Talk:Variety (universal algebra)

message: 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)
replies: text: Thanks! The definition could be simplified using the notion of "term" (see term (logic)#Formal definition). 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)
thread_title: Someone please define variety

message: 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)
thread_title: Someone please define subvariety

message: What is arity? LilHelpa (talk) 19:44, 3 March 2009 (UTC)
replies: text: 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)
thread_title:  arity? 

message: 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)
replies: text: 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)
thread_title:  Unclear variable in the Category Theory section 

message: 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)
replies: text: No one commented, I have deleted the sentence.Paolo Lipparini (talk) 18:26, 6 March 2017 (UTC)
thread_title:  Finite algebras 

message: "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)
replies: text: 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)
thread_title:  "Identities" are not defined 

message: 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)
replies: text: Are there sources stating that? Paradoctor (talk) 03:12, 15 December 2020 (UTC)
replies: text: 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)
thread_title:  Fields and varieties 

message: 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)
replies: text: 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)
thread_title: problem on chrome not firefox