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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)
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)
What is arity? LilHelpa ( talk) 19:44, 3 March 2009 (UTC)
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)
There is a line
in the category theory section. Where does come from? I understand that this is defined in terms of in a canonical way, but how? Is where 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)
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)
"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)
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)
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)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
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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)
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)
What is arity? LilHelpa ( talk) 19:44, 3 March 2009 (UTC)
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)
There is a line
in the category theory section. Where does come from? I understand that this is defined in terms of in a canonical way, but how? Is where 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)
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)
"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)
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)
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)