Talk:List of algebraic structures

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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)

[edit] 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)

[edit] 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)

[edit] 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)