Talk:Algebraic structure

Graph (in graph theory)

Is the graph an algebraic structure or not? Mx1024 (talk) 09:37, 22 October 2017 (UTC)

• In any obvious way it is not a set with operations. Which set could it be? Wikisaurus (talk) 11:50, 7 January 2018 (UTC)

Deciding if a graph is an algebraic structure depends on the definition that is given to "algebraic structure", and to "operation". In fact, a graph is a set (of vertices) with an operation with values in {true, false}, the result being true if the two vertices are connected by an edge. The incidence matrix of a graph is thus the table of this operation. The graph is undirected if and only if the operation is commutative.

More precisely, there is no commonly accepted definition of "algebraic structure". This does not cause trouble for mathematicians, as most of them do not work on algebraic structures in general, but consider only specific structures that have been given a name (such as groups). Although a graph is rarely viewed as an algebraic structure, graph theory is generally considered by mathematicians as a part of algebra. D.Lazard (talk) 14:01, 7 January 2018 (UTC)

Two uses of the term "algebra"

The lead now uses the term "algebra" in two senses. My textbooks define an algebra as a vector space V over a field F with a binary operation of multiplication of vectors satisfying three additional axioms. If the two meanings of algebra in the lead are distinct and valid, I think they should be clearly defined.—Anita5192 (talk) 16:39, 5 November 2019 (UTC)

Good point. I have tryed to fix it. D.Lazard (talk) 19:10, 5 November 2019 (UTC)

Thank you! —Anita5192 (talk) 19:22, 5 November 2019 (UTC)

The explanation of "involving multiple sets" has become misguided.

I think at some point this article went the wrong way with explaining how algebraic structures involve multiple sets. This became clearer to me when someone recently made an edit to the effect that an algebraic structure "can have more than one underlying set" which I do not think is true. I have a hard time believing the exposition in this article accurately reflects what universal algebraists have established.

To take vector spaces as an example, the article currently unzips the algebraic structure of the scalar field as if it was on an equal footing with the operations on the vector space. This doesn't make any sense: the field furnishes operations on the abelian group. It provides unary operations on the group, not more underlying sets and more binary operations.

So we shouldn't be saying "a vector space is two sets with five operations" we should be saying "a vector space is one set with a binary operation and many unary operations." And indeed, this is already the case at the universal algebra article.

Unless I see some compelling counterarguments and citations, I may be bold and revise accordingly. Rschwieb (talk) 14:50, 15 October 2020 (UTC)

I find the advice to pretend that, for instance, the scalar multiplication of a vector space or, more generally, the ring action on a module should be primarily (or even exclusively, as sugested above!) viewed (or even defined) as a loose collection of operators without acknowledging the respective algebraic objects, just an obsolete obfuscation of the algebraic structure.

Therefore I object to the above proposal. We would be gaining nothing by hiding the obvious intrinsic structure and by artificially narrowing down the general definition and, thus, misrepresenting the algebraic nature of algebraic structures and creating confusion.

Filozofo (talk) 15:35, 15 October 2020 (UTC)

I agree with above Filozofo's post, but I disagree with their edit. IMO, one must consider structures such as vector spaces as structures that are parametrized by other structures. I believe that this is one of the great advantage of category theory over universal algebra to allow an easy definition of parametrized structures. Rschwieb's point of view is misleading because it does not correspond to the common way of thinking of mathematicians, and requires an infinity of operations, which is strange for pure algebra.

I have thus edited the lead for explaining informally this point of view. Also, I have removed the mention of abstract algebra and universal algebra, as the concept is encountered in all mathematics, and is formalized also in category theory. D.Lazard (talk) 16:43, 15 October 2020 (UTC)

D.Lazard I get your point that we don't usually explicitly phrase it this way, but with respect I do not think it is as unusual as you say: I think it's the essence of thinking of an R module M as R being mapped into the endomorphism ring of M. I do not think it is a common way of thinking of mathematicians to say that an algebraic object has "multiple underlying sets," and I sense we agree on that point. I'm willing to bend toward some middle point.

It's OK, perhaps to say that the structure "has" multiple sets and operations, but I wouldn't dream of calling them all underlying sets, or saying that all the operations involved belong to the structure at hand. There ought to be some way to say it, as you preserved in my edit, that keeps the other algebraic structures used as separate entities. Or perhaps it should be said "from the point of view of universal algebra <what I said earlier about unary operations>". Perhaps the real issue is that we can't take the POV of universal algebra throughout the article.

