Talk:Universal algebra/Archive 1

Archive 1

Range and organization of subject

It seems to me that this article deals with algebras in the sense of universal algebras (algebraic structures) rather than universal algebra as a branch of mathematics. I think that the current content should be merged with the content of the article algebraic structure. The article about the branch of mathematics should not define and describe algebraic structures but present the history of the branch and its important results, and its title perhaps should be "Theory of universal algebras" to avoid confusion. Andres 08:33, 12 Apr 2004 (UTC)

Disagree. History can be added to the page, but the suggested merge isn't an improvement, in my opinion.

Charles Matthews 11:25, 12 Apr 2004 (UTC)

Let me explain this again. There are different concepts, such as group and group theory or topological space and topology. Analogously, universal algebra aka algebra aka algebraic structure is different concept from universal algebra as a branch of mathematics, and therefore I think they deserve different articles. Currently, in the present article most talk is about universal algebras. I think this part of the article should be merged with the content of the article Algebraic structure. True, there still is a Bourbakian concept of algebraic structure, but nothing effectively is said about it in that article. There is a terminological mess in this field but I think the first step could be such a reorganization of material, the second step would be finding the adequate titles. Then a formal definition of a (universal) algebra could be given involving signatures. And further, more information about the topic could be given. But I would not go for it before the organization of material is clear. Please explain why you think this wouldn't be an improvement. Andres 14:31, 12 Apr 2004 (UTC)

I do know the distinction you are making. But if 'universal algebra' is a little ambiguous, we should still discuss this all on one page. The situation is similar with tensor algebra. This can mean two things. In the end the page might need to be split up; but there is no hurry about that.

Charles Matthews 15:51, 12 Apr 2004 (UTC)

This article is too brief to indicate the breadth of the results of universal algebra Yes it is... but that is no excuse not to add more to indicate this breadth - Wikipedia is not a paper encyclopedia. Tompw 00:03, 23 December 2005 (UTC)

I removed the section on modules, containing only a confusing link to a redirected page. Spakoj 09:49, 12 January 2006 (UTC)

Details

Currently, the entry I've created for Yde Venema is likely to be deleted on the grounds that it's an insignificant biographical entries. Thoughts?

jtvisona 03:13, 7 Aug 2003 (UTC)

Yes, a thought! Who is "Yde Venema" and what makes him or her worth quoting? This quotation, if kept, does not belong in the rather nice introduction. Zaslav 18:37, 25 June 2006 (UTC)

Article removed from Wikipedia:Good articles

This article was formerly listed as a good article, but was removed from the listing because the article lists none of its references or sources --Allen3 ^talk 20:36, 18 February 2006 (UTC)

I failed the current GA nomination because the intoduction needs to be fleshed out, the connections to other mathmatical topics expanded, the history of the concept mentioned, and more sources cited. Ideally the explanations and examples could also be made more clear for the non-specialist, but that is very difficult in these specialized topics. Eluchil404 01:06, 26 April 2006 (UTC)

This seems to me one of the most outstandingly (and exceptionally) accessible math articles. It is written for someone who isn't already knowledgeable about its subject (though not for a person without any mathematical knowledge; that is too much to expect in such a specialization). Congratulations to the writer. Zaslav 22:50, 12 February 2007 (UTC)

An algebra structure, once defined, either has axiom(s) or it doesn't.

"After the operations have been specified, the nature of the algebra can be further limited by axioms, ..."

Consider a set and a specific function which maps x + y to z, where x, y and z are members of the set. It may be the case that something like, x + y = y + x, holds true for this structure, or it may not. You can't impose an axiom on a specific structure for which the axiom doesn't hold.

Perhaps something like the following?

Given an algebraic structure it is important to identify interesting axioms which hold for the given structure.

By interesting I mean something general like x+y=y+x and not 1+2=2+1 for two specific function ?executions/applications? which happen to be equal. Dan Wood 24.6.80.219 07:33, 1 April 2007 (UTC)

I think you are looking at a different point of view. In universal algebra, you are interested in studying all structures of a certain kind. If you specify a binary operation and nothing else, then you are saying you are interested in studying all sets with a binary operation on them. If you further specify the axiom "x+y=y+x", then you are saying that you are interested in studying all sets with a binary operation that satisfies that identity. If you were staring at one that does not satisfy it, you are not "imposing" the axiom on that set and operation, rather, you are removing it from consideration. You are not given the algebra, you are given the collection of things the algebra should satisfy. But your comment suggests that you are given the algebra and then you try to identify interesting things about it. While this is something some people do, it is not really the concern of universal algebra. Magidin 18:15, 1 April 2007 (UTC)

