Talk:Closure (mathematics)

message: Hi, I speak spanish, so forgive my english. This article is not linked with spanish version. the spanish version is "Ley de composición interna". I don't know how to link the pages. Thanks
:This request is over two years old! Sigh.....Added.----occono (talk) 02:18, 22 August 2009 (UTC)
replies: text: And now it is not there any more: I see it was a link to es:Operación_matemática#Ley de composición interna — does that not work with the new mechanism? PJTraill (talk) 22:24, 24 April 2014 (UTC)
thread_title:  Link to Spanish wiki 

message: I think that the notion of "closure without qualifier", i.e. Closure (topology), referring to closed sets, should be made a little more visible. (Maybe some kind of "disambig list" at the end of the introduction.)
Also, I find that this article is written in a way which is somehow unnecessarily complicated... — MFH:Talk 22:12, 21 March 2006 (UTC)
replies: text: Are you referring to the rewrite I did a few days ago?  What about it do you not like?  My main goal was to make it so that this page describes both the property called closure (a set satisfies this property if it is closed), as well as the closure operator (which maps each set to a closed set).  If you have some specific complaints, I could try to address them. -lethe talk [ +] 13:55, 24 March 2006 (UTC)
thread_title:  I think... 

message: The current version of the article states that.
:An operation of a different sort is that of taking the limit of a sequence.  A set that is closed under this operation is usually just referred to as a closed set in the context of topology.
This is not true. In a topological space a set is closed if and only if it is closed under taking the limits of nets or filters. Limits of sequences aren't sufficient in general.
I would also say, that the article should refer to closure operator in the part about abstract closure operators. --Kompik 11:31, 12 April 2006 (UTC)
replies: text: I agree with your points and have attempted to address them in the article.  How do you like it now? -lethe talk [ +] 14:46, 12 April 2006 (UTC)
replies: text: The improvements appear to have been deleted, now. The uncorrect statement about closure in topology is still here.--78.15.196.42 (talk) 15:24, 25 February 2012 (UTC)
thread_title:  Closed sets 

message: Closure property is a property which a set either has or lacks with respect to a given operation.A set is closednwith respect to the operation if the operation can always be completed with elements in the set .  —Preceding unsigned comment added by 69.111.182.182 (talk) 01:54, 4 February 2010 (UTC)
thread_title:  Closure Properties 

message: A discussion of how this relates (or doesn't) to closures in functional programming languages might
be useful.  I don't consider myself qualified to discuss this yet.  Although the concept of mathematical
closure has been brought up in some lectures on Lisp I've seen, the closures in the language seem
like a more concrete, perhaps separate concept from mathematical closure... but I could be wrong.
In other words, is "closure" in functional programming overloaded to mean something else, or not?  —Preceding unsigned comment added by 98.207.0.180 (talk) 17:55, 26 February 2010 (UTC)
replies: text: Somewhat late answer: “closure” in programming means something else (“having no open bindings” rather than “containing all results of an operation on its members”, which can only apply to a set), but I don’t think it is worth mentioning — Closure (disambiguation) is referenced, and that seems enough. PJTraill (talk) 22:56, 6 March 2014 (UTC)
thread_title:  Closures in Functional Programming Languages 

message: performance of that operation on members of the set always produces a unique member of the same set
I'm not a mathematician, but surely this is not right. Surely the operation could only produce a unique  member if it always produced the same member, regardless of the arguments. The only other meaning I can ascribe is that this is in distinction to operations on sets which produce different results when the operation is applied on different occasions - as when there is an internal state which affects the result of a function.
Looks to me like the word unique is not merely redundant, but wrong.
If you are enough of a mathematician to know, please either clarify what unique might mean in this context & remove this comment, or eliminate the offending word.  — Preceding unsigned comment added by 82.0.88.8 (talk) 20:20, 23 March 2012 (UTC)
thread_title:  Why 'Unique' 

