related to y, no?  —Preceding unsigned comment added by PhaseQ (talk • contribs) 14:44, 16 January 2010 (UTC)
thread_title:  Another definition 

message: What does it mean for a relation to be total? There are two different definitions in the article, one under special relations (for all x in X there exists a y in Y such that xRy) and one under relations over a set (for all x and y in X it holds that xRy or yRx); the latter one I have seen before. If the former one is also used in the literature, then it should be clarified that there are different definitions. -- Jitse Niesen 14:57, 16 Jun 2005 (UTC)
replies: text: For the first meaning the term entire may be preferable. I used this term in the Axiom of dependent choice article, but I don't remember where I got it from. --Zundark 15:42, 16 Jun 2005 (UTC)
thread_title:  Definition of total? 

message: What is the "correct" definition of composition. Some hours ago, an anonymous changed it from
:S o R = { (x, z) | there exists y ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S },
to
:R o S = { (x, z) | there exists y ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }.
Now, Paul August has changed it back. I seem to remember that it has also changed some time ago,
The first definition has the advantage that it agrees with function composition (as Paul notes). However, I looked at some web pages found via Google (which is of course heavily slanted towards computer science), and it does seem that the second definition appears regularly. What should we do? Should we mention both? -- Jitse Niesen (talk) 21:35, 11 October 2005 (UTC)
replies: text: I'm afraid there's no "correct" definition. Conventions differ, though in my experience the second definition is more common. The fact that it conflicts with function composition is not necessarily a problem; also, it is sometimes resolved in other ways, either by changing the order of arguments in function composition, or by swapping domain and range in the representation of functions by relations (i.e., f : X → Y is identified with {(f(x),x); x ∈ X }). I guess any definition would work here, provided we use it consistently, and mention the other possibility as well. -- EJ 19:13, 12 October 2005 (UTC)
thread_title:  Definition of composition 

message: JA: I humbly (but most sensibly) suggest that the class/set distinction, however important it may be in the long run, is definitely not 'line 1' material, since it can't be defined or discussed without getting into a controverted  variety of different axiomatizations for set theory, and that the (non-pejoratively speaking) 'naive' reader should probably be given a (however illusorily) 'solid' foundation in naive set theory before being led down that particular garden of forking paths.  Jon Awbrey 17:16, 18 January 2006 (UTC)
replies: text: I disagree:  the remark about membership and inclusion as examples of relations (not to mention general equality) requires some qualification if classes are excluded. Randall Holmes 17:40, 18 January 2006 (UTC)

text: A concrete example:  if proper classes are not considered, the statements in the article on equality are wrong unless everything is restricted to a specific set. Randall Holmes 17:43, 18 January 2006 (UTC)
replies: text: I moved the discussion of class/set to a subsection in binary relations (as requested by Stolfi) and omitted all reference to it in function (mathematics); I think a mention is needed in binary relations, but it doesn't need to be at the outset, and not even a mention is needed in the function article.  When you're right, you're right. Randall Holmes 18:28, 18 January 2006 (UTC)
thread_title: Distinction between class and set

message: If relations have frames (as described in the relation (mathematics) article), then a binary relation is a specific case of the more general concept of k-ary relation.  However, if a relation is identified with its graph, then k-ary relations are special cases of binary relations (in fact, a (k+1)-ary relation is a species of k-ary relation...)
I am myself an unregenerate advocate of simplicity:  the relation is its graph.  Adding the frame to  the relation as a component is analogous to sticking type-labels to things:  objects should not have to carry type labels around with them, as the context in which they appear should provide cues sufficient to tell what type we are currently assigning to them.  This is not a proposal to change the article (the usage described is unfortunately common), just a philosophical comment... Randall Holmes 23:46, 20 January 2006 (UTC)
replies: text: No, I'm not confusing anything.  I am standing up for the old-fashioned view that a binary relation from A to B is best identified with its graph (a subset of the cartesian product of A and B) and not complicated with indications of its domain and codomain.  If binary relations are defined in this way, then ternary relations (for example) are a kind of binary relation (I don't tout this as a virtue of the old approach -- it's merely an observation that things are different there).  If relations are encumbered with explicit indications of domain, then this ceases to be true (binary relation is then a special case of k-ary relation).  The domain and codomain labels are indications of type (indications of how the graph is intended to be used); my view (in general) is that the context in which an object is used rather than intrinsic features of the object should give type information (the context should not be part of the object, but just that -- its context).  Please note though that this is a philosophical grumble (because in my writing I keep having to nod to the (IMHO misguided) general practice), not a proposal that articles should be written differently.  Randall Holmes 00:36, 23 January 2006 (UTC)
thread_title: Which concept nests in which?