Whitehead's treatise

In 1898 Alfred North Whitehead published A Treatise on Universal Algebra. At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander MacFarlane (mathematician) said "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures." At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities. Given the modern references, this article sets out a theme used by some to describe their mathematical niche. Evidently the term has evolved in usage; it would be a challenging but useful service to sketch its evolution since Whitehead.Rgdboer 23:22, 16 July 2007 (UTC)

Ok, I pulled in some material from a book review that brings us up to the 1960s. I used what you wrote, and added to it from a book review (as you can see). Best, Smmurphy^(Talk) 23:53, 16 July 2007 (UTC)

By the way, is "calculus of extensions" the same as exterior algebra? More precisely, would it be ok to link to exterior algebra for the Grassman contribution? Best, Smmurphy^(Talk) 23:58, 16 July 2007 (UTC)

I have expanded the history section somewhat, based on the introduction to Grätzer's Universal Algebra. Grätzer indicates in a footnote that Whitehead credits Sylvester with coining the term, so I've modified the attribution. I also added some more information on the development through the early 60s. Magidin 19:21, 17 July 2007 (UTC)

You're far better informed (read:smarter) than I, thanks a lot. Smmurphy^(Talk) 19:30, 17 July 2007 (UTC)

Vector spaces as universal algebras

I removed the following new (and at least partially false) claim from the article:

"Note, though, that universal algebra is not truly universal, in the sense of being able to represent all kinds of algebras. For example, a vector space cannot be represented as an universal-algebraic structure."

For every scalar k, multiplication with k can be seen as a unary function on the vector space. In model theory, these unary function symbols together with 0 and + are the standard signature for vector spaces. I would expect that the same is true in universal algebra.

E.g. Freese ("Finitely based modular congruence varieties are distributive", 1994) talks about "varieties of vector spaces". --Hans Adler (talk) 02:21, 19 January 2008 (UTC)

You can represent modules over a (fixed) ring that way; Burris &Sanka [section 2.1] give that example. You cannot get the field axioms. 217.42.16.135 (talk) 05:40, 19 January 2008 (UTC)

Thanks for the pointer to Burris + Sankappanavar. Because the ring is fixed there is nothing that prevents us from choosing a field; which, after all is just a ring that happens to be a field. Because the ring is fixed there is no need to express the field axioms (which, of course, we cannot do using only equations). I see that Arthur Rubin (a universal algebraist) has already reverted your change. Sorry for misleading you by mentioning model theory. --Hans Adler (talk) 09:57, 19 January 2008 (UTC)

I have re-reverted, for the reasons given. I believe that it is important to mention about fields, etc.: otherwise, some readers are bound to be misled. What constructive suggestions for alternative formulations do you have? 217.42.16.135 (talk) 19:11, 20 January 2008 (UTC)

I was working on Arthur Rubin's version while you re-reverted. In general improving is better than reverting, although often one just doesn't have the time to do it instantly. As the same thing was now being said twice, I have removed your sentence again. I hope that is OK. You are of course welcome to edit it further until we have a version that the three of us can agree on. --Hans Adler (talk) 19:24, 20 January 2008 (UTC)

It is also not entirely correct that "a field" cannot be represented in universal algebra. It's no problem to represent an individual field in the language of rings. The problem is that the class of fields is not a variety and doesn't even come close. --Hans Adler (talk) 19:35, 20 January 2008 (UTC)

I now understand the problem with the paragraph where you made your addition. "... can be proved once and for all for every kind of algebraic system" is inappropriate grandiloquence. I am not sure how exactly to tone it down, though. --Hans Adler (talk) 19:38, 20 January 2008 (UTC)

Thanks, yes, that statement seems inappropriate, and the next part ("for every kind of algebraic system") is invalid. I do like what you just added though; one suggestion would be to include the word other: "i.e. functions but no other relations except for equality"—especially given the definitions on the page for relations. 217.42.16.135 (talk) 20:57, 20 January 2008 (UTC)

