Talk:Outline of algebraic structures

Opening heading

Melchior, thank you for taking an interest in this entry. You have materially changed the stuff on modules, vector spaces, algebras over rings/fields, in ways I have no grounds to dispute. I refined lots of details using Birkhoff and MacLane (and Herget and Michel), but the fact remains I am no authority whatsoever; in higher math and logic, I am wholly self-taught. I come to this topic fascinated by Boolean algebra and lattices, and with a working knowledge of linear algebra. Otherwise, I pretty much only reword what I find in printed sources.

Burris and Sanka are very weak on modules, vector spaces, and algebras. Much of the structure you changed I inherited from Algebraic structures as it stood 4+ months ago. (Incidentally, Stan Burris is quite annoyed by the Wiki definition of "algebraic structure.") The whole variety/nonvariety dichotomy is another legacy from the past.

Jipsen confirms that vector spaces form a variety, but I do not see how, given that the real field is not a variety. I have not yet fully grasped just what it means for a structure to be a variety, and so the discussion at the start of section 2, re axioms that are not identities, will have to revised.

Two objectives I have set for myself:

• If a structure has its own Wiki entry, it should be mentioned in this list. This does not preclude including structures for which there is no entry (yet);
• The definition of a structure given in this entry must be consistent with the definition given in the linked Wiki entry. If that entry is wrong, that should be addressed first.

Too bad you are not an authority on multilinear algebra; there are suprisingly few books on the subject, the ones I have handy are less than pellucid, and the relevant Wiki entries don't meet my standard of clarity either. Finally, I am disappointed that we cannot agree on a home for that creature taught to undergraduates everywhere, called linear algebra. It's more than a vector space over the reals and deserves a pigeonhole, but where?132.181.160.42 04:17, 18 July 2006 (UTC)

Well, all vector spaces together don't form a variety, or really even a sensible category of any kind. After, all, what would a "linear map" between a vector space over the reals and a vector space over the field with 7 elements look like? But the category of vector spaces over a given field is a variety, and you can check that the axioms are all identities. To keep it simple, a vector space over the field with 2 elements is a set with one 0-ary operation (the zero vector), three unary operations (additive inverse, multiply by 0, and multiply by 1) and one binary operation (addition). The axioms are all identities among these five operations. The field structure dictates what these identities look like, but they're still identities.

As for "multilinear algebra" and "linear algebra", these aren't algebraic structures, so they don't merit entries in a list of algebraic structures. They're just fields of study, like universal algebra, abstract algebra, and middle-school algebra.

I am reluctant to agree. Linear algebra is vector spaces plus linear transformations over vector spaces, as characterized by matrices and determinants. I have been rather surprised to discover that the literature is silent about any formal axiomatic structure for matrices and determinants.

The multilinear algebras, legacies from Algebraic structures, do have formal axiomatic structures, although the presentation of those structures in texts I can access leaves something to be desired. Birkhoff and MacLane give these fair attention. Clifford and geometric algebras excite physicists. I am drawn to exterior algebra because fascinated by their inventor, Hermann Grassmann.

BTW, I noticed only 30 minutes ago that you killed my paragraph on free modules. I wrote that paragraph to answer objections someone raised 6-8 weeks ago, carefully distilling it from Birkhoff and MacLane. I was rather proud of the result. As far as I could determine, Wikipedia does not do free modules justice elsewhere.

Yes, I deleted it in this edit for two reasons. First, it was misplaced: the category of free modules (over some ring) is not a variety, and Algebra (ring theory) does not state any freeness conditions. Second, every algebraic structure in the variety section has a free algebra concept, so why should modules be given an extra entry for theirs? It's better to keep this list concise and do full justice to the concept at the Free module article, isn't it? Melchoir 03:50, 19 July 2006 (UTC)

I think the main editorial value of the variety/nonvariety dichotomy is that the variety section has a clear and unambiguous circumscription, while the nonvariety section is undefined and can contain practically anything.

True, although I am quite comfortable with the scope of the entry as it now stands. I did give thought to adding Hilbert and Banach spaces, the topological spaces T0, T1, and T2... and blinked. Algebraic structures mentions "pointed unary systems" and then stops; I immediately saw that it was only one small step short of a Peano system. Then why not Peano arithmetic? And thus the whole section titled Arithmetic was born. I am having trouble getting a handle on Skolem arithmetic, so much so that I have Emailed the living authority thereon.

I have had nothing to do with the content of "Allowing additional structure"; that's 100% legacy.