message: FYI, in case anyone is reading this page and not Talk:Relation (mathematics), I have proposed that we consider moving Binary relation to Relation (mathematics) and moving what is currently in Relation (mathematics) to n-ary relation.  See the discussion at Talk:Relation (mathematics); in the interest of consolidating, let's have the discussion over there. Mangojuice 19:08, 25 January 2006 (UTC)
replies: text: Please don't. How would that help, at all? Binary relations are an important special case, but are certainly special. Charles Matthews 19:51, 25 January 2006 (UTC)
thread_title:  Renaming things 

message: I think there word "binary" in front of word "relation(s)" is missing somewhere. In other case if someone read those paragrafs out of text context it can be confusing and  maybe false. 
--User:Vanished user 8ij3r8jwefi 14:09, 10 February 2006 (UTC)
thread_title:  Missing word "binary" somewhere 

message: The text i'm using uses the terminology "on" a set instead of "over" a set.. would it be appropriate to add a note mentioning the alternative usage?--Keith 00:42, 12 April 2006 (UTC)
thread_title:  Relations "over" a set vs. "on" a set 

message: According to my experience, a "binary relation" is usually a relation from one set to itself; a "relation" is usually a (dyadic) relation from one set to another, and never an n-ary relation unless this is explicitely stated.
Thus, all (...) could be dropped in the following:
* a function from E to F is a (functional) relation from E to F (m-maybe "on" E x F)
* a relation (on E x F) is a (dyadic) relation from some set E to some set F
* a (partial) order on E is a binary relation on E, i.e. a dyadic relation on E x E or from E to E
* a n-ary relation on E_1 x E_2 x ... x E_n is an n+1 tuple R = (E_1,...,E_n, G) where G is a subset of E_1 x E_2 x ... x E_n, called the graph of R.
Do you really know (one? / several? / many?) places where an author speaking of a "relation" tacitely understands relations which are not dyadic relations (i.e. more than 2 arguments), without saying so explicitely?
If not, I would strongly argue for making some minor changes explaining more clearly this (IMHO) universial  use of terminology. — MFH:Talk 16:14, 13 October 2006 (UTC)
thread_title:  relation vs binary relation 

message: I'm going to delete the mods made anonymously that mention usage of binary relation in linguistics and computer science.  Here's why... if the usage(s?) is (are?) distinct from the mathematical usage, then it (they) should be in a separate article treated through Wiki-style disambiguation.  If, on the other hand, it (they?) is (are?) a special class of (isomorphic to?) the usage already described in this article, then the references to its use in linguistics and computer science should make this identity explicit.  Until then, the comments confuse and cruft up this article.—PaulTanenbaum (talk) 13:17, 28 January 2008 (UTC)
thread_title:  Yanked linguistics/CS comments 

