Talk:Universal algebra

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)

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)

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

"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)

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)

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)

Universal algebra vs. Abstract algebra

I would say that Abstract Algebra concerns itself with specific instances of Universal Algebra (studying groups, studying rings, studying modules, studying fields, etc), whereas Universal Algebra concerns itself with studies that cut across all such subjects (varieties, quasivarieties, congruences, etc). Perhaps an analogy: Abstract Algebra is the study of specific languages, whereas Universal Algebra is the study of the linguistics common to all those languages. They are certainly used differently; while a group theorist might say he is "doing" abstract algebra, he would probably never say he is "doing" universal algebra. Magidin (talk) 13:50, 11 March 2008 (UTC)
I agree, I came onto the Universal Algebra Wikipedia page hoping to find something, and I found something else (which very much resembles the description of Abstract Algebra in my books). I propose keeping the Universal Algebra moniker for the specific algebraic structure, and merging all the current content into the Abstract Algebra article. Aqualung (talk) 21:52, 8 November 2012 (UTC)
Sorry, but what do you mean "the specific algebraic structure"? The point is that the term "Universal algebra" refers to the study of algebraic structures, not the study of a specific algebraic structure. What "specific algebraic structure" are you refering to? "Abstract algebra" is a generic term for the study of specific algebraic structures (ring theory, module theory, group theory, lie theory, etc). Universal algebra is the study of all the things that all these specific instances have in common. To repeat the analogy I already made: abstract algebra is to universal algebra like a Department of Language Instruction is to a Department of Linguistics. Linguistics is not a part of teaching French, English, Latin, etc; and the study of English, French, Latin, etc. is not linguistics. Magidin (talk) 20:33, 9 November 2012 (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)

Varieties only?

Varieties are a very important subject in universal algebra. But I don't know any mathematicians (or specifically: mathematicians working in universal algebra) who claim that varieties are the only objects studied in universal algebra. (The very idea is a bit contradictory -- when you study a variety, you have to study some of its members occasionally.)

And I strongly disagree with the conclusion the article draws, for example

"Universal algebra cannot study the class of fields, because there is no type in which all field laws can be written as equations."

By the same argument one could claim that "universal algebra cannot study the class of all free algebras (of a given type/signature)", because they do not form a variety. But even those who concentrate all their work in understanding varieties are specially interested in free algebras.

--Aleph4 (talk) 12:42, 19 March 2009 (UTC)