message: * The 1st sentence says "The notion of a closure can be generalized for an arbitrary binary relation". But no proper generalization is neccessary to apply the notions from the introductory sectiion to binary relations. Therefor, I replace "generalized" by "applied".
* The link "reflexive transitive closure" (meanwhile?) refers to the very same article it starts from; it should be adjusted or deleted. Similar for "reflexive transitive symmetric closure" and "equivalence closure".
* I suggest to add the usual lemmas about closures of binary relations in this section. A first draft version is the following.
---
Some important particular closures can be obtained as follows:
* cl_{ref}(R) = R \cup \{ \langle x,x \rangle \mid x \in S \} is the reflexive closure of R,
* cl_{sym}(R) = R \cup \{ \langle y,x \rangle \mid x \mathrel{R} y \} is its symmetry closure,
* cl_{trn}(R) = R \cup \{ \langle x_1,x_n \rangle \mid n >1 \land x_1 \mathrel{R} x_2 \mathrel{R} \ldots \mathrel{R} x_n \} is its transitive closure,
* cl_{emb,\Sigma}(R) = R \cup \{ \langle f(\ldots,x,\ldots),f(\ldots,y,\ldots) \rangle \mid x \mathrel{R} y \land f \in \Sigma \} is its embedding closure with respect to a given set \Sigma of operations on S.
We say that R has closure under some cl_{xxx}, if R = cl_{xxx}(R); for example R is called symmetric if R = cl_{sym}(R).
Any of these four closures preserves symmetry, i.e., if R is symmetric, so is any cl_{xxx}(R).
Similarly, all four preserve reflexivity.
Moreover, cl_{trn} preserves closure under cl_{emb,\Sigma} for arbitrary \Sigma.
As a consequence, the equivalence closure of an arbitrary binary relation R can be obtained as cl_{trn}(cl_{sym}(cl_{ref}(R))), and the congruence closure with respect to some \Sigma can be obtained as cl_{trn}(cl_{emb}(cl_{sym}(cl_{ref}(R)))). In the latter case, the nesting order does matter; e.g. for S being the set of terms over \Sigma = \{ a, b, c, f \} and R = \{ \langle a,b \rangle, \langle f(b),c \rangle \}, we have \langle f(a),c \rangle in the congruence closure of cl_{trn}(cl_{emb,\Sigma}(cl_{sym}(cl_{ref}(R)))), but not in cl_{emb,\Sigma}(cl_{trn}(cl_{sym}(cl_{ref}(R)))).
---
I know, this draft certainly needs to be improved. The notion of congruence closure should be explained better, and switching between xRy and \langle x,y \rangle \in R should be avoided, and some source should be cited, and some application example from term rewriting (where congruence closure is an important notion) should be given.
Jochen Burghardt (talk) 07:25, 23 May 2013 (UTC)
thread_title:  About section "P closures of binary relations " 

message: I think part of the problem we are having here is there is no apparent universal set that contains all the subsets in question, including the closures.  The definitions of closure and closure operator surely require one? Given the first sentence of this section, I was at first under the impression that X was such a set.  But now the last sentence suggests it isn't -- so what is?  Moreover, the closure of a subset of X is now ambiguous: is the closure also a subset of X? And if not, then what is the point of talking about subsets of X?  I'm not saying I don't think there could be a reason; just that the section is too ambiguous without addressing these points. Daren Cline (talk) 15:27, 26 April 2014 (UTC)
replies: text: My edits have, so far, been based solely on the definition of the first sentence of the article. However, you have raised a point that does need to be addressed, so I've now looked at the article as a whole, including its history, in order to respond. "Closure" is a concept that has been used in several mathematical areas with slightly different meanings, but the same general "flavor". This article started out to be an umbrella article to cover all these specifics with fairly general language to emphasize the commonality. This meant that it needed to be described set-theoretically with respect to some generalized property P which would vary from one application to another. Then, in 2006, in an attempt to widen the applicability of the concept, the language about closure with respect to an operator on a set was introduced. Essentially this operator replaced the property P terminology, and in order to fit the new framework, some classical terms had to be mangled. I am not convinced that this was a good thing to do. To get back to your specific point. In some cases there is a universal set that arises naturally. For instance, in topology, the whole topological space is a closed set and so, for any set X in that space, the closure of X will be defined (in general, it is a superset of X and can't be any smaller than X itself). However, in other situations, there may not be a clearly identified universal set. For instance, in dealing with the algebraic closure of a field, this, generally larger, unique field is constructed from the original field. In the original formulation (pre-operator), a closure operator was defined on the power set of a set (satisfying certain conditions, one of which was that the set itself was closed) but this had to be modified to fit the operator framework. The result is that portions of both approaches appear in the article, and sometimes these are in conflict, leading to the lack of clarity that you have perceived. I don't see a quick fix to this problem – certainly it will take more than just adding new material. I would be in favor of getting rid of the operator framework and going back to the original formulation (and since that was unsourced material anyway, I see no problem with doing so.) Bill Cherowitzo  (talk) 18:13, 26 April 2014 (UTC)
thread_title:  Closure Operator 