message: I reverted the changes from the curly to the straight LaTeX symbols because the former are for general binary relations (of the given kind), the straight ones for particular binary relations. For example  a \preceq b  stands for -any- partial order, but  a \le b  stands for the particular 'less-than' relation for numbers.
Also,  \approx   does not stand for an equivalence relation (approximation does not satisfy transitivity), and  \equiv  is a specific equivalence relation on integers. Hahahaha4 (talk) 19:21, 25 September 2008 (UTC)
replies: text: Sorry, but what you are saying is complete nonsense. In order theory and algebra (lattice theory), orders (and preorders) are always denoted \le by default, other symbols being employed only to avoid confusion when several distinct orders are considered at the same time, and even then it is more common to use indices like \le_1, \le_2 rather than odd symbols like \precsim. In fact, I can't remember ever seeing \precsim used for such purpose, unlike \preceq or \sqsubseteq. The reality is exactly opposite to what you claim, generic orders are usually denoted by \le, whereas \preceq is typically used for specific (pre)orders (e.g., elementary submodels in model theory). And yes, both \approx and \equiv are used to denote various equivalence relations, not just the two specific relations which you mention. — Emil J. 14:13, 26 September 2008 (UTC)
replies: text: It was my impression that more recent usage favors the curly symbols for abstract relations and straight for specific relations. But I could have narrow experience that is also skewed by LaTeX/CS culture. Hahahaha4 (talk) 20:16, 26 September 2008 (UTC)
replies: text: I agree with EmilJ, and I and I don't think it's a question of publication dates. More likely a matter of different conventions in different fields. I would imagine that many school teachers agree with your comments (as less confusing for the kind of children, in the same way that the same letter in school physics always has the "same" meaning), but the universal practice in the mathematical community is as described by EmilJ (as much more convenient). --20:24, 26 September 2008 (UTC)

text: Although I've seen mostly just \le, only occasionally \sqsubseteq, and rarely \preceq, perhaps there should be a short section listing all of these symbols, stating that they are all sometimes used to denote ordering? linas (talk) 13:29, 27 September 2008 (UTC)
thread_title:  Symbols for Binary Relations 

message: The term "extended" is not explained in the section about "Relations over a set" (bottom of the section)
FredrikMeyer (talk) 10:39, 1 September 2009 (UTC)
replies: text: It was overlooked when Hans Adler corrected the terminology in that section a few months ago.  Thanks.  —Dominus (talk) 13:28, 1 September 2009 (UTC)
replies: text:  What did it even mean? It was also mentioned in inverse relation with a link to here (where it isn't defined [now]), but I couldn't find it in the literature, so I've deleted the sentence about that over there. 86.127.138.67 (talk) 07:47, 19 April 2015 (UTC)
replies: text:  I see it meant serial. 86.127.138.67 (talk) 07:50, 19 April 2015 (UTC)
thread_title: Extended? (Or extendable?)

message: I had expected to see something about "functions between relations on a set" in this article. Namely, given two relations (R on X) and (S on Y), we can consider functions f:X->Y which are relation-preserving in the sense that xRx' implies f(x)Sf(x'). This yields a category whose objects are pairs (X,R) with X a set and R a relation on X; an arrow (X,R) -> (Y,S) is a relation-preserving function X->Y.
A special case of relation-preserving functions is when the relations are posets: then relation-preserving functions are the monotonic ones.
145.97.197.129 (talk) 16:12, 7 August 2010 (UTC)
thread_title:  Functions between relations? 

message: In the section http://en.wikipedia.org/wiki/Binary_relation#Is_a_relation_more_than_its_graph.3F there is a sentence that goes "Most authors insist on distinguishing between a function's codomain and its range." I think this should be "a function's domain and its range or co-domain." Or am I mistaking what is being said here?
Neil (talk) 13:47, 12 July 2011 (UTC)
replies: text: You're mistaken.  Some authors do not distinguish between the codomain and the range (e.g., Jean E. Rubin, Set Theory for the Mathematician).  Most do.  — Arthur Rubin  (talk) 16:45, 12 July 2011 (UTC)
thread_title:  Error in codomain/domain terminology? 

message: The section https://secure.wikimedia.org/wikipedia/en/wiki/Binary_relation#Restriction defines restriction only for a relation on a set S, i.e. over S×S. But a restriction may be defined for a relation over S×T, and in such case the current definition is wrong. See https://secure.wikimedia.org/wikipedia/en/w/index.php?title=Restriction_%28mathematics%29&oldid=451456911 and http://www.proofwiki.org/w/index.php?title=Definition:Restriction/Relation&oldid=66420.  — Preceding unsigned comment added by Rosivaldo (talk • contribs) 16:37, 24 October 2011 (UTC)
thread_title:  Restriction definition 

message: Isn't there a notion of homomorphism of relations similar to that of orders?
i.e. for relations R from M_1 to M_2 and S from N_1 to N_2, a pair of functions h = (h_1 : M_1 \to N_1, h_2 : M_2 \to N_2) with \forall m_1 \in M_1: \forall m_2 \in M_2: R(m_1, m_2) \Rightarrow S(h_1(m_1), h_2(m_2))?
Shouldn't something like this be mentioned in the article?
-- 132.231.198.153 (talk) 10:08, 7 December 2011 (UTC)
thread_title:  Homomorphisms? 