Filozofo You have put a lot of words in my mouth and come to the wrong conclusions about what I hope to accomplish. I'm not completely married to expressing it as I did in that first draft, but I do object to the way it is currently written because I've never seen it in print that way. I don't have access to the citations, but if you can help me determine which ones might support the point of view that an algebraic structure is "a union of the algebraic structures its composed of", I could try to obtain the text.

Next time we talk, kindly restrain the artistic license that you've taken with my words. It is quite incorrect and not constructive. I didn't hide or narrow anything, and I think my revision (however imperfect) is far less misleading than what I revised. Rschwieb (talk) 19:48, 15 October 2020 (UTC)

Rschwieb I am sincerely sorry if the tone of my earlier (well-intentioned) objection came through as "unrestrained". I did not intend it as such. Please accept my apology!

However the essense of my objection still stands. I think, it is common enough to think of various types of algebraic structures/systems as an ensemble of multiple more "elementary" structures combined in a certain way and, therefore, such algebraic structure is based on multiple supporting sets. It is not always obvious or reasonable/natural/productive to consider one of such constituents as principal and the other(s) as accessory. A simple and commonly known example of such algebraic structure would be the concept of group ring.

D.Lazard probably had in mind similar considerations in the above comment.

So I suggest that it makes sense to eliminate the artificial narrowing down of the allowable assortment of what is naturally considered an algebraic structure (or, in another terminological tradition, an algebraic system).

Sincerely, Filozofo (talk) 01:30, 16 October 2020 (UTC)

I never suggested that one of the "constituents is principal and the other(s) are accessory". Auxiliary is not accessory. To know whether one of the constituents is principal, it suffices to consider the membership relation. If $x\in S$ is commonly used with S being a structure, it generally means that x belongs to the principal component. IMO, it is fundamental to define algebraic structures in a way that $x\in S$ does not require a specific definition. This means, for example, that the definition of a group as a pair of a set and an operation must be avoided in elementary texts (it may be needed in formal definitions required by computer aided proofs). I have recently rewrote Affine space, and one of the writing difficulties was that an affine space is constitued by a set and a vector space, and the phrase "x belongs to the affine space" means "x belongs to the set".

However, there are algebraic structures without "principal constituent". The most common such structure, is probably the structure of a graph: although a graph can be defined as a set with a binary operation taking its values in $\{0,1\},$ there is no commonly accepted definition of "x belongs to a graph". In other words, neither the set of vertices, neither the set of edges is a principal constituent. IMO, these considerations are too subtle for appearing in the lead of the article without making it confusing. D.Lazard (talk) 10:13, 16 October 2020 (UTC)

Definition given in short description

Since there have been two reverts on the short description recently I'm opening up a discussion of the short description because this seems to represent an unresolved disagreement as to the article's intended content. In reverting my edit User:D.Lazard said that the only classes of algebraic structures which are not defined by some axioms are sets and magmas. I would first of all like to point out that this is false, since the class of all algebras of any fixed signature would also work. Moreover, the actual point under consideration is, to my mind, whether or not an algebraic structure itself should include a set of axioms which it obeys (whether identities in the sense of universal algebra or other axioms). I am of the opinion that including axioms in the definition is akin to saying that a topological space is a set equipped with a topology which obeys certain axioms. While it is true that the topological spaces which are typically considered obey some axioms (they might be Hausdorff or separable for example) a space itself is not given along with some axioms it might satisfy. Saying that some axioms should be included along with an algebraic structure itself seems to stem from the first examples of algebras being groups or members of other classes so defined, but this would be like saying that since the first examples of spaces one sees in an undergraduate course are metric spaces that a topological space should carry with it some axioms that it satisfies. The literature is consistent with this viewpoint, and the only place where algebraic structures are rigorously defined in the article are in the context of universal algebra, where the definition certainly does not include any particular axioms beyond a set with an indexed collection of finitary operations. caterpillar_tree (talk) 20:38, 24 January 2022 (UTC)

Please read WP:short description. Short descriptions are not here for providing definitions, but for helping readers to decide whether they are interested in the article, generally found with a research engine. The definition of an algebraic structure given in the first paragraph of the lead mention axioms explicitely. So, before removing axioms from the short description, you must get a consensus for removing them from the lead. There is no hope for that, as you have no reliable source supporting your views. D.Lazard (talk) 21:14, 24 January 2022 (UTC)

