Talk:Algebraic structure

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: B Class Mid Priority  Field: Algebra

Contents

[edit] Just what is an algebraic structure?

Would it be correct to say that "field" and "group" are algebraic structures, or that "the field of reals" and "the group of integers under addition" are algebraic structures? The article flips back and forth between the two in a confusing fashion. (The definition at the top of the article implies that it is specific instances which are algebraic structures; but the last paragraph implies that "group" is an algebraic structure.) Cwitty 03:18, 8 Nov 2003 (UTC)

I must confess I never heard algebraic structure to mean a class of algebraic structures.The article is too confusing in this sense! One group is just one group, and is an algebraic structure. A class of groups (or even the class of all groups) is a class of algebraic structures. The ambiguity is even more confusing in the sense that the duality between syntax and semantics (see Gödel's completeness theorem) is completely lost with such a terminolgy! I suggest that algebraic structure should mean just ONE algebraic structure, and the locution class of algebraic structures should be used for many algebraic structures. This is not inconvenient, since most classes of algebraic structures under consideration are varieties, hence the word variety can be used in this case.Popopp 16:43, 13 February 2007 (UTC)


Division should be disambiguated in some way on this page. It is not correct to implie that "inverse element" and "division is always posible" are equivalent - see the short definition of group. This is concrete thinking in an abstract world. There is no such thing as division that is understood as an operator on tables, chairs, cars or bags - yet all these things can be seen as sets and with apropriate operators made part of a group. Without defining a division operator, that is. What I am trying to say is that "inverse element" is NOT to be thought of as "one divided by the element". If a is the element then 1/a is NOT the inverse element, unless we are dealing with numbers and the groups composition rule is multiplication. What is 1/a if the element a is a car? Still, it is perfectly posible to define an inverse to the element a! (213.112.153.244)

Where do you claim to find the implication that "inverse element" and "division is always posible" are equivalent? I don't see it. Melchoir 00:23, 27 September 2005 (UTC)

[edit] Reorg

Okay, I've separated out different senses of algebraic structure. The article still needs help, though! Melchoir 02:53, 16 November 2005 (UTC)

I found the previous arrangment confusing, saw field appeard to be missing, added it then saw it was mentioned below, and merged the two setions. I think it would be more helpful to have a discussion on what it means to a univeral algrbra, the current distinction is not clear to me. Also how come a module over field counts as UA but field does not? --Salix alba (talk) 10:27, 20 February 2006 (UTC)
I'll revert to the previous version, since integral domains, division rings, and fields are not algebras in the sense of universal algebra; the current version does not internally make sense. Maybe you can help me make this clearer in the article: the algebraic structures studied in universal algebra are defined by identities. On the other hand, the definition for an integral domain contains the axiom "0 ≠ 1", which is not an identity. Worse still, the multiplicative inverse is not a unary operation defined on the whole of a division ring; it excludes 0, which would not be allowed in universal algebra. These defects have negative consequences; as I've written, "Although these structures undoubtedly have an algebraic flavor, they suffer from defects not found in universal algebra. For example, there does not exist a product of two integral domains, nor a free field over any set."
As for vector spaces, a vector space is not a field, nor does it contain a field. The vector space axioms are all identities; the field axioms are not. Melchoir 10:46, 20 February 2006 (UTC)
While I was typing that, you made some more changes; I hope you won't be too insulted if I revert anyway and wait for you to read this... Melchoir 10:49, 20 February 2006 (UTC)
I guess this depends on quite what the topic of this article is. Is it about algebraic structures in general, or is is mainly about algebraic structures in the sense of universal algebras. Motivation for edit is that Algrbra is current mathematical colaboration of the week and I'd like to use this article as a sub page of that fleshing out the technical details not appropriate for the rather general and accesable main article. I'm afraid I know very little about the latter so please forgive any errors I make. In particular, we have two definitions of Division ring (the inverse operation is not defined for the additive identity) and (the inverse operation is not defined on the whole set). Are their divison rings which exclude more than just the additive identity? --Salix alba (talk) 12:12, 20 February 2006 (UTC)
Vector spaces. Currently vector space is just defined a module over a field. Does it have structure beyond this? --Salix alba (talk) 12:24, 20 February 2006 (UTC)
First of all, thanks for all the additions you've made!
The topic of the article is pretty broad, which is why I think it's necessary to break it into sections. This was the state of the article before I found it. Arbitrary relations were allowed at all points of the article, which was inconsistent with the title, "Algebraic structure". One could have added ordered sets or topological spaces just as easily. I wanted to preserve the generality of the article, but still retain some meaningful restrictions in stages. And that's the current layout.
Your definition of a division ring is correct.
Vectors spaces do not (necessarily) have any structure beyond being a module. I'll take the liberty of moving them back to the first section... Melchoir 20:41, 20 February 2006 (UTC)
Like the new edit. I think I've been a bit confused over why these did not count a universal structures, its taken a bit of thinking to work out what the diference in my mind. I've added a section expanding on why the 0≠1 condition differs. What do you think? --Salix alba (talk) 00:38, 21 February 2006 (UTC)
Well, I'm not sure what you're trying to say. The sentence "The above structures are all completely constructive abstract structures, and they do not require any mathematical results in their definitions." doesn't make any sense to me. All of the structures are abstract, and "mathematical results" is awfully vague. Could you take another crack at it? Melchoir 08:06, 21 February 2006 (UTC)


