Talk:Closure operator

Useful - my thanks to Axel for pulling this together.

Charles Matthews 09:09, 23 Feb 2004 (UTC)

An errata?
Where say:

they satisfy $$\operatorname{cl}(X_1 \cup\dots\cup X_n) = \operatorname{cl}(X_1)\cup\dots\cup \operatorname{cl}(X_n) $$ for all integers n ≥ 0

should not say? either:

they satisfy $$\operatorname{cl}(X_1 \cup\dots\cup X_n) = \operatorname{cl}(X_1)\cup\dots\cup \operatorname{cl}(X_n) $$ for all integers n ≥ 1

or

they satisfy $$\operatorname{cl}(X_0 \cup\dots\cup X_n) = \operatorname{cl}(X_0)\cup\dots\cup \operatorname{cl}(X_n) $$ for all integers n ≥ 0 —Preceding unsigned comment added by 189.216.40.253 (talk) 20:16, 31 December 2008 (UTC)


 * Actually, that does make sense for n = 0. If you make the convention that the union of 0 sets is empty, then it reads:
 * $$\operatorname{cl}(\emptyset)=\emptyset.$$
 * — Arthur Rubin (talk) 06:47, 2 January 2009 (UTC)

Proposed merge from consequence operator
I'm proposing that consequence operator be merged into this article (at closure operator). The only significant difference I see as that the operator may be required to be finitary, in that
 * (3) $$X = \bigcup\{C(Y)\mid Y \in F(X)\}$$,

then $$C$$ is (finitary, an operation of finite character (although not precisely the definition we're presently using), etc.)


 * (Although I'm sure there's a better characterization of finitary.) &mdash; Arthur Rubin |  (talk) 20:01, 16 November 2006 (UTC)

The finite consequence operator is much more than a mere closure operator. It is the major notion used in the subject universal logic. It is a major investigation of the Polish logicians due to its algebraic properties. The basic definition is correctly stated as it was done by Tarski. There are two types but the finite one is characterized by the general logic-system, which, like the consequence operator, is a set-theoretic generalization of a formal logic system. It is used to investigate natural languages, as is the main goal of universal logic. Raherrmann 19:12, 2 December 2006 (UTC)

Mathematically, a finite (I'd call it finitary) consequence operator is exactly a closure operator of finite character. As the latter concept has applications in (finitary) universal algebra, and there's little that can be said about a finitary consequence operator in the abstract that would not apply to a finitary closure operator, or the operation of the closure of a set with respect to the operations of a partial algebra. And Tarski is in closure operator already. &mdash; Arthur Rubin | (talk) 19:48, 2 December 2006 (UTC)

I am aware of all of this, but the finite consequence operators do not form a complete lattice. They do form a join-complete lattice. Using universal logic notions, when this is applied to any collection of physical theories, then one obtains the sup. unification, one that does not alter any of the actual physical theories. This fact has just recently been completely related to "general" logic-systems. Obviously, this page is a further refinement of the rather small introduction that appears as but an example of a closure operator as used in logic. There is also a more general class of objects called deductiuve processes that have consequence operators as a major subclass. 4.249.39.6 17:52, 3 December 2006 (UTC)

That should all be here, or in a separate article on finitary closure operators. That they are "consequence operators" is irrelevant. (Besides, "consequence operators" doesn't seem to be a standard name....) &mdash; Arthur Rubin |  (talk) 17:56, 3 December 2006 (UTC)

I'll discuss below whether they are indeed irrelevant in their properties. In 1981 Dziobiak, changed the name to "finite consequence operations." In 1983, I changed it to "operators" for their use as set-maps. Apparently, this is now being used by those in universal logic. Even though it is true that they are finitary closure operators, it is shown in a paper just published that they when the notion of the general logic-system is carefully defined, that the finite consequence operators they generate are of two general types. When one can use general logic-systems to generate finitary closure operators, then they also would be of two general types. These types depend upon whether the rules of inference as set-theoretically defined are finite or not. The finite ones are sufficient for standard formal logic such as propositional and predicate. Of course, closure operators need not be restricted and related to say ZF set theory. Now to obtain a non-finite (general) consequence operator via the notion of rules of inference, it was shown many years ago that you need an infinite product space notion. The language one uses can have different effects upon the operators. The greatest effect is upon collections of operators. When languages are specially constructed, like restrictions of the propositional language, there are significant collections of consequence operators using these languages that are not meet-complete. These are very specifically defined wwf and the result are relative to the wwf's forms. I don't believe that this sort of thing happens for finite closure operators that are not so language specific as are some consequence operators. So, there probably is a difference when the closure operator is view as a language specific consequence operator.