Heh, I'm afraid that rather inelegant legacy is partially mine; I created the section to deal with sets with structure other than operations that had been included by a still earlier editor. Such is the evolution of an article... Probably it should be recombined with "Structures that are not varieties", which already deals with such non-operations as norms, inner products, and gradings. Melchoir 03:55, 19 July 2006 (UTC)

I would be fine with combining them, as long as there remains a clear visual distinction between the two: perhaps colored flags for the entries? Melchoir 07:13, 18 July 2006 (UTC)

I would prefer a richer categorization suggested by Jipsen's webpage: varieties, quasivarieties, first order, etc.132.181.160.42 03:48, 19 July 2006 (UTC)

I haven't seen it. Got a link? Melchoir 03:56, 19 July 2006 (UTC)

The fate of "Allowing additional structure"

I have had nothing to do with the content of "Allowing additional structure"; that's 100% legacy.

Heh, I'm afraid that rather inelegant legacy is partially mine; I created the section to deal with sets with structure other than operations that had been included by a still earlier editor. Such is the evolution of an article... Probably it should be recombined with "Structures that are not varieties", which already deals with such non-operations as norms, inner products, and gradings. Melchoir 03:55, 19 July 2006 (UTC)

I would be fine with combining them, as long as there remains a clear visual distinction between the two: perhaps colored flags for the entries? Melchoir 07:13, 18 July 2006 (UTC)

I would prefer a richer categorization suggested by Jipsen's webpage: varieties, quasivarieties, first order, etc.132.181.160.42 03:48, 19 July 2006 (UTC)

I haven't seen it. Got a link? Melchoir 03:56, 19 July 2006 (UTC)

It's the sole link at the bottom of the entry! Go ahead and merge in "Allowing additional structure" in any way you see fit.

Let me return to free modules. It is very curious (to me) that modules are varieties but that free modules are not. My orginal description a module mentioned an optional basis. An editor objected that there is no module analog of a vector space basis. A bit of browsing of Birkhoff and MacLane, and I found free modules, which fit the bill. Wikipedia does include an entry named free modules, and my preference is to include in this list every structure described somewhere in Wikipedia. And thus I created a paragraph describing free modules. It is true that there is a free variant of all sorts of algebraic structures, and perhaps this entry should include 1-4 sentences talking about that. Are free algebras properly discussed anywhere in Wiki?

Jipsen mentions structures that I don't quite know how to pigeonhole: hoops come to mind. But all such structures in Jipsen have no Wiki entry to date, so I omit them from this list with a clear conscience!202.36.179.65 11:08, 25 July 2006 (UTC)

Let me try to explain what's going on with free algebras. The concept of a free X isn't a "variant" of the concept of an X in the same way a commutative X or an X-with-identity is. Free groups are, for example, comparable to alternating groups. Yes, we can talk about alternating groups in the plural, but the actual definition of an alternating group requires you to pick an n, and once you do so, you're left with a unique object: A_n. So "alternating group" is just the name of a list of examples. Now, before I go further, just to make sure we're on the same page: would you want to include "Alternating group" in this article? Melchoir 19:38, 25 July 2006 (UTC)

Alternating group reveals that it is a type of permutation group of finite order. It is simply a concrete instance of a group and not a distinct algebraic structure. Hence what makes an alternating group distinct from other groups is not its universal algebra structure (the variety group includes alternating groups), but its model theoretic structure. So I see no reason for including it in this list. Good thing, because group theory texts I've been browsing of late reveal a zoo well-stocked with species!202.36.179.65 11:04, 30 July 2006 (UTC)

Combinatory algebra

Would someone well-versed in combinatory logic give my definition of combinatory algebra a close critical reading? The combinatory logic I am to algebraize is, of course, the classic one with S and K primitive. I was inspired to do this by the section "Symbolic Systems" on p. 1172 of Wolfram's A New Kind of Science, but I may be misreading the work of Wolfram and his collaborators.132.181.160.42 01:46, 7 September 2007 (UTC)

Matroids, antimatroids: where?

Matroids and antimatroids are pretty clearly not varieties. Do they belong here and if so, where? I have put them under lattices that are not varieties, but am very open to moving them elsewhere.132.181.160.42 03:48, 23 October 2007 (UTC)

Requested move "List ..." -> "Outline ..."

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: page moved per request. - GTBacchus^(talk) 19:18, 14 September 2011 (UTC)

---

List of algebraic structures → Outline of algebraic structures –

There has recently been a lot of discussion regarding pages named "List of ..." or "Outline of ...": see for example Wikipedia_talk:WikiProject_Mathematics#Undiscussed_List_-.3E_Outline_moves and Wikipedia:Administrators' noticeboard/Incidents#Gamewizard71 and Wikipedia:Village_pump_(proposals)#RfC:_Elimination_of_outline_articles.