[edit] Dreaming of a periodic table for algebra

I've done a great deal of work on this entry in recent weeks, because doing so has given me an opportunity of advance a dream I've had for decades, namely to do for math what the periodic table does for chemistry. As of this writing, the entry defines, however briefly, 54 structures. I've read that it is believed that circa 200 structures have been discussed in the literature as of the 1990s.

I draw your collective attention to the following points:

  • Because a vector space presupposes the definition of a field, I treat vector spaces in the same fashion as fields, namely as structures not definable in universal algebra. I suspect that my choice here will not please some who have made comments above.
  • Just how does the the plain vanilla linear algebra taught to undergrads fit into this overall scheme? That linear algebra has a bit more structure than a vector space.
  • I added the two references. When Springer dropped Burris and Sankappanavar, Stan Burris took the exemplary step of scanning it and putting the result on the web. As of 2003, his book had been downloaded more than 50,000 times. While he foregoes royalties, he enjoys the considerable satisfaction of knowing that his text is widely used in the Third World.
  • The Wikipedia definition of a ring is that of Birkhoff and MacLane (1979: 85), but not that of Burris & Sankappanavar (1981: 24), who define a ring as a set that is an Abelian group (with distinguished member 0) under addition, and a semigroup under multiplication, with multiplication distributing over addition. This is the entry's definition of a Rng. If multiplication is a monoid, B&S call the result a "ring with identity." The B&S way means that the vector part of an algebra constitutes a ring rather than a rng. The possible value of doing things the B&S way is not evident to me.
  • The section on multilinear algebras is not finished. I am not even confident that "multilinear algebra" is the best name for this collection of mathematical systems. My main difficulty here is that I cannot find a clear coherent treatment of these algebras written in the spirit of this entry. The closest thing is Birkhoff and MacLane (1979), which is very hard going in parts. All other references I've found treat of only a subset of these algebras. Getting this right is not merely an idle curiosity; interest in Grassmann's work is rising of late, and there is a growing feeling that a great deal of physics and engineering can be recast into Clifford and geometric algebras. Moreover, it is claimed that this recasting yields substantial insight and simplification. Watch this space!
  • It would be nice if someone knowing some category theory would spice up this entry by contributing, say, a few hundred words.132.181.160.42 08:11, 29 May 2006 (UTC)

I continue working on this deeply fascinating yet frustratingly hard entry. It now touches on more than 60 structures. The entry now mentions varieties, and makes clear that the main way fields, vector spaces, and other interesting structures are not varieties is the requirement that S be nontrivial. I have recently added plain old linear algebra as a species of associative algebra with matrices as the multivectors, but am not confident that doing so is correct. Only today did I read in Birkhoff and MacLane that modules have bases, but that a module cannot have an orthonormal basis because it lacks an inner product. I've concluded that relation algebras are proper extensions of interior algebras but no printed source mentions that; am I mistaken? I've chanced on Weyl algebras but don't know how to describe them concisely.132.181.160.42 08:17, 14 June 2006 (UTC)

[edit] Thank you, Kuratowski's Ghost

for introducing me to interior algebras by slipping into this entry a mention thereof. I have shifted that mention a bit.

