Talk:Algebraic structure

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Field: Algebra

Inner product space ...

is listed under "Algebra-like structures". Shouldn't it be listed under "Module-like structures"? An inner product space has V × V → F, but not V × V → V. — Preceding unsigned comment added by Herbmuell (talkcontribs) 06:23, 5 December 2013 (UTC)

Algebra-like structures...

are described in the article as "composite systems defined over two sets, a ring R and a free R module M. Counting the two ring operations and the single module operation, this can be viewed as a system with three binary operations." Aren't you just describing a module here? I thought an algebra always has a multiplication, which makes 4 binary operations (2 in R and 2 in M). - Apart from that, many thanks to all the people who have contributed to this very readable article. — Preceding unsigned comment added by Herbmuell (talkcontribs) 05:36, 4 December 2013 (UTC)

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)

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)
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)
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)

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)

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.

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)

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)

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)

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)

Is it me or this is a contradiction? On the first line, you read, "an algebraic structure consists of one or more sets closed under one or more operations," while some lines below, it is said that the set is a degenerate algebraic structure having no operations. What exactly means degenerate? Saying a set is an algebraic structure is a strong affirmation to me. Bogdanno (talk) 23:17, 1 July 2009 (UTC)

The intro paragraph is good. I also found the algebraic structure table and the examples useful. But that whole listing of structures is too long and replicates an existing article List of algebraic structures. The first paragraph is also strange as well because its tone is completely different from the intro paragraph. Angry bee (talk) 22:38, 14 April 2011 (UTC)

Three or four binary operations?

This article seems to contradict the article Algebra over a field, on the number of binary operations. The current article places Algebra over a field under the subsection "Four binary operations." But the referenced article says, "An algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field." The way I count, multiplication, addition, and scalar multiplication are only three binary operations. What is the fourth binary operation? —Preceding unsigned comment added by 134.79.192.112 (talk) 22:09, 13 October 2010 (UTC)

algebraic structure = class of algebraic structures?

Several editors have complained about the confusing notation that uses the word "structure" in two senses:

1. The integers with addition are an algebraic structure, namely, a group. In fact, every group, field, etc, is an algebraic structure.
2. The class of all groups is an algebraic structure. In fact, every variety, quasivariety, etc is an algebraic structure.

I have now boldly reworded (hopefully) all references to this second concept. A structure in the sense of this article is now a "single" structure -- a set (possibly many-sorted) together with operations.

--Aleph4 (talk) 15:20, 31 October 2011 (UTC)

Expect lots of changes

Hello: it recently came to the attention of the math wikiproject how much work was needed on this page. There will probably be huge changes to deal with the organization of the page, which seems to be entirely from a universal algebra standpoint, and is not accessable to laypeople. Some of the content is fine, but the major task is to avoid the "identities" "not by identities" dichotomy, and to limit repetition. There are also some really questionable organizational issues like discussing domains and fields and ringlike structures separately... that could just be me though.

1. The intro right now is not terrible, but it should at least be looked at, and the following (longer) section should contain a solid description of what makes an algebraic object an algebraic object (ala D.Lazard's comments on the math project page)
2. Next section could be full of examples, wlinks and blurbs, using the scheme of "no operations" "one operation" "two operations" "composite systems" and "even more structures".
3. The category theory section does not do either category theory or universal algebra justice, so we need an expert to sum up how they are related to the article title.

I'll begin work on #2, but #1 and #3 are best explained by someone other than me :) Rschwieb (talk) 14:18, 1 February 2012 (UTC)

Please be aware of the rather similar article Outline of algebraic structures. See also #Proposal to scale back this entry above. There seems to be a huge amount of duplication, which should be addressed before you do too much work here. — Quondum 15:16, 1 February 2012 (UTC)
Thanks for bringing this to my attention! That page appears to be no better organized than this one... I'm going to bounce this off the math project page fast. I was hoping Algebraic Structure could be vastly abbreviated, and more readable as an introduction article. Well I'm going to dig in sometime soon, and if anything goes wrong you can just cut and paste from that horrible outline. Rschwieb (talk) 16:30, 1 February 2012 (UTC)

… one or more binary operations defined on it

Should we mention that constants such as 0 and 1 are "nullary operations", by the analogy to situation with logical connectives? Incnis Mrsi (talk) 10:16, 12 March 2012 (UTC)