message: Page 1331 of Encyclopedic dictionary of Mathematics seems to define correspondence just like relation without the left and right total bit, in fact no real difference from abinary relation. Dmcq (talk) 13:27, 14 February 2012 (UTC)
thread_title:  Correspondence definition 

message: change 'The concept of function is defined as a special kind of binary relation' to 'A function with one domain variable is a special kind of binary relation' ?  — Preceding unsigned comment added by 72.160.223.249 (talk) 12:01, 23 June 2012 (UTC)
thread_title:  Intro 

message: I cannot believe that "left-unique" means injective instead of functional, and that "right-unique" means functional instead of injective.
I can't verify the claim, because I don't have the referenced book, and it costs a shitton of cash. Can someone verify that the terms "left-unique" and "right-unique" are not mixed up on Wikipedia?  — Preceding unsigned comment added by 31.46.91.77 (talk) 13:49, 15 January 2013 (UTC)
replies: text: I can see the referenced page through Google books, and it agrees with our article. In any case, I find the terminology perfectly natural: right unique means that elements on the right-hand side corresponding to a given element are unique (i.e., the relation is a partial function), and symmetrically for left-unique.—Emil J. 14:11, 15 January 2013 (UTC)
replies: text: And I find an opposite terminology perfectly natural: right-unique means that elements on the right-hand side are unique (i.e., an element appears on the right-hand side at most "once", i.e., the relation is injective).  — Preceding unsigned comment added by 81.182.174.55 (talk) 17:38, 15 January 2013 (UTC)

text: Are these terms standard terminology? They seem natural to me, but I had never heard them before reading wikipedia, and I wonder if the article should really mention them at all. Are there any books other than M. Kilp, U. Knauer, A.V. Mikhalev, "Monoids, Acts and Categories" () that use these terms? – Tobias Bergemann (talk) 14:26, 15 January 2013 (UTC)

text: For what it's worth, there's a discussion at MathOverflow at http://mathoverflow.net/questions/17854, where people wonder about these terms. Specifically the comment by Harald Hanche-Olsen (associate professor at the Department of Mathematical Sciences at the Norwegian University of Science and Technology): "I certainly did not know the terms left-unique and right-unique, and moreover when I tried to guess what they meant, I ended up with the opposite meanings. Left-unique, I reasoned, must mean that a pair in the relation is uniquely determined by its left member, i.e., functional. But that is the definition of right-unique. Go figure." –Tobias Bergemann (talk) 14:35, 15 January 2013 (UTC)
replies: text: A web search suggests that various authors use these terms, it’s certainly not just one book. I think it does not do any harm to mention them, even though it is a minority usage.—Emil J. 14:49, 15 January 2013 (UTC)

text: I also never saw those usages in mathematics, but there are lots of hits on Google Books for "left unique" relation. The same issue with "left" and "right" not having clear meanings also happens in other places, e.g. group actions. — Carl (CBM · talk) 15:03, 15 January 2013 (UTC)

text: Well, I can't find it the terminology as such in Bourbaki, so the authors of the pdf above might just mean that the notions are defined there (but not necessarily in that terminology). But I found at least "left total" defined in a pretty ancient 1967 book  Lattice Theory by Helmuth Gericke. I can only see snippets in Google though, and he doesn't seem to define the left/right unique but he does use the term "bi-unique" for bijective. Some1Redirects4You (talk) 05:19, 20 April 2015 (UTC)
thread_title:  Left- vs right- uniqueness 

message: It's written that "The transitive relation is irreflexive if and only if it's asymmetric" how is that? if I defined a relation xRy that holds if both X and Y are even numbers, so it is clear that it is transitive - if xRy is valid and yRz is valid then xRz is valid - and symmetric, if xRy is valid then yRx is valid too, however it is irreflexive in the same time, for example 3R3 is false! so it is Symmetric, Transitive and Irreflexive in the same time. 
Mo-Cubed (talk) 22:53, 22 October 2014 (UTC)
replies: text: Your example is not irreflexive since that property must hold for all x and you have that 2R2 is true. Bill Cherowitzo  (talk) 03:45, 23 October 2014 (UTC)