I see this particular page as a good model for what Wikipedia:WikiProject Outlines ought to be doing: unlike the other "List of ..." articles, this page is much more than a bare list of links, and contains some useful expository prose. (The page Wikipedia:Manual of Style/Stand-alone lists links to some examples of "annotated lists". The information in List of algebraic structures goes well beyond such annotations.) Therefore I think that renaming this page to include outline in the title would set a constructive precedent. Jowa fan (talk) 02:57, 6 September 2011 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

"Is a variety"?

I do not think that this sentence is correct:

In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety.

If we accept this definition, the neither the field of rational numbers nor the class of all fields would be an algebraic structure.

I do agree with the terminology in Algebraic structure:

An algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms.

(Although I do not agree with the emphasis on the axioms.)

I also do not understand what the editor meant with this sentence:

An algebraic structure consists of one or two sets closed under some operations, functions, and relations

1. First, it is true that one- and two-sorted structures account for the bulk of algebraic structures that appear in mathematics (e.g. groups are one-sorted, vector spaces two-sorted). But why not allow more sorts?
2. What is the intended difference between "function" and "relation"?
3. What does it mean to be "closed under a relation"?

--Aleph4 (talk) 11:06, 24 October 2011 (UTC)

Recommendations to improve this outline

Hi. I'm guessing many of the problems I am about to mention are probably a result of the "list/outline" switch done earlier. I think the article does not carry out the first "main purpose of outlines" (1. present subtopics...provide understanding especially to those unfamiliar with the topic.) This outline is too technical, too overly detailed, and completely misorganized for helping one learn about algebraic structures. I propose the following changes:

1. A brief intro about the presence of algebraic objects in mathematics, and a description about how they are usually presented (first groups, fields and vector spaces, and lattices, only later the harder stuff).
2. Remove the big list of examples. It does not seem appropriate on an outline, and they are certainly present in all the articles for which the outline acts as a navigation tool. A handful of elementary examples might be well-placed in the intro.
3. There are nonstandard appearing which could be mistaken for standard. Two examples: "ringoid" and "shell". The shell article has even been created and deleted by WP:Math already.
4. A subsection devoted to universal algebra (the abstract study of algebraic objects).
5. Relegate the variety/nonvariety split to a subsection on universal algebra. The organization of objects should be less Linnean and more conceptual, as was recently implemented in Template:Algebraic structure.
6. I know "outlines are not lists" but certainly part of this outline will end up being a list, out of necessity. We need to at least reduce the number of structures mentioned.
7. Graphs are borderline algebraic, and WP:Math is discussing whether or not they really belong. Same can be said for sets with no operations.

I'll avoid editing this page for a week to wait for feedback. Rschwieb (talk) 14:12, 12 March 2012 (UTC)

It's been an entire month, so I'm going to move ahead with changes along the lines described above. Rschwieb (talk) 13:42, 13 April 2012 (UTC)

As can be seen I have greatly reduced the content. It has not been totally deleted, it still exists in storage for those who care to try to read it. The outline can certainly regain structures that were removed, but we really need to avoid including every single name and detail about every structure that can be found. Not only is it an eyesore, but also nobody is going to read it. According to the guidelines at WP:outlines the two main purposes of outlines are:

1. Since outlines present a subject's subtopics and how they are related to each other by where they are placed on the outline's tree structure, outlines provide understanding, especially to those who are not completely familiar with the subject.
2. Since the subtopics in outlines are linked whenever Wikipedia has articles about them, outlines serve as a table of contents or site map to its subject's coverage on Wikipedia. In this respect, each outline is a navigation aid, for its subject and Wikipedia's coverage of it.

I'm contending that the previous article was bungling #1 by including too much material and organizing it in an oversophisticated way. We should strive to keep the organization as simple as possible, and minimize the space taken up by overly obscure structures. Rschwieb (talk) 20:23, 13 April 2012 (UTC)

I used to come to this page all the time for its encyclopedic (ha!) nature. I feel that it has now been gutted. I **liked** the huge list, especially the inclusion of non-standard items. I am not against make this page an 'outline' based on fairly stable content -- I am against the deletion of the old content from Wikipedia altogether (I know it is still available in the page history, that's not my point). Basically I want to know where the content of the old page should go (on Wikipedia). When you say above that "nobody is going to read it", I both agree and disagree: I agree with the sentiment that most won't read it, but disagree as I am one of the few that really did read it, in detail. Anyways: I am not against the changes that have been done, they might even be an improvement of this page. What I am really after is a proper location for the old content. JacquesCarette (talk) 18:08, 27 April 2012 (UTC)

Hi, thanks for making your request so understandable. Firstly, if you can recommend items that got the axe that you would like to see back again, I would be happy to try to fit some back in, provided they seem to be a reasonably established terms. I did cut the kudzu way back, but I am looking for ways to reintroduce some deleted terms. That said, the excessive detail on each entry simply cannot return.