The problem with your formulation is that in this context every universal algebraist or model theorist will read "function" or "relation" in the sense of signature (mathematical logic). For us syntax is more important than it is in the rest of mathematics, because we are dealing with syntactical constructions like equalities as mathematical objects. There is an extension of universal algebra that can handle relations. An "equality" or "identity" involving a ternary relation is then something like Rxyz or Rxxy. Or if we also have a unary function symbol then Rxyf(x) is also an equation. But the restrictions are too serious for this to be very useful in mathematics. (It is extremely useful for modeling databases, however.) There are also various extensions that can handle partial functions. Using that, one can (essentially) model relations by functions, which is actually more natural in a UA context than doing it the other way round as usual. Because of all this, any language that implies that a function is a relation is unacceptable to universal algebraists. --Hans Adler (talk) 21:10, 20 January 2008 (UTC)

That makes sense, but it still seems to leave a problem with the article here. Your user page says that you "found that some Wikipedia articles around model theory … used inconsistent terminology", and yet that is the problem that your change introduces here: the definition of "relation" used in the article is inconsistent with the definition used in the article on relation. 217.42.16.135 (talk) 07:45, 25 January 2008 (UTC)

Perhaps I should update my user page, because the inconsistencies were much worse than what we are talking about here and they are mostly fixed now. I have changed the statement; please revert or edit if you don't prefer the new version. The underlying problem here is that in universal algebra and in model theory we have to distinguish syntax and semantics, individual structures ("a group") and classes of structures ("all groups" as opposed to "all magmas). There are two reasons not to make the distinction: The writer and all expected readers have thoroughly understood it and automatically read "function" as "function" or "function symbol" depending on context. Or when writing for an audience that is not used to the distinction and might not get the unrelated main points if the distinction is stressed too much. I think in this respect universal algebra strikes a relatively good balance, while algebraic structure blurs the distinction to the point where I consider the article fundamentally flawed. --Hans Adler (talk) 11:49, 25 January 2008 (UTC)

Perhaps there should be a paragraph discussing all this in the article? Also, I changed "signature" to "type" because type is the term used earlier in the section (and also preferred by Burris & Sanka)—okay?   217.42.16.135 (talk) 13:46, 25 January 2008 (UTC)

I agree absolutely with your change. Perhaps I needn't update my user page yet: I wasn't aware that this article still uses a "type" defined as a list of arities. I am worried that an early discussion of the details of signatures might detract from the main points; but more general universal algebra with relations or partial functions should be mentioned somewhere towards the end, perhaps under "Further issues". --Hans Adler (talk) 14:16, 25 January 2008 (UTC)

I think the statement about fields not forming a variety is really much stronger. Probably something like there is not even a signature that may have relations, such that the homomorphisms are the correct ones for fields and the fields form a quasivariety. But I would have to look this up in my office. Btw, we seem to have some of the confusion between individual algebras and varieties (that algebraic structure is badly affected by) even in this article. --Hans Adler (talk) 19:42, 20 January 2008 (UTC)

Homomorphisms are not a problem here. Any ring homomorphism between two fields that maps 1 to 1 will automatically be 1-1 (if ab=1, then f(a)f(b)=1, so f(a) cannot be 0) and will respect division.

Aleph4 (talk) 22:04, 20 January 2008 (UTC)

Are you sure? The point here is that we don't yet know what the signature looks like. As in, a priori it's not clear whether the signature for groups should contain inversion or not. If we allow relations, then we can model + as a ternary relation R. But then a priori nothing prevents us from replacing R by its complement, leaving us with an unnatural and useless notion of "homomorphism". --Hans Adler (talk) 22:20, 20 January 2008 (UTC)

"not allowed"

Please do not claim that the definition of a universal algebra only allows universally quantified equations. This is simply not true.

1. A universal algebra is a set equipped with operations. These operations may or may not satisfy certain "laws" (universally quantified equations), and they also may or may not satisfy certain other properties.
2. It is true that in the field of Universal Algebra (I write it with capital letters to distinguish it from the objects, the universal algebras), varieties play an important role, and that varieties are classes of algebras defined by universally quantified equations.
3. It is also true that in universal algebra it is often more convenient to define groups as structures with signature (2,1,0) rather than as structures of type (2) (group multiplication), because that makes them into a variety.
4. But mathematicians working in the field of Universal Algebra are also interested in many classes of universal algebras that are not (and in fact cannot be) defined by equations. Off the top of my head: fields; complete lattices; atomic or atomless Boolean algebras; locally finite structures; subdirectly irreducible elements of your favorite variety; etc.

Aleph4 19:24, 25 June 2006 (UTC)

I’ve changed it to specify that some but not all authors restrict universal algebra to varieties: Universal_algebra#Varieties – any references would be appreciated!

Nbarth (email) (talk) 23:43, 9 July 2008 (UTC)