I am a largely self-taught amateur logician and mathematician, living in the southern hemisphere. I earn my living teaching something other than mathematics and philosophy, my true loves, for financial reasons, as you aptly put it. I also love Physics.

I have time for classical and Biblical history. I am a bad Catholic and occasional Anglican. I do not know what "maximal" and "minimal" mean in the context of the Bible. I think up tunes all the time, but do not bother writing them down. I like classical music and pre-1970 jazz. Like you, I have little time for deconstructionists and post-modernists. The world is close to the point where anyone who wants an honest education in the history of our civilisation and its ideas, will have to acquire that education on his own.

[edit] Proposal to scale back this entry

I have moved most of the content of this entry to a new entry titled List of algebraic structures and continue to edit and expand that list; it contains about 70 items. I've only recently discovered the existence of Wiki entries giving lists of mathematical topics; the value of such lists is evident.

I propose that the scope of this entry be drastically cut back to two things: a careful definition of the term "algebraic structure," and a bit of friendly talk re category theory (seen as a close and healthy rival of universal algebra). I should add that Burris of Burris and Sankappanavar (1981) tells me he is quite unhappy with the definition of algebraic structure set out in the entry. Fortunately, section 2.1 of B&S contains ample material for an improved definition.132.181.160.42 06:16, 12 July 2006 (UTC)

Seems reasonable to me. Green light. You should probably keep 2 or 3 examples of algebraic structures in this article, and link to the other with a tag like {{main|list of algebraic structures}}. -lethe talk + 06:24, 12 July 2006 (UTC)
Revisiting this page for the first time in 6 months I'm suprised about how big it has grown. It now is more of a list than an article and its getting harder to see the wood for the trees. I think revisiting the above comments would be a good idea. --Salix alba (talk) 17:08, 19 October 2006 (UTC)

[edit] Rationale for modification of characterization of Group

To say that "a group is a monoid with unary operation, inverse, giving rise to an inverse element equal to the identity element" makes no sense. It could have meant "giving rise to an inverse element which when binop'ed upon with the original element is equal to the identity element" OR "giving rise to an inverse element and by the way the result of this unop on the identity element is equal to the identity element", but neither of those statements should be put here for different reasons (namely the first is just repeating the definition of inverse element for one thing and the second could/should go in a 'simple deductions' section)--Netrapt 12:58, 10 February 2007 (UTC)

[edit] Positive definite

This article references the term positive definite, which is a disambiguation page. Please review this usage and determine which of the articles at the disambiguation is intended and adjust as appropriate. Chromaticity 02:21, 7 May 2007 (UTC)


Done. Paul August ☎ 02:52, 7 May 2007 (UTC)

[edit] First paragraph is seriously misleading

This article seems to attract people from many different backgrounds. Among other things, it appears that it tries to play the role of an article on algebras in the sense of universal algebra, but IMHO it does so very poorly.

The first paragraph defines the general setting of the article as universal algebra. What happens if we take this seriously, encouraged by later references to universal algebra?

The second paragraph seems to claim that an algebra is the same thing as a variety (universal algebra); which is just nonsense. The confusion between individual structures/algebras and classes of structures which (i.e. the classes) are defined by a certain set of axioms continues throughout the entire article.

The explanation is that the universal algebra content was added relatively late to this article. In particular, the roots of the second paragraph are older than the first paragraph: http://en.wikipedia.org/w/index.php?title=Algebraic_structure&oldid=50029042 . Note that the original setting was essentially exactly the opposite of the current one: "In higher mathematics, "algebraic structure" is a loosely-defined phrase [...]". In this original setting of classical mathematics and its fuzzy use of language, structure has a completely different meaning, and the entire article suddenly makes sense.

Perhaps someone with more Wikipedia experience can replace the first paragraph by something more appropriate? --Hans Adler 16:48, 13 November 2007 (UTC)