I don't think that any portion of WP would be a good home for the excessive detail and terms of dubious commonality would be welcome. Let's see if we can replant a bit of what was lost here. Rschwieb (talk) 21:41, 28 April 2012 (UTC)

Merge proposal

Strong oppose: Do not under any circumstances put any material from the outline into Algebraic structure. They were largely identical until recently. As you can see, I was involved earlier in a significant rewrite of algebraic structure to make it more accessible and less rambling. Moving any information from the outline would cause duplication and reintroduction of the very problems that were being addressed. While the articles of course share content, the missions of outlines are very different from those of regular articles, so they probably both need to keep existing (unmerged). I will be happy to take action to distinguish the two articles. Rschwieb (talk) 13:33, 13 April 2012 (UTC)

Response to Strong oppose: Agreed. I did notice that that Algebraic structure has large flat lists of topics, whereas Outline of algebraic structures has (as it should) trees of these topics. Would it be prudent / in-style to remove the former's flat-lists, and simply point readers of the former to the trees of topics in the latter? (Excuse me, I'm new to both higher math and editing Wikipedia.) Yangjerng (talk) 13:58, 13 April 2012 (UTC)

That's OK, welcome to WP! I don't think we should complately remove the lists from algebraic structure. They are currently acting as examples, which typically appear in math articles as lists along with a little detail.

I think the outline article, however, should probably have longer lists and no detail on individual points, so as to function as a "sitemap" as the WP:outlines guidelines seem to encourage. Since you've drawn my attention to the outline again, I've decided to start working on making the outline a little more useful (I had almost forgotten about it!) Rschwieb (talk) 18:14, 13 April 2012 (UTC)

User:JP.Martin-Flatin

Edit Rejected

Hi! I saw you made an extremely large edit containing some good changes recently. Unfortunately, there were also a lot of questionable changes mixed in. Since the edit was too monolithic to split between the two types, I have reverted it and requested your presence here. It would be nice to carry out the least controversial things first.

• For sure, I wish the changes to the references could be reproduced.
• You changed a lot of headers from something like "Two X's and two Y's" to "Two Y'x and two X's". I suppose this is possible, but superficially it seems like a change that doesn't do anything. I could be persuaded, though.
• Massive rewordings of text by a single author are usually viewed with suspicion because they may be heavily influenced by personal views. I don't mean to say you can't write anything like that, but it would be more constructive if we could do a chunk at a time and not every section at once.

Sorry to slow down your groove... I hope we can reinstate several parts of your edit soon. Rschwieb (talk) 14:18, 12 February 2013 (UTC)

Answer

What specific changes pose you a problem? I did not change the semantics anywhere. I just tried to improve the wording and fixed a few simple classification problems. For instance, in Section "Three binary operations and two sets," the entry "Bialgebra" states "There are actually four operations for this structure," which obviously clashes with the section heading. To avoid this minor problem, I moved this entry to a new section dedicated to algebraic structures with two sets and four operations. Another example: I transferred the Hazewinkel reference from "External links" to "References," where it belongs. Another example: I made the introduction crisper by transferring some material further down and rewording a few sentences. Which of these changes made you reject the entire edit?

Feel free to keep the section headings unchanged. I proposed to replace "N operations on/and M sets" with "M sets and N operations" because Lang, Hungerford, Rotman, Bourbaki, etc. all define algebraic structures as sets with operations, rather than operations on sets. Mentioning sets before operations seems more natural to me, but I won't fight for it. If that is the only change that makes you feel uncomfortable, please validate the rest and undo this one.

Jp.martin-flatin (talk) 14:22, 13 February 2013 (UTC)

Quick explanation of Wikipedia outlines

"Outline" is short for "hierarchical outline". There are two types of outlines: sentence outlines (like those you made in school to plan a paper), and topic outlines (like the topical synopses that professors hand out at the beginning of a college course). Outlines on Wikipedia are primarily topic outlines that serve 2 main purposes: they provide taxonomical classification of subjects showing what topics belong to a subject and how they are related to each other (via their placement in the tree structure), and as subject-based tables of contents linked to topics in the encyclopedia. The hierarchy is maintained through the use of heading levels and indented bullets. See Wikipedia:Outlines for a more in-depth explanation. The Transhumanist 00:02, 9 August 2015 (UTC)

"Group action" is missing, it fits well into "module like" structures.

Group action is missing from the list. Since it shares many properties with module-like structures, such as being a set acting in another, such as vector space wich is a field acting in "the vector set" or module wich is a ring acting on "the modules", the group action is a group acting on a set. Now, the set doesn't have to have a operation within. Is that why is not incuded? — Preceding unsigned comment added by Santropedro (talk • contribs) 18:56, 31 May 2017 (UTC)