Talk:Variety (universal algebra)

Someone please define variety
It would be great if this article contained a formal and self-contained definition of "variety of algebras". Right now you have to guess and also look at other pages.

I'll add one, which may not be perfectly polished. John Baez (talk) 04:31, 10 September 2018 (UTC)


 * Thanks! The definition could be simplified using the notion of "term" (see term (logic)). Otherwise, the link tree should be changed to tree (data structure). I tried to find a reference to Birkhoff, but it seems he didn't use the name "variety". - Jochen Burghardt (talk) 10:14, 10 September 2018 (UTC)

Someone please define subvariety
In this entry, the word "subvariety" appears without explanation. Would someone please define it? Better yet, would someone create a brief entry for the term, and blue link it to this entry? It is fascinating to read that groups are not necessarily subvarieties of semigroups, but that abelian groups are all subvarieties of groups. Are Boolean algebras subvarieties of lattices? Of commutative monoids?202.36.179.65 09:20, 25 September 2007 (UTC)

I rewrote the definition, did that make it any clearer? Lattices lack the complement of operation of Boolean algebras, and commutative monoids have only one binary operation whereas lattices have two, so all three of these classes have different signatures and can't be sublattices of one another by that definition. --Vaughan Pratt (talk) 08:04, 25 March 2011 (UTC)

arity?
What is arity? LilHelpa (talk) 19:44, 3 March 2009 (UTC)
 * Arity is the number of arguments an operation takes. Constants are 0-ary, negation and inversion are 1-ary (unary), addition, subtraction, multiplication, and division are 2-ary (binary), and the triple product is 3-ary. JackSchmidt (talk) 20:12, 3 March 2009 (UTC)

Covariety is unexplained
The lead mentions "covariety" with a link to "coalgebraic structures", but which redirects to Coalgebra. However as far as I can tell a coalgebra is something that is not in the realm of universal algebra, since it supposes vector spaces over a field. So please explain, or remove the phrase. Marc van Leeuwen (talk) 16:29, 2 February 2011 (UTC)
 * I think what happened here is just the usual confusion between universal algebras and algebras over a field. Apparently there are coalgebras in both senses, related to each other just like the two kinds of algebras. The link was probably supposed to go to universal coalgebras, but the article is about coalgebras over a field. I agree it should be explained, but the best place for that is probably a new article universal coalgebra. There is a redirect to this article from covariety, so that explains why it's in the lead and bold. Hans Adler 17:15, 2 February 2011 (UTC)
 * Turns out the requisite article already exists, under the awkward name F-coalgebra, so I redirected the link accordingly.
 * Since associative coalgebras are a pretty special case of coalgebras (I've written a fair about coalgebras but never had occasion to write about associative coalgebras), perhaps the following moves should be made:
 * Coalgebra --> Associative coalgebra
 * F-coalgebra --> Coalgebra (with a hatnote to associative coalgebra)
 * The associative coalgebra crowd could reasonably argue that the name belongs to them since they had it first. However it may be worth proposing at those pages to see what people think. --Vaughan Pratt (talk) 15:52, 25 March 2011 (UTC)


 * Agreeing on ambiguous terminology will always be difficult. I'm pretty sure the coalgebra people will object to moving to Associative coalgebra, and one good reason for that is that associativity is not the heart of the matter (although the current article assumes that). Their term "coalgebra" is based on dualizing either algebra (ring theory) or algebra over a field, so the essential point is that a coalebra is an additional structure defined on a module or vector space. But as far as I could see the examples at F-coalgebra never mention a ring or a field. I agree though that the name "F-coalgebra" is awkward (what to do if the functor is named G?). Although I'm no expert in either field, it seems that F-coalgraba is something dual to universal algebra, so why not something like "Universal coalgebra" or "coalgebraic variety"? As for the current coalgebra, one could propose a move to coalgebra over a field to distribute the pain fairly, but I'm not sure people will like that either. Of course the real error was to let the term algebra become as ambiguous as it is in the first place. Marc van Leeuwen (talk) 12:05, 26 March 2011 (UTC)
 * One could argue that "coalgebra" was more appropriately applied to locales than to "co-operations" of the form X &rarr; F(X) as dual to operations of the form F(X) &rarr; X, since these are special cases of the more general form F(X) &rarr; G(X), whereas there is no comparable common generalization of frames and locales.  An equational theory presented as the commutative diagrams in a category C can accommodate any mixture of all of these notions simultaneously, regardless of whether C is an ordinary category with discrete homsets or an additive category whose homsets are abelian groups (in which case the endomorphisms of any object form a ring).  From this perspective coalgebras and F-coalgebras are different facets of the same very general subject having in common the operation form X &rarr; F(X), with F(X) = X &otimes; X in some additive category C in the case of coalgebras.    --Vaughan Pratt (talk) 00:44, 27 March 2011 (UTC)