The point with fields is not merely that they don't "form a variety" (see the extensive prior discussion about fields above your comments). The issue with fields is that multiplicative inverses cannot be given by an operation on the algebra (since they are only defined for some elements), but they are an integral part of the definition of fields. Hence there is no type in which the field laws can be written as equation (that's not the same as saying "fields don't form a variety"! You can describe solvable groups with a particular signature and a disjunction of equations they must satisfy, but the solvable groups don't form a variety either). I agree that varieties are not the sole concern: there's also quasivarieties and the like. Really, the key idea is that of a signature and equationally defined laws. Fields can be studied qua fields only if you extend the idea of signature and operations to include the notion of "partial operations" (which some do). You can always try your hand at improving the article. Magidin (talk) 13:29, 19 March 2009 (UTC)
It is true that there is a difference. But my point remains valid even if I change "not a variety" to "not a variety, even in an expanded signature". --Aleph4 (talk) 14:17, 25 March 2009 (UTC)
If by "my point" you refer to "varieties are not the sole concern", I continue to agree with you; that needs to be fixed. Or if you meant that the explanation for why the study of fields is not generally considered part of universal algebra, again I agree: that needs to be explained better. I think it would be difficult to be really clear on just what the problem with fields is, as again you can probably see with the discussions in the talk page, yet they are the "typical" example of an algebraic structure whose study is not considered to be part of Universal Algebra. Even if you allow partial operations, you end up allowing things that are not fields in (since types with partial operations do not prescribe the domains of the partial operations; see for example Gratzer's book. So one does not define a type with a partial operation that is required to be defined on "all elements except for the image of this particular unary operation"). However, I disagree with you about your claim that the (poorly written) sentence would indicate that we cannot study free algebras; "there is no type for which all the field laws can be written as equations" is not quite the same thing as "they do not form a variety", at least the way I read it. Magidin (talk) 15:21, 25 March 2009 (UTC)
Fields are commutative rings which satisfy certain additional axioms (namely, 0 ≠ 1 and ${\displaystyle \forall x\,(x\neq 0\to \exists y\,x\cdot y=1)}$), in the signature of rings. There is no reason whatsoever why multiplicative inverses should be included in the signature, or why their inability to do so because of being partial should be a problem. Also, you seem to claim that solvable groups can be described by disjunctions of equations. This cannot be true, because they do not form a variety, but they are closed under finite products (hence a disjunction of equations can be valid in all of them only if one of the disjuncts is). In fact, by the same argument solvable groups are not closed under ultraproducts, hence they cannot be axiomatized by any first-order theory. — Emil J. 13:58, 19 March 2009 (UTC)
I don't understand universal algebra at all, but solvable groups are really quite often defined by group theorists as those groups whose elements satisfy one of a countably infinite list of identities (namely that some element of their derived series is the identity, which is just a statement that some iterated commutator is the identity). Maybe you assumed a finite disjunction? JackSchmidt (talk) 14:06, 19 March 2009 (UTC)
In the context of universal algebra by disjunctions we usually mean finite disjunctions, because only finite formulas behave really well. --Hans Adler (talk) 14:41, 19 March 2009 (UTC)
Sorry I wasn't clearer; yes, I meant an infinite disjunction. The class of solvable groups forms a pseudo-variety (a class closed under substructures, quotients, and finite direct products) (an instance of "things other than varieties" that are often of interest to universal algebraist; as long as I'm mentioning that, there are also qhe quasi-varieties, classes closed under substructures and arbitrary direct products). Now, as to fields: yes, of course fields are "rings with extra axioms". And yes, we can study structures in which we have axioms instead of operations-satisfying-identities, or with partial operations or relations instead of simply operations (Gratzer includes a chapter on Partial Algebras in his Universal Algebra book). However, the study of partial algebras and of relational algebras is usually separate from the study of algebras; as someone put it (I believe it was Brian Davey), "many beautiful but false theorems can be proven for partial algebras if one is not very careful." For example, with partial algebras, if you have partial algebras A and B, and a homomorphism ${\displaystyle f\colon A\to B}$, then f(A) need not be a partial sub-algebra of B (one requires that the subset be closed under the restrictions of the partial operations to the suitable intersection of their domain with a power of the subset). So most of the basic theorems go out the window, though we can sometimes prove analogues. We can study these partial or relational algebras, but they do not usually get folded into Universal Algebra. Though included in Gratzer's book, Burris and Sankappanavar only introduce partial unary algebras en passant in order to show that the languages accepted by finite state acceptors are exactly the regular languages. I don't remember whether they are included in ALVIN or not (and I don't have a copy).
Naturally, we can study fields, whether as an axiomatically defined ring, or as a kind of partial or relational algebra, or any other number of ways; but when doing so the usual techniques and the standard theorems of Universal Algebra no longer apply, which means that we don't usually consider that study as being part of Universal Algebra (which of course is not the same as saying we don't study them). Just like there is no reason why we should exclude associative rings with 1 from the role of scalars in studying the vector space axioms; however, when we do that, we no longer call it Linear Algebra and instead call it Module Theory. Magidin (talk) 16:14, 19 March 2009 (UTC)
I agree this statement is problematic, although read in context it's clear that it's not meant as strongly as it sounds in isolation. But it's certainly worth fixing.
It was added by Nbarth in July. In the same series of edits, Nbarth also wrote something that is now in the first lede sentence: "studies algebraic structures themselves, not examples ("models") of algebraic structures". I think this is highly problematic. There is some confusion caused by people not referring to the class of all groups as if they were talking about an individual group, etc. This is particularly bad in the algebraic structures article. Now here we have this confusing language in the most prominent position, and where the distinction is crucial. I think it can hardly get worse.
I don't remember seeing the kind of language used at algebraic structure (As an abstraction, an "algebraic structure" is the collection of all possible models of a given set of axioms.) anywhere outside Wikipedia, although apparently 1 1/2 years ago I was under the impression that this was once common. I am thinking about bringing this up at WT:WikiProject Mathematics, since nobody responded on the article talk page (Talk:Algebraic structure#First paragraph is seriously misleading). --Hans Adler (talk) 13:46, 19 March 2009 (UTC)
How do universal algebraists feel about integral domains?
They seem to be very naturally defined by an *in*equation rather than an equation, so it might make the same point as fields while being more natural. Also fields might be a bad example, since they are so fundamental to mathematics, I would not be surprised what loops algebraists might go through in order to discuss them. Fields have also benefited greatly from model theory (which is lumped in with general algebra in my head), especially such silly things as viewing the complex numbers as the "universal" field since it has so many indeterminates.
On the off chance that u.a. is comfortable with integral domains rather than disinterested, then a field is just a von Neumann regular integral domain. I'm not sure about homomorphisms, but just singling out the VNR rings from all rings is definitely done by a very simple equation and unary operator. For fields, you just defined 0^-1 as 0, and require xx^-1x = x. This works if you happen to know the thing is an integral domain. JackSchmidt (talk) 14:37, 19 March 2009 (UTC)

Type

I suggest someone put in a definition of "type" and some examples. It has been overlooked. Zaslav (talk) 03:09, 6 June 2010 (UTC)

Thank you pointing out the lack of basis for the term. A (piped) link has been placed to Outline_of_algebraic_structures#Types_of_algebraic_structures where the most common types are listed.Rgdboer (talk) 02:38, 17 January 2014 (UTC)

Derived Operations

Hi Arthur Rubin, please explain what you think was incorrect about the definition of "derived operation". I will be happy to fix it. If you want a citation, add p281 of Bergman's book [1] 128.32.45.247 (talk) 00:39, 11 February 2011 (UTC)

Better: Definition 1.6.1 here [2] 128.32.45.247 (talk) 00:41, 11 February 2011 (UTC)

It's not in any of my texts on universal algebra. If you want to include it as an alternate definition of equations and equality, go ahead, but leave the existing text alone. Per WP:BRD, please do not continue to add the material without consensus. — Arthur Rubin (talk) 01:52, 11 February 2011 (UTC)
Part of my Ph.D. thesis was in universal algebra. I supposed it's possible the "standard" definitions have changed in the intervening 32 years, but it doesn't seem likely. — Arthur Rubin (talk) 16:26, 11 February 2011 (UTC)

Equational Reasoning

This term redirects to this page but it is not discussed in the article. 70.210.14.17 (talk) 15:19, 16 January 2014 (UTC)

In the section Basic idea, the term equational laws is in bold. However, the inclusion of the algebra of logic in UA is discussed in the following section: Varieties. If you have a source on "equational reasoning", please bring it here to support improvements.Rgdboer (talk) 02:27, 17 January 2014 (UTC)