Talk:Universal algebra

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Universal algebra was a good articles nominee, but did not meet the good article criteria at the time. There may be suggestions below for improving the article. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.

Article milestones
Date	Process	Result
April 26, 2006	Good article nominee	Not listed

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Universal algebra vs. Abstract algebra

(I have asked these same questions over at Talk:Abstract algebra#What questions should this page answer?). 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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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#Types_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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

Assessment comment

The comment(s) below were originally left at Talk:Universal algebra/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs information on applications, major theorems; plus further information on homomorphism and more examples Tompw 15:11, 5 October 2006 (UTC)[reply]

Last edited at 22:54, 19 April 2007 (UTC). Substituted at 02:40, 5 May 2016 (UTC)

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That didn't work, but I think I've fixed it. DavidHobby (talk) 02:45, 22 July 2016 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

"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)[reply]

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)[reply]

Bluelink on class of groups

@Jochen Burghardt the language in this page that references "the class of groups" makes it clear that we are referring to the class of all groups and not merely a single class whose elements are groups. The article class of groups is about the latter, which is not what is being referenced. However, we still bluelink to it. Can you explain why you support this bluelink? Mathwriter2718 (talk) 12:59, 1 September 2024 (UTC)[reply]