Talk:Universal algebra

Universal algebra vs. Abstract algebra
(I have asked these same questions over at Talk:Abstract algebra). Do the meanings of Abstract algebra and Universal algebra truly differ from each other? Isn't Universal algebra in Alfred North Whitehead's A Treatise On Universal Algebra simply another way of saying Abstract algebra? Alternatively, are there still unresolved problems in the reconciling of Abstract algebra and Universal algebra as there still are in the reconciling of Category theory and Set Theory ? A quote from Pierre Cartier, "Bourbaki got away with talking about categories without really talking about them. If they were to redo the treatise [Bourbaki's not Whitehead's], they would have to start with category theory.  But there are still unresolved problems about reconciling category theory and set theory." --Firefly322 (talk) 09:55, 11 March 2008 (UTC)
 * 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)

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 $$\forall x\,(x\ne0\to\exists y\,x\cdot y=1)$$), 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 $$f\colon A\to B$$, 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). --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 where the most common types are listed.Rgdboer (talk) 02:38, 17 January 2014 (UTC)
 * The section "Basic idea" concludes with the following statement:

One way of talking about an algebra, then, is by referring to it as an algebra of a certain type \Omega, where \Omega is an ordered sequence of natural numbers representing the arity of the operations of the algebra.
 * Although the linked page Outline_of_algebraic_structures does list and qualify many common types of algebra, it makes no reference to any definition of type defined as \Omega, where \Omega is an ordered sequence of natural numbers representing the arity of the operations of the algebra. The article that describes the types does, however, classify algebra types by subheadings specifying count of sets together with count and arity of operations.  What remains unclear about this numeric type \Omega are these things:
 * what structure it might have, apart from being an ordered sequence of natural numbers; and
 * whether it is the same thing called the signature in earlier comments, and if so,
 * which terms are presently used, and by whom, for it.


 * Whilst we want Wikipedia articles to be as informative as possible to the interested layperson, I feel we should use and define those technical terms that have proven most relevant and useful to specialists (which I suspect the sentence I've quoted may be shying away from doing), particularly so whenever using a suitable technical term would make the exposition clearer. yoyo (talk) 14:15, 7 November 2015 (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 128.32.45.247 (talk) 00:39, 11 February 2011 (UTC)

Better: Definition 1.6.1 here 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)

The same goes for the term equational theory. --213.55.184.208 (talk) 06:17, 29 March 2018 (UTC)

Assessment comment
Substituted at 02:40, 5 May 2016 (UTC)

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Article unclear on the question: Are lattices universal algebras?
The subsection "Other Examples" lists some examples of universal algebras and then says "Examples of relational algebras include semilattices, lattices or Boolean algebras." This seems problematic because (1) the term "relational algebra" is not defined in this article and (2) It leave unclear whether lattices, for example, are universal algebras of a special kind called "relational algebras" or are not universal algebras but instead something else called "relational algebras" NathanReading (talk) 16:54, 9 November 2017 (UTC)

The same subsection states "Most algebraic structures are examples of universal algebras." Then proceeds listing examples of algebraic structures that are universal algebras. Given the lead sentence, certainly a far more interesting list would be examples of algebraic structures that are not universal algebras. --5.186.55.135 (talk) 11:39, 11 December 2021 (UTC)

Possibly false ellegance
...An n-ary operation on A is a function that takes n elements of A and returns a single element of A. then ¿All universal algebras must be closed on themselves? then integers are not universal under division. Also a root is not a function as it takes one or more outputs from a two inputs.--167.56.134.58 (talk) 21:54, 8 July 2018 (UTC)
 * That is correct; by definition an operation on a set is closed. The integers with division are not an algebra, because division is a partial operation, not an operation. In fact, fields are not a universal algebra, because the multiplicative inverse is not defined for all elements. The multivalued root function is not an operation, it is a multi-valued function. (There are contexts where one can talk about a unique root; e.g., a torsionfree $n$-divisible group has a well-defined $n$th root operation). P.S. What is an "ellegance"? Magidin (talk) 22:28, 8 July 2018 (UTC)

"Identities" are not defined
"A collection of algebraic structures defined by identities is called a variety or equational class." However, there is given no precise definition of identity.

This bug is common for Universal_algebra and Variety_(universal_algebra) articles. --VictorPorton (talk) 07:54, 16 January 2020 (UTC)


 * The immediately prior paragraph says "After the operations have been specified, the nature of the algebra is further defined by axioms, which in universal algebra often take the form of identities, or equational laws." and "identities" is hyperlinked to a page that contains a precise definition. Wikipedia frowns on repeated links, so close together, for the same term. The article for Varieties includes a direct link to the same definition in its very first sentence. Magidin (talk) 03:10, 18 January 2020 (UTC)