Thank you for pointing out the purpose of a short description. I still stand by eliminating the mention of axioms in the short description, however, as "satisfying given axioms" may be appended to the short description of any large class of mathematical object, as in the example I gave before of topological spaces, but doesn't add any useful information. You are correct that I would like to get a consensus for removing the discussion of axioms from the lead. It seems a bit prejudicial to say that I have no reliable sources for doing so when the only citation in the introduction is to a universal algebra text which gives the concrete definition I mentioned previously. It would seem that the onus is on those who would like the page to remain they way it is to provide sources for this treatment of algebraic structures. At the moment this page appears to be trying to be a page on algebras in the sense of Universal algebra (and indeed the section on algebras on that page links to Algebraic structure) but it is at the moment largely an unsourced list of various classes of algebraic structures, broadly construed. I agree with you that the term "algebraic structure" has a broader meaning within mathematics, but without a clear source on what exactly that is it seems better to remake this page into one solely on the universal algebra notion and point people to the well-written disambiguation page Algebra (disambiguation) for other meanings. Trying to lay down what exactly an algebraic structure is outside of the universal algebra definition without producing an original authoritative source would be in violation of Wikipedia:No_original_research. caterpillar_tree (talk) 03:57, 25 January 2022 (UTC)

You are pointing a difficulty that arises rather commonly when writing encyclopedic articles in mathematics: the term "algebraic structure" is commonly used in mathematics, mathematicians knows what is an algebraic structure, but there is no consensus for a formal definition. Universal algebra is a tentative for such a definition, but fields are not algebraic structures for it, as the field axioms require an existential quantifier. Category theory provides another definition, which does not distinguish algebraic structures from non-algebraic ones. Also, there is no consensus whether partial orders and graphs are algebraic structures. I agree that Algebraic structure is not well sourced and requires improvements. In particular, it lacks a discussion of the various definitions that have been given, and examples illustrating the differences between these definitions. Nevertheless, IMO, there is not too much to change in the lead. D.Lazard (talk) 11:16, 25 January 2022 (UTC)

I confer that the current introductory paragraph may be considered misleading for various reasons.

For instance, "algebraic structure" very often (probably more often than not) refers to the structure itself (the notion roughly equivalent to the class of the algebraic systems embodying it) than to a specific instance of such a structure.

Furthermore, algebraic structure is a notion more abstract than a specific way to present it with a concrete set of axioms. For example, group (or field) as an algebraic structure may be defined using different signatures.

And, by the way, signatures of algebraic structures may contain not only operations, but also relations: such are, for instance, the more common, natural definitions of (various types of) lattices and semi-lattices. Filozofo (talk) 11:51, 25 January 2022 (UTC)

I agree that it is ambiguous whether "algebraic structure" refers to a structure (group in general) or to a specific realization of the structure (a specific group). Imo, the distinction is too technical for the lead, but a specific section would be needed for discussing this.

The last paragraph of your post points the fact that there is no consensus on the exact meaning of "algebraic structure": If relations are accepted, a partial order and a graph are algebraic structures. Otherwise they are not. This is the implicit choice of the article, and this requires to be clarified. D.Lazard (talk) 13:55, 25 January 2022 (UTC)

Mentioning the adjective "Closed" in the definition

@D.Lazard: Hi again. No, every operation is not closed within a set, for example "minus" is an operation, but is not closed within natural numbers, e.g., 2 and 3 are natural numbers but -1 (2-3=-1) is not a natural number.

Here https://www.geeksforgeeks.org/groups-discrete-mathematics/ explicitly mentions the adjective "Closure" for algebraic structures. And here www.javatpoint.com/algebraic-structure-in-discrete-mathematics implicitly says an algebraic structure must be closed. But explicitly says that the operation must be closed (closure property) for Semigroup, Monoid, Group, and Abelian Groups. This way, I really think that this definition lacks an important property of algebraic structure's operation, and in the definition, the adjective "closed" must be preceded the word "operation". Hooman Mallahzadeh (talk) 15:15, 30 May 2022 (UTC)

Minus is not an operation in the sense of Operation (mathematics). However, it is a partial operation. I know that some US teachers insist on "closure", but this seems specific to North America and to low level courses. Here, "operation" is linked to a mathematical definition, which, as far as I know, is the most common one; if you add "closed", this means that you consider another definition of "operation"; this may clearly be confusing for some readers. In other words, adding "closed" would need to first get a consensus for changing the definition given in Operation (mathematics).

Even if one considers, as you do implicitly, that an operation may be a partial operation, "closed" must not be added here, as this would imply that, in a field, division is not an operation (see § Non-equational axioms). So, even if there may be some ambiguity in the meaning of "operation", disambiguating here is more harmful than keeping the ambiguity. D.Lazard (talk) 17:35, 30 May 2022 (UTC)