In 1963, Robinson embedded a formal language into a nonstandard structure. (The hyphen has not been used for 20 years.) He developed a nonstandard semantics and applied it to model theory. In particular, he shows the need for a dual approach to what constitutes a truth-function, there being two acceptable but different such notions. I did the same thing for proof theory. But, this is done in a more general way via the consequence operator. Now the nonstandard consequence oeprators that extend the finite consequence operator are not usually finitary in the usual sense. Further, although they have many interesting properties, the extended operators are usually not a closure operator unless one restricts the sets upon which they are defined. The extensions can be extended by definition to a consequence operator that makes every "external" set inconsistent. These ideas and many more complement the Robinson results. I also point out that once again nonstandard consequence operators can differ in that certain significant properties are relative to specific language constructions. Of course, one can consider the general theory of finite closure operators within ZF set theory and then investigate the usual nonstandard stuff and today such a general investigation would be considered as rather. It is when you get much more specific properties that things become interesting. But, the basic question is whether Wikipedia is interested in listing such specific categories in a more refined manner? 4.249.12.31 22:15, 3 December 2006 (UTC)


 * Don't merge is my initial gut reaction. Although these are all clearly very similar concepts,their domain of application is different. If one is not careful, one will then get the urge to merge in Galois connection and also merge praclosure, which are also very similar concepts. Before long, we'll merge in all of topos theory into here. Its probably more useful to keep the articles separate, according to the classical divisions between the subjects, and then go to great lengths to demonstrate, in each article, how they are closely related to this other concept coming from this other branch of mathematics. This is in part because students will be reading textbooks that are still written along the classical subject distinctions; you want to open their minds to the possibilities; but you don't want to just confuse them. But that's my gut reaction. A well-written, well-done merge would change my mind. linas 14:59, 13 April 2007 (UTC)


 * I'm afraid I don't see the distinction between the concepts, while Galois connection seems quite different to me. The only differences I see between what should be in the respective articles is:
 * Field of reference:
 * consequence operator lives on subsets of a fixed set
 * closure operator lives on an arbitrary partially ordered set
 * Closure operator separates out the finitary operator, which only makes sense (to me) in the domain of subsets of a fixed set. However, that's still only an additional section to this article.
 * Consequence operator has sections on "non-standard" consequence operators; however, those sections do not seem to me to have adequate context to have any meaning. (For what it's worth, I've done some work in non-standard analysis.)
 * consequence operator has a section on the lattice/semi-lattice of all consequence operators. I don't know if there are corresponding results in the more general closure operator context.
 * consequence operator has a lot of references by Herrmann, who apparently created the concept of general logic-system which has since been merged elsewhere. (No comment as to whether this is a good idea, but it is a difference.)
 * For what it's worth, most of what used to be in general logic-system now lives in this article in #Closure operators in logic (in which the closure operator also lives on the set of all formulas in the language.)
 * Also, for what it's worth, the hyphen in "non-standard" (v. "nonstandard") is used a lot more than the hyphen in "logic-system" (v. "logic system").
 * My present take is that the "nonstandard" sections of consequence operator should be moved somewhere near non-standard analysis, while the sections relating to specific properties of "consequence operators" on the set of all subsets of a set and finitary closure operators should be moved to new sections of this article. However, I'd certainly consider a statement in this article that a closure operator on the poset of the the set of all subsets of a given set is also known as a "consequence operator", if, indeed, that is correct.

Pseudo-closed sets
I added the definition of pseudo-closed sets. Note that it is indeed a correct recursive definition, although it is self-referential. For brevity, I did not include that pseudo-closed sets are essential in the definition of the canonical basis in Formal concept analysis, which in turn is crucial for the knowledge acquisition method called attribute exploration. I also did not mention that pseudo-closed sets are of recent interest in research because the complexity of finding them is not fully known. --Bernhard Ganter (talk) 09:17, 14 December 2017 (UTC)