message: Modulo systems can contain no number equal to or greater than their modulus, so all modulo systems are closed by definition. It'd be nice to see some mention made of this (even if only in passing)... The Grand Rascal (talk) 06:15, 28 September 2020 (UTC)
replies: text: No.  (Perhaps you should read, in some order, WP:RS and the lead section of the article.) --JBL (talk) 11:29, 28 September 2020 (UTC)
thread_title:  What About Modulo Systems...? 

message: This artice is very confusing because it tries to present two different concepts as if they were the same concept.
First concept: subset closed under an internal operation. Apparently the main related result is not even mentioned anywhere in WP: If an algebraic structure  belongs to a variety (that is, all axioms are purely universal), then every subset that is closed under all operations belongs to the same variety. For example, a subset of a group is a subgroup if it is closed under the group operation (arity 2), the inverse operation (arity 1) and identity (arity 0).
Second concept: closure operation: Let  be a property of subsets of a set  such that, if all members of a family of subsets of  satisfy , then the intersection of the family satisfies also . Under this hypothesis, the closure of a subset  of  is the intersection of all subsets of  that contain  and satisfy . In other words, the closure of  is the smallest subset of  that contains  and satisfies . Common examples of such closure operations include topological closure and structures or spaces defined by a generating set.
A Closure operator is a third concept that is presented in the article, and can be viewed as a generalization a closure operation. The problem is that, very often, the closure operator is much more difficult to specify than the closure operation. For example, the affine subspace generated by a set of points is trivially defined as a closure operation, while the corresponding closure operator requires the concept of affine combination. Also, the Zariski closure of a set of points is the smallest algebraic set that contains these points, while the corresponding closure operator can hardly be defined. 
Moreover, even if the results that I have sketched above are implied by the content of existing articles, these articles are much too technical for a non-expert who needs them.
My suggestion is to have several articles:
* Redirect Closure (mathematics) to Closure (disambiguation)#Mathematics, a disambiguation page
* Create Closed set under an operation that could be a redirect to a new subsection of Operation (mathematics)
* Create Closure operation for expanding the above concept
* Keep Closure operator, and add to it closure operations as motivating examples. 
Many variations are possibles, but we lack clearly of elementary treatment of these subjects. Also, a large part, if not all, of Generator (mathematics) could be merged into Closure operation. D.Lazard (talk) 19:19, 24 March 2022 (UTC)
thread_title:  Two different topics for a single article 

message: "The closure of a subset under some operations is the smallest subset that is closed under these operations" -- should this be "smallest superset of that subset"? Ilya239 (talk) 19:22, 4 March 2023 (UTC)
replies: text: Indeed. I changed the text accordingly. Thanks for noticing. - Jochen Burghardt (talk) 07:33, 5 March 2023 (UTC)
thread_title:  Imprecise definition of closure?