text: You are right in that your relation isn't reflexive, since not 3R3. But it isn't irreflexive, since 2R2. I added a note to the article about this issue. - Jochen Burghardt (talk) 04:18, 23 October 2014 (UTC)
thread_title:  Error in the transitivity in relations 

message: It should probably be merged with (or at least turned into a subsection of) the section on [types of] endorelations. Arguably functions are very common examples of relations too... Some1Redirects4You (talk) 03:49, 20 April 2015 (UTC)
thread_title:  Section titled "Examples of common binary relations" 

message: The definition of transitive offers a statement about transitive relations (connecting their irreflexiveness and asymmetry) and then gives an example of two relations, both of which are irreflexive and asymmetric, but one of which is transitive and the other of which is not. These two sentences should be reversed.
Not sure if this is worthy of the talk page or if I should just make the change, but I've only ever done small edits.
clahey (talk) 00:05, 6 October 2015 (UTC)
replies: text: Be bold, you should have done this yourself. If you make a mistake, someone is very likely to spot it and correct it, so don't let that hold you back. The trick is to not involve your ego with your edits − you do the best you can, knowing that you can't always be perfect. Bill Cherowitzo  (talk) 01:53, 6 October 2015 (UTC)
thread_title:  Confusing Sentence Order in definition of Transitive Relation 

message: There is a tag on the section "Is a relation more than its graph?" saying it is confusing and requires clarification. I just reverted a change where someone was saying a relation was always a correspondence as in the formal definition section before rather than the looser definition often use din logic. So it looks like the tag is needed an a bit more clarification is in order but I can't see how to phrase it better. Dmcq (talk) 21:18, 4 January 2016 (UTC)
replies: text: That section is not simply misleading, but plain wrong: A Graph consists of both a set of vertices and a set of edges. In particular, a graph can have vertices without edges. The section here claims a graph would consist of its edges. --84.153.18.145 (talk) 08:19, 8 October 2016 (UTC)
replies: text: It's graph of a relation, not graph (discrete mathematics). I repeated the wikilink in the article's section. - Jochen Burghardt (talk) 13:58, 8 October 2016 (UTC)

text:  This question isn't well posed, because it presumes that a relation is conceptually distinct from it's graphical representation, and they are not. An algebraic OR a graphical description of a given mapping may be unclear; but this is usually due to the fact that the writer implied information that the reader failed to infer.
A graph of a relation only exists at the points over the domain and range where the relation is defined. At the co-ordinates within the cross product of the domain and range, a graph depicts the point at that co-ordinate with one color if it belongs to the relation; otherwise, it marks that point in another color. 
Outside of the area where the relation is defined, there is no point to mark one way or another; it's not part of the graph being drawn.
For maximum clarity, the parts of the page where points on the graph do not exist should be visually distinct from the points where the graph exists (for example, drawn in a different color); but mathematicians traditionally saved time and ink by only marking the points on the page where the relation is defined. In cases where it matters, the domain and range are typically stated; where they are not, a specific domain and range is typically implied by context within a given branch of mathematics: anything left unstated within a mathematical formula is implied by context to be "usual one" for the branch of math being studied (eg. the domain and range of a function are real numbers when doing Real analysis, Complex numbers with Complex Analysis, etc.)  — Preceding unsigned comment added by 206.108.127.16 (talk) 19:36, 20 December 2018 (UTC)
thread_title:  Tag on section "Is a relation more than its graph?" 