From reading other math articles (e.g. arity, signature (logic) and universal algebra) it seems pretty normal to include nullary operations in the signature of an algebraic structure. They seem to be reserved for "special" elements such as the ones you listed, and may be treated as on par with operations with higher arities. I would thus support such a mention, especially in the context of this article. A pointed set is an example that may be defined as with "one nullary operation defined on it". — Quondum 12:43, 12 March 2012 (UTC)
As far as exposition goes, it certainly isn't standard fare. IMO it just doesn't really contribute anything, unless you are destined to become a universal algebraist. It's well suited to be a blurb in a subsection centered around universal algebra, or in a footnote, but it doesn't make any sense to mention it in the intro or overview. Most of the recent work was dedicating to eliminating such information, in the interest of clarity. Rschwieb (talk) 13:11, 12 March 2012 (UTC)
I agree with Rschwieb about 0-ary operations. But 1-ary operations need more care: The article says (3d line of section "Overview") that the axioms do not contain existential quantifiers. Thus, for a group being an algebraic structure, the existence of inverse elements has to be replaced by providing the inverse as a unary operation. D.Lazard (talk) 13:37, 12 March 2012 (UTC)
… or an unary inverse operation has to be replaced with a binary division operation. Incnis Mrsi (talk) 16:22, 12 March 2012 (UTC)
Does either case solve the axiom problem? It would seem just as difficult to define an inverse operation and a division operation in terms of multiplication without existential qualification. You can't just define two operations, one just "happening" to be the inverse of the other. You have to define on as the inverse of thwe other, which a mounts to saying "the operation that yields the element that is the solution to an equation". — Quondum 20:04, 12 March 2012 (UTC)
Why existential quantifiers? I do not realize which axioms may be required for the (right) division but a ∕ a = e, a ∕ e = a, (a ∕ b) ∙ b = a, (a ∕ b) ∕ c = a ∕ (c ∙ b) , and a ∙ (b ∕ c) = (a ∙ b) ∕ c = a ∕ (c ∕ b) (note I do not claim that this set of axioms is not redundant). What did I miss? Of course, division in division rings will not be so happy, but not due to proscription of existentials. Just opposite, because the division becomes a partial function. Incnis Mrsi (talk) 07:25, 13 March 2012 (UTC)
I'm sorry, I'm approaching it very informally, and if someone is missing something, it is no doubt me. In any event, this seems to be off-topic. — Quondum 15:00, 13 March 2012 (UTC)

Set with no binary operation

I would like to call the "no binary operation" section into question. Brief consultation with my common sense says that a set alone is not an algebraic object. I feel the same way about a pointed set. A case might be made for keeping a set with a unary operation, but we would need more evidence. Overall this little subsection seems a little shaky, so it should be talked about. Anyone have references which would lend credibility to a set with unary operation (without binary operations) as being an algebraic structure? Rschwieb (talk) 13:17, 12 March 2012 (UTC)

The natural integers are defined by Peano axioms as a set with two unary operations ("successor", defined for every integer, and "predecessor" defined for non zero integers). The classical binary operations are derived from the unary operations and do not appear in the definition. Is that a credible algebraic structure? D.Lazard (talk) 13:49, 12 March 2012 (UTC)
Sure, certainly more convincing than set/pointed set. Can you comment on set/pointed set? Rschwieb (talk) 14:15, 12 March 2012 (UTC)
IMO, the mathematical topics which have no application outside themselves are of no interest. IMO again, this is the case of universal algebra and any classification of algebraic structures: Who really care of what is and what is not an algebraic structure? The important thing is to know which part of mathematics can be done inside the first order logic. IMO this is the right definition of algebra. This is important because this is the basis of constructive and computational mathematics. Thus, for me, set and pointed sets are algebraic structure, but it is of no importance to classify them as such or no. A more interesting example is Real closed field. I ignore if the universal algebraists classify it or not as an algebraic structure. However, Tarski's theorem on quantifier elimination shows that the first order theory of these fields is complete, and thus that this structure captures all the algebraic properties of the real numbers. It follows that a computational real algebraic geometry is possible (and becomes an important part of recent real algebraic geometry). D.Lazard (talk) 15:19, 12 March 2012 (UTC)
I'm not sure this was fully resolved. Basically the only example for "set with no binary operations" was the Peano axiom example, but those give rise to the two binary operations, so we are more likely to use it in that section. I'm not sure that it lends sufficient importance to including a "set with no binary operations" as an algebraic structure. Rschwieb (talk) 13:40, 13 April 2012 (UTC)

Interpretation vs. representation vs. model

From what I can tell, a lot of the confusion here comes from the fact that some editors are coming from a model theory background, where a variety is just the olde-fashioned name for an equational theory, while other editors are coming from a category theory background, and want to see everything as a class. Most readers probably don't have a background in either, and so get stumped by the statement: "an algebraic structure is a set with binary operators". But what is that set? So, for example: consider the set V={x,y,z,...} whose elements are interpreted as variables. Then (V, 0, +) would seem to be an "algebraic structure" according to this article. But other readers imagine a set Z={0,1,2,...} and so imagine that (Z,+) is an "algebraic structure". I think this confusion needs to be clarified first. It seems that Structure (mathematical logic) provides the correct definition of an algebraic structure, but this article does not seem to link to that one... or, at least, doesn't underscore that an algebraic structure is a special case of a structure.

