Talk:Closure (mathematics)

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)Reply

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)Reply

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)Reply

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)Reply

Natural numbers are not closed in subraction because one (natural number) minus (natural number) is zero (not a natural number). —The preceding unsigned comment was added by Ieopo (talk • contribs) .

You're right about that. I'm glad you agree with the article, which says in the first sentence that "For example, [...] the natural numbers are not [closed under subtraction]". Thank you for your help. -lethe ^{talk +} 19:11, 11 April 2006 (UTC)Reply

He’s right that they are not closed, but not that 0 is not a natural. PJTraill (talk) 22:24, 24 April 2014 (UTC)Reply

-Side comment from random person: while 3 - 8 is indeed not a natural number, it may be a bit confusing because natural numbers are often confused with integers. I would recommend changing this to 3 / 8, as this would make it clear that it's not a natural number nor integer. — Preceding unsigned comment added by 99.231.244.178 (talk) 20:53, 24 April 2014 (UTC)Reply

I have left 3 - 8, but added text to clarify for beginners why -5 is not natural. PJTraill (talk)

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)Reply

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)Reply

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)Reply

If a set were to be {0, 1}, would that be closed under addition? 0+1 = 0, ok, 0+0=0, ok, but 1+1=2. Does this mean that it is not closed under addition? All the examples I can find use natural numbers or integers as sets. These don't matter for addition or division. —Preceding unsigned comment added by Nswartz (talk • contribs) 15:05, 16 August 2008 (UTC)Reply

As you point out, 1+1 = 2 ∉ {0,1}, so, no, our set is not closed under the usual addition operation for integers. In fact, no finite, nonempty set of integers other than {0} is closed under integer addition: let S be such a set; by assumption, S contains some nonzero member e. Then e + e + . . . + e = ne ∉ S for some integer n, since e ≠ e + e and S is finite.

On the other hand, define an operation + on our set {0,1} by

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0

("addition mod 2"). In this case, the answer is "yes". For closure properties of sets of integers under addition more generally, the entries on modular arithmetic and cyclic groups seem like reasonable places to start.

75.184.118.88 (talk) 16:33, 16 August 2010 (UTC)Reply

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)Reply

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)Reply

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)Reply

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)Reply

• 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.

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Some important particular closures can be obtained as follows:

• ${\displaystyle cl_{ref}(R)=R\cup \{\langle x,x\rangle \mid x\in S\}}$  is the reflexive closure of ${\displaystyle R}$ ,
• ${\displaystyle cl_{sym}(R)=R\cup \{\langle y,x\rangle \mid x{\mathrel {R}}y\}}$  is its symmetry closure,
• ${\displaystyle cl_{trn}(R)=R\cup \{\langle x_{1},x_{n}\rangle \mid n>1\land x_{1}{\mathrel {R}}x_{2}{\mathrel {R}}\ldots {\mathrel {R}}x_{n}\}}$  is its transitive closure,
• ${\displaystyle cl_{emb,\Sigma }(R)=R\cup \{\langle f(\ldots ,x,\ldots ),f(\ldots ,y,\ldots )\rangle \mid x{\mathrel {R}}y\land f\in \Sigma \}}$  is its embedding closure with respect to a given set ${\displaystyle \Sigma }$  of operations on ${\displaystyle S}$ .

We say that ${\displaystyle R}$  has closure under some ${\displaystyle cl_{xxx}}$ , if ${\displaystyle R=cl_{xxx}(R)}$ ; for example ${\displaystyle R}$  is called symmetric if ${\displaystyle R=cl_{sym}(R)}$ .

Any of these four closures preserves symmetry, i.e., if ${\displaystyle R}$  is symmetric, so is any ${\displaystyle cl_{xxx}(R)}$ . Similarly, all four preserve reflexivity. Moreover, ${\displaystyle cl_{trn}}$  preserves closure under ${\displaystyle cl_{emb,\Sigma }}$  for arbitrary ${\displaystyle \Sigma }$ . As a consequence, the equivalence closure of an arbitrary binary relation ${\displaystyle R}$  can be obtained as ${\displaystyle cl_{trn}(cl_{sym}(cl_{ref}(R)))}$ , and the congruence closure with respect to some ${\displaystyle \Sigma }$  can be obtained as ${\displaystyle cl_{trn}(cl_{emb}(cl_{sym}(cl_{ref}(R))))}$ . In the latter case, the nesting order does matter; e.g. for ${\displaystyle S}$  being the set of terms over ${\displaystyle \Sigma =\{a,b,c,f\}}$  and ${\displaystyle R=\{\langle a,b\rangle ,\langle f(b),c\rangle \}}$ , we have ${\displaystyle \langle f(a),c\rangle }$  in the congruence closure of ${\displaystyle cl_{trn}(cl_{emb,\Sigma }(cl_{sym}(cl_{ref}(R))))}$ , but not in ${\displaystyle cl_{emb,\Sigma }(cl_{trn}(cl_{sym}(cl_{ref}(R))))}$ .

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I know, this draft certainly needs to be improved. The notion of congruence closure should be explained better, and switching between ${\displaystyle xRy}$  and ${\displaystyle \langle x,y\rangle \in R}$  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)Reply

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)Reply

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)Reply

Sure - Have at it. My interest is incidental anyway and I was just thinking to clarify the section. But I can see how that was a bit limiting. Better someone who has a broader view do it, especially toward making the whole article consistent. Daren Cline (talk) 15:28, 27 April 2014 (UTC)Reply

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)Reply

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)Reply

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 P be a property of subsets of a set S such that, if all members of a family of subsets of S satisfy P, then the intersection of the family satisfies also P. Under this hypothesis, the closure of a subset X of S is the intersection of all subsets of S that contain X and satisfy P. In other words, the closure of X is the smallest subset of S that contains X and satisfies P. 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)Reply

• Just some comments: For a WP article about a main result w.r.t. the first concept, try Birkhoff's HSP theorem. Also, the first and the third concept are related, see 1st sentence in [[Closure_(mathematics)#Closure_operator]; I'm not familiar at all with the second concept. And I don't yet have an opinion about your split suggestion. - Jochen Burghardt (talk) 19:37, 24 March 2022 (UTC)Reply

I was confused by the previous version of the article. I have cleaned the article up, and it appeards from the resulting version that there is no need to split the article. On the opposite, one could discuss a merge here of closure operator. D.Lazard (talk) 16:30, 17 April 2022 (UTC)Reply

"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)Reply

Indeed. I changed the text accordingly. Thanks for noticing. - Jochen Burghardt (talk) 07:33, 5 March 2023 (UTC)Reply