message: If X is the set {1,2,3} then is the relation {(1,1),(2,2)} a reflexive binary relation "on" the set X?
The definition of "reflexive" in the current article is: "for all x in X it holds that xRx."   By that definition, a reflexive relation "on" {1,2,3} is required to contain the ordered pair (3,3).   
Is the intent of the definition of "R is a reflexive relation on X" to require that each element of X must appear as the first member of an ordered pair of R ?
Tashiro~enwiki (talk) 23:05, 4 November 2016 (UTC)
replies: text: Yes, that is the intent. This is the standard definition and without that condition "reflexive" would be a rather useless property. Bill Cherowitzo  (talk) 02:53, 5 November 2016 (UTC)
thread_title: Definition of "reflexive"

message: There are two names given for the same type of relation given in the current version of this article. Both left-total relation and serial relation are described in different sections. In fact, both names have references supporting them, now mentioned on Serial relation. Consolidation of the descriptions is recommended for reading consistency. — Rgdboer (talk) 03:31, 13 April 2018 (UTC)
replies: text: I agree with these clarifications. However, I suggest to keep warning footnotes about the old terminology, both at total order (there explaining the name "total" by historical reasons), at connex relation (there hinting at a former name that is still in use);  possibly also at the "connex" item in binary relation#Relations over a set. - Jochen Burghardt (talk) 07:33, 24 May 2018 (UTC)
thread_title:  Serial/Total 

message: Currently the article contains this paragraph:
:In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology is justified by the fact that when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block diagonal matrix with rectangular blocks of true on the (asymmetric) main diagonal.  Other authors however use the term "rectangular" to denote any heterogeneous relation whatsoever (Winter 2007).
These sources have been confirmed to use the term rectangular relation as described. However, Jacques Riguet has used the term in another sense (as described in his article). His usage has been taken up by Gunther Schmidt in his books on relations. Suggestions are requested for properly digesting these various uses of the same term in the study of relations. — Rgdboer (talk) 21:42, 9 June 2018 (UTC)
thread_title:  Rectangular 

message: This redirect page is potentially confusing, since relation can refer to other finitary relations instead of binary relations. I've seen several Wikipedia articles that link to this page in error, when they should link to finitary relation instead. Should we re-target it to Relation#Mathematics to avoid this misunderstanding? Jarble (talk) 06:35, 19 June 2018 (UTC)
replies: text: In mathematics, binary relation is clearly a primary topic for relation (unless otherwise specified, relation means generally binary relation). So, if "relation" would not have meanings outside mathematics, Binary relation would be simply named "Relation", with a disambiguating hatnote. Here the ambiguity is solved exactly as usual, with a disambiguating hatnote. If you know articles containing wrong links, the simplest way is to fix them directly. D.Lazard (talk) 08:19, 19 June 2018 (UTC)
thread_title:  ''[[Relation (mathematics)]]'' redirect page 

message: The following was removed and abstract rewriting system added to See also:
=== Algebras, categories, and rewriting systems ===
:Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relation algebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finite and many-sorted).
:For heterogenous binary relations, a category of relations arises. Winter 2007
:Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstract rewriting system.
This long-neglected subsection has been superseded by heterogeneous relation and improvements to algebraic logic. — Rgdboer (talk) 23:53, 25 July 2018 (UTC)
thread_title:  Algebras 

message: See Talk:Binary operation#Merge with Binary relation?. --Ipatrol (talk) 17:48, 26 November 2018 (UTC)
thread_title:  Merge discussion 