This article also evokes other, related ideas that serve only to confuse, so for example, the relationship between a representation (say, of a group) and the abstract group itself (both have the same "algebraic structure"(?) but are not the same). Similarly, the relationship between the interpretation (model theory) and the model itself. That is, since this article implies that "algebraic structures" are like universal algebra, but with binary operators only, this really just implies that algebraic structures are just a special case of an equational theory. So there should be at least a tiny hint, in this article, that there is a model, and that there is a theory of all of the valid interpretations of the model. Otherwise, one risks, potentially, of having "algebraic structures" which have interpretations that don't fit the theory, yeah? (I'm hand-waving here, but see the point?). And... on a completely different vein, the relationship between this and standard category theory should be clarified. Anyway, all these should be given at least a nod, so as to draw a sharper distinction. linas (talk) 15:20, 13 April 2012 (UTC)

Hmmm. I was alerted to problems in this article. I hate to say it, but the versions of this article in late 2011 seem to be more clear, more direct, more understandable. The current article is a bit more tedious and soporific; the assorted hand-waving and inexact statements makes it harder to read and grok. The old article stood straight up, got to the point and said it clearly, directly. I dunno. linas (talk) 15:50, 13 April 2012 (UTC)
As an example of the objectionable hand-waving, I removed this piece of bullshit:
The main motivation behind the study of algebraic structures is that many seemingly unrelated concepts can be related in terms of their algebraic properties. This provides links between them and gives a more in depth understanding of the mathematics involved. Although algebraic structures are mainly studied in pure mathematics, they do have applications in other fields, for example mathematical physics.
First of all, one historical motivation is that its interesting. Another one is that there are real-world applications in proof theory, computer algebra, machine learning, genetic programming, satisfiability modulo theories, databases, and knowledge representation in general. All of these are more or less "applied mathematics" and not "pure mathematics" and the overlap to "mathematical physics" is almost zilch. When MIT announced that it had developed a computer program that learned F=ma by watching a video of a double-pendulum, this was a comp-sci achievement, not a "mathematical physics" achievement.linas (talk) 16:49, 13 April 2012 (UTC)
I agree with Linas that Structure (mathematical logic) should be a starting point for "algebraic structure", but not "that Structure (mathematical logic) provides the correct definition of an algebraic structure". IMO, an algebraic structure is a "structure (mathematical logic)" with a set of axioms. In other words, the signature of an algebraic structure should include the axioms. This leads to the question of what is an axiom?. It may be simply an equation, a formula of propositional logic or contain existential quantifiers. However existential quantifiers may be removed, like in the definition of a group where the existence of an inverse may be replaced by the function "inverse".
By the way both articles make the distinction between structures whose signature contains relations from those whose signature contains only functions. The difference is very tiny, as soon as one considers multi-sorted structures, as a relation is a function into the booleans. This should be mentioned in both articles.
Nevertheless the main difficulty in writing this article is that it should give a definition of "algebraic structures" which corresponds to what most algebraists mean and to relates this definition to the various theories of such a notion. More, none of the related theories, like universal algebra of type theory(ies) give a definition which is convenient for algebraists. An example is Axiom (computer algebra system), which is the first software attempting a general implementation of algebraic structures; lacking of a convenient terminology, their authors called "category" what could be a general definition of algebraic structures.
D.Lazard (talk) 17:42, 13 April 2012 (UTC)
@linas I'm not sure what version from late 2011 you were looking at, but I'm having a hard time believing you were looking at the same article as I saw. IMO the level of tedious junk has been decreased by at least half from that period. The article was cut by thousands of bytes. It included all manner of unnecessary detail. Formerly it required a degree in universal algebra, but now it is aimed at anyone with basic set theory (the standard basis to learn about algebraic structures). I suppose one can view set theory through model theory, but in any case basic set theory is the way to go.
I imagine by Z={0,1,2,..} you meant N={0,1,2,...}, but even then, I am still interested in knowing why (Z,+) (N,+) shouldn't be algebraic structures. We can agree on the handwaving in the lead sentence: it was inserted by a non-expert at the beginning of the rewrite. Rschwieb (talk) 18:00, 13 April 2012 (UTC)