Unclear variable in the Category Theory section
There is a line
 * $$K:A\to \mathbf{Set}^{\mathbb{T}}$$

in the category theory section. Where does $$\mathbb{T}$$ come from? I understand that this is defined in terms of $$U$$ in a canonical way, but how? Is $$\mathbb{T}=UF$$ where $$F$$ is the free functor for our algebraic category? Also, there is no article on (finitary) algebraic category, and this is pulled out of nowhere in this article. Cheers Cmknapp (talk) 19:29, 27 November 2012 (UTC)
 * ✅ you were right, $$\mathbb{T}$$ is indeed the composition of the free group functor and the forgetful functor. I have clarified it in the article. AxelBoldt (talk) 08:55, 23 April 2018 (UTC)

Finite algebras
The sentence It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case seems patently false, due to David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics 1988: Volume 76, American Mathematical Society. Should we delete the sentence?Paolo Lipparini (talk) 14:15, 14 November 2016 (UTC)
 * No one commented, I have deleted the sentence.Paolo Lipparini (talk) 18:26, 6 March 2017 (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) 07:53, 16 January 2020 (UTC)


 * ✅ I added a brief formal section "Logic and universal algebra" to identity (mathematics) and re-targeted the "identity" links to there. I hope this is sufficient for now. - Jochen Burghardt (talk) 12:21, 16 January 2020 (UTC)

Fields and varieties
The article contains the statement "The fields do not form a variety of algebras; the requirement that all non-zero elements be invertible cannot be expressed as a universally satisfied identity." It seems natural to expand this to mention the smallest variety that contains the fields, namely the commutative strongly von Neumann regular rings, AFAICT. This shows that invertibility of all non-zero-divisors can be expressed equationally. What is the general feeling about including this? —Quondum 02:44, 15 December 2020 (UTC)


 * Are there sources stating that? Paradoctor (talk) 03:12, 15 December 2020 (UTC)


 * I have come across papers that deal with the topic, such as https://link.springer.com/chapter/10.1007/978-3-540-78127-1_10, or by the same authors: https://arxiv.org/pdf/0901.0803.pdf. These particular ones might not be sufficient to meet WP criteria.  The arXiv paper has the statement "Perhaps the clearest way to specify the class of meadows is as the smallest variety containing all zero-totalized fields".  The cringe-worthy terminology ("meadow") gives something to search on, but it also narrows down the selection of authors considerably.  I would be surprised if this is not a standard result in more general work on strongly von Neumann regular rings, since this is a well-known class that appears to be exactly the "skew meadows" (with "meadows" being the restriction to the commutative case).  In my mind there is little doubt in the validity of the claim, but others will be better than I at checking the validity and sourcing it reliably.  —Quondum 15:27, 15 December 2020 (UTC)

problem on chrome not firefox
Under examples there is a superscript -1. For me the - is not visible in Chrome, is ok in firefox.

Sma045 (talk) 01:08, 3 January 2021 (UTC)


 * The only expressions such as you describe in that section are usually generated as pictures (though this might depend on your preferences settings). This suggests that the problem might be with your display being too small to render the detail.  Try expanding the display size and see whether the superscript minus is still invisible.  —Quondum 01:53, 3 January 2021 (UTC)