message: An editor has asked for a discussion to address the redirect ≘. Please participate in the redirect discussion if you wish to do so.  D.Lazard (talk) 16:20, 18 June 2019 (UTC)
thread_title:  ≘ listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: Pinging the earlier participants: 
Despite the lengthy discussion at #Spltting off to Homogeneous relation and #Schmidt comments, the article structure is still not satisfying. In particular, everything said in Heterogeneous relation applies in general to (and thus belongs to) Binary relation, except for the requirement A≠B. Adding the latter does not add any new results to the theory. For these reasons, I suggest the merge.
Much of the earlier discussion was about the definition of a Homogeneous relation: whether it should (1) require A≠B, or just (2) allow A≠B. The article currently uses definition (1). The current merger proposal, however, is independent of this issue, since there is not much to say about a Homogeneous relation in the sense of (1), except that it is a binary relation on two different sets. Just to say the latter, an own article is hardly needed.
Beyond that, and risking a re-opening of the earlier discussion, I note that I'd favor to split off Homogeneous relation (handling A=B as a special case), and leaving the rest in Binary relation (handling the general case A=B or A≠B). - Jochen Burghardt (talk) 16:15, 16 May 2021 (UTC)
replies: text: Support with the merger with heterogeneous relation and split-off of homogeneous relation: I am very confused with Jochen Burghardt's rendering of the current situation. My understanding is that the current article takes a view that a "binary relation" is synonymous with "heterogeneous relation"; both terms refer to "not-necessarily-homogeneous relation" (and this is the view that can be backed by the sources, if we like it or not). But the use of the term can emphasize the fact the domain and codomain might be different for the benefit of the audience who are more familiar with homogeneous relation. Since a homogeneous relation is a binary relation that is more frequently used by the general readers, it makes sense to have a separate article for it. -- Taku (talk) 03:22, 19 May 2021 (UTC)
replies: text:  Just to clarify myself: I share your view that "heterogeneous relation" should be synonymous with "binary relation" (what I called def. (2) above), but I understand the current version of Heterogeneous relation to use definition (1), as I read "distinct" as "≠". I would have no objection to change the 1st lead sentence e.g. to "where A and B are allowed to be different sets". - Jochen Burghardt (talk) 08:53, 19 May 2021 (UTC)
replies: text: Ah, I see: in your original post, you meant to say "Heterogeneous relation" not "Homogeneous relation". This is why I couldn't follow your argument. I think we are in agreement (on possible organizational changes). -- Taku (talk) 21:17, 19 May 2021 (UTC)
replies: text:  Oops - yes, you are right. I changed my original post to fix that. Sorry for the confusion. - Jochen Burghardt (talk) 21:39, 19 May 2021 (UTC)
thread_title:  Merge with Heterogeneous relation 

message: "≙" redirects to this page. However, I could not find it in the article text. --mafutrct (talk) 00:14, 2 March 2024 (UTC)
replies: text: This symbol is not used in Wikipedia. I nominated the redirect for deletion; see below. D.Lazard (talk) 10:25, 2 March 2024 (UTC)
thread_title:  ≙ is not explained? 

message: The redirect ≙ has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached.  D.Lazard (talk) 10:12, 2 March 2024 (UTC)
thread_title:  "[[:≙]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: A technicality. In the sentence "A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.", "subsets" shouldn't be used, since what the sentence intends to call "concepts" are not sets, nor is the class of all relations (what will be "decomposed") ... Unless one intends to study relations on small categories only. This can be avoided with a simpler sentence, like "A deeper analysis of relations involves studying particular [or special] classes of relations." Thatwhichislearnt (talk) 18:28, 14 March 2024 (UTC)
replies: text: The sentence does not define what are these concepts. In any case this meaning of a concept seems very specific to a narrow area of computer science (data base theory, I think, but I may be wrong). These concepts are not used in the mainstream of mathematics, and I recommend to remove the sentence. D.Lazard (talk) 18:59, 14 March 2024 (UTC)
replies: text: They are referring to the  types of relations, clearly. Thatwhichislearnt (talk) 19:03, 14 March 2024 (UTC)
thread_title:  "Concepts" of relations are not subsets 

message: In the sentence "Uniqueness and totality properties (only definable if the domain X and codomain Y are specified)", it is understandable that is trying to be succinct. But the "and" inside the parentheses does not apply to both cases of the subject. It does not apply to the property of being total. Thatwhichislearnt (talk) 18:53, 14 March 2024 (UTC)
thread_title:  Total does not require codomain 

message: Regarding the sentence "A binary relation R over sets X and Y is a subset of X\times Y". It begins defining "binary relation over X and Y". Good. Most of the article consistently uses the fully qualified name, by the surname "over X and Y". The problem is that then the sentence says that what is being defined is an object from a category that forgets stuff, a set, a subset of X\times Y. That subset no longer remembers X and Y. A warning sign should have been that the very next sentence has to issue a correction (without acknowledging that it is a correction). Another warning sign should be those references 2 and 8 being used. Those are very derivative (and also dated) texts. Compare also to the wording in Finitary_relation#Definitions. There, they did a slightly better job. They have the sentence saying that the relation is the subset, but it is presented as "not yet the definition". Note the use of "is given", instead of "is". Then the correction following acknowledges that it is a correction with the "may be more formally". Thatwhichislearnt (talk) 20:52, 14 March 2024 (UTC)
thread_title:  The definition forgets what is trying to define.