Reply to D. Lazard: Re "axiom": from the model theory point of view, "axioms" are just "equations", so the "axioms" of group theory are just "equations" that equate the left and right hand sides. Now, model theory, in genreal, allows n-ary relations, and not just binary equations. Thus, an algebraic structure disallows all n-ary relations, and allows only 2-ary equations; this is why an algebraic structure is a special case of a structure.
The word "axiom" is dangerous, because it implies that you can use quantifiers; for an algebraic structure, you cannot. As soon as you start using quantifiers, you are talking about something else, typically, about first-order logic, such structures are no longer algebraic structures.
The word "axiom" is dangerous for another reason: in computer algebra systems, one emphasizes reduction rules, as opposed to equations. Thus, x+0 reduces to x, and so one has all the problems of unification (computer science), and word problem (mathematics), etc. That is, although one can abstractly define an algebraic structure in terms of a quotient term algebra, one still has the problem of identifying the members of the cosets, and as the word problem shows, this is not computable; thus a certain focus on reduction as opposed to equation. To be fair, this article should mention this topic, at least very briefly.
Finally, this article doesn't get across the importance of the interpretation. So, for example, given operator OP and neutral element NE, we have the algebraic structure x OP NE = x This has multiple interpretations "x+0=x" ... "x or F = x" ... "x and T = x" ... "x.1 = x" all of which are valid interpretations. Many of these are ruled out only when additional axioms are added. Those additional axioms are added by means of mathematical-logic-and so interpretations narrow as more axioms are and-ed or or-ed or negated into the system. This is why the article list of algebraic structures has a big diagram that looks like a poset: its because, from the mathematical-logic point of view, algebraic structures form a kind-of-poset, as additional axioms are added. ...and one may have countably many axioms, etc. all of the usual concerns and issues that arise. We do a bit of a dis-service by not hinting at any of this. Then again, its possible that I have simply drank too much coffee, and am hyperactive at this instant. :-) linas (talk) 18:10, 13 April 2012 (UTC)
Reply to Rschweib: the general issue about whether or not (Z,+) or (N,+) or whatever, is algebraic structure is that it obscures the difference between identity and equality. Please recall that an identity holds in all interpretations, whereas an equality only holds in some. So, for example, x+0=x is an identity that holds for all abelian groups, whereas, for Lie algebras, [L_i, L_j] = e_ijk L_k is merely an equality that happens to hold only for su(2) and o(3) but not for other Lie algebras (because other Lie algebras have structure constants c_ijk that are not, in general, equal to e_ijk). Likewise, there are many equalities on (N +) known to number theorists that do not hold, in general, for all algebraic structures with this signature. Thus, (N +) is really just one possible interpretation of the structure; there are others, e.g. all abelian groups. (Or, you could say that 2+2=1 is an equality that holds for Z_3 but not abelian groups in general. Or you could say that 6+6=0 is an equality that holds for Z_12, then if we add 6+6=0 as an axiom, then there is an interpretation of (Z_12, +, 0) by Z_3 (since 6+6=0 holds in Z_3 as well) However, this now starts smelling suspiciously of being a "representation" as in a "representation of a group", and, once again, one has equalities that may hold in a representation that will not hold for the group which is being represented. That's why I blurbled about this initially, but now I wonder whether bring this up is a wise idea, or whether it will only confuse the reader yet more. Hmmmm.... linas (talk) 18:55, 13 April 2012 (UTC)

Reply, part 2: I have three books that define algebraic structures: Paul Cohen, "Universal algebra", page 48ff Baader & Nipkow, "Term rewriting" page 44ff and Wilifred Hodges, "A shorter model theory" starting on page 2. The 2011 version of this article Some earlier version of this article, that I can no longer find, defined something that, at least vaguely, if incompletely, resembled what these books say. The current article does not do even that.

This is not an appropriate place to perform "original research" as to the definition of an "algebraic structure". Many authors have already defined this for us. We need merely reproduce what they have written. An "algebraic structure" is not supposed to be a vague, intuitive thing: its supposed to be a crisp definition; one is supposed to be able to go off and state theorems about them, etc. linas (talk) 00:09, 14 April 2012 (UTC)

I looked first at Universal Algebra by P.M. Cohn just before rewriting this article, and was reassured by sentence #1 of Chapter II: Algebraic structures:

An algebraic structure on a set A is essentially a collection of finitary operations on A...

I will go ahead and cite that to allay any of your OR fears for the time being.
Googlebooks will not show me either of the pages in the other books you mentioned, and the inbook search didn't seem to yield any results for "algebraic structure", but of course I could be searching for the wrong thing. Rschwieb (talk) 01:29, 14 April 2012 (UTC)
Clearly, I had too much coffee to drink the other day. I was merely suggesting that this article should explain that the concept of an algebraic structures is actually useful for something other than creating a big list. The articles in this entire subject area are all woefully short, incomplete and/or confusing, and I've had vague plans for fixing these up for a while now. Maybe soon. Maybe a few weeks... The Cohn book is 50 years old now. I'm certain that every book on model theory will define an algebraic structure. I remember seeing a very nice whirl-wind tour of the subject in some recent proceedings of SMT solvers, online. Later, then. linas (talk) 14:47, 16 April 2012 (UTC)
I can agree with that: it would be very nice to have an "applications" section highlighting the uses of algebraic structures (outside of list populating). Rschwieb (talk) 14:56, 16 April 2012 (UTC)