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I would like to propose removing Magma, Abelian group, Ring with unity, commutative ring, and domain: they are too specific. I would like to propose adding algebra, module and lattice: they are glaring omissions. Rschwieb (talk) 20:55, 5 March 2012 (UTC)[reply]
I'm changing my position a bit. User:Jowa fan has been proposed that the specific rings and specific groups be moved into collapsed sections. I like this because it allows fields and integral domains to stay, but to be off equal footing with "ring" itself. I still support removing "magma" and "ring with unity", and the addition of the glaring omissions. Rschwieb (talk) 02:08, 6 March 2012 (UTC)[reply]
Yes, that would fit under module in the proposed scheme. I created a starting point below. I'm struggling on what to do with semigroup and monoid. I recognize their importance, but I don't know if or how they should be included. Either semigroup would be the parent with group underneath, or group would be the parent of its own branch. Comments? Rschwieb (talk) 14:32, 6 March 2012 (UTC)[reply]
Presently, the template considers only algebraic structures in which only one set is involved. If this choice is kept, this should appear clearly and explicitly, and, at least, lattice should be added. If structures involving more sets are to be added, the template should list not only module, vector space and algebra, but also ordered set, graph and tree, which are all algebraic structures which may defined by a binary operation with values in {true, false}. It is a pity that algebraic structure does not mention these important algebraic structures. D.Lazard (talk) 14:36, 6 March 2012 (UTC)[reply]
Certainly they can be added, as long as we're sure there is some consensus they are algebraic. If there isn't any controversy I'll stick them in. I cannot immediately see how the treeness of a graph is defined algebraically. Please see/help with the draft below. Rschwieb (talk) 21:24, 6 March 2012 (UTC)[reply]
I cannot immediately see how the treeness of a graph is defined algebraically. There is a point here. Thus I agree to remove tree from the template. But this point deserves to be further analyzed. The starting question is "What does mean defined algebraically?" Thinking about that, I convincing myself that, while most mathematicians well know what is and what is not algebraic, there is no explicit consensus for a formal definition. The same is true for algebraic structure. I have reread this page. It appears that the definition given there suppose that the axioms of the structure should be formulas of propositional calculus, i.e. quantifier free formulas. For many mathematicians, in particular those which are also involved in computer science, predicates of first order logic are acceptable for defining algebraic structures. For the definition of the page, tree and Archimedean fields are not algebraic structures, although they are for most mathematicians and they may be defined with axioms of the first order logic. Thus, in this sense, Algebraic structure is original research (or non neutral point of view). To solve this, I think that the lead of Algebraic structure should be rewritten to quote that this is a notion which is informally used by most mathematicians, and which has been formalized in several ways. One of these formalization occurs in universal algebra and is described in the page. Another formalization is the theory of abstract data types.
I'll try, later, to propose such a clarification of the page
It's seeming like orders and graphs are the most debatable of the structures below, so I put question marks next to them. I think everything else though will appear universally in algebra books, so they are not likely to be challenged.
I find it amazing that a google search for "What is an algebraic structure?" turns up so few results. Here is a concrete one (in a very simple looking book) that might serve as a reference, but I'd prefer more. The two pages I found addressing the question directly seemed to agree that an algebraic structure was a set with n-ary operations and algebraic identities. That can probably be expressed categorically somehow in terms of a set and diagrams. Maybe a category theorist would be a good person to ask. Rschwieb (talk) 14:44, 8 March 2012 (UTC)[reply]
P.M. Cohn's Universal Algebra book defines an algebraic object to be "a set with finitary operations", but I can't see enough of the preview to decide how modules fit into that. Rschwieb (talk) 17:39, 8 March 2012 (UTC)[reply]
Nope for ordered set and graph – the former is an order structure, not algebraic. Simple graph is no more an algebraic structure than any binary relation-based one; though multigraphs use natural numbers instead of truth values. But lattice is usually classified as algebraic, which is reflected in the term "Boolean algebra (structure)". Incnis Mrsi (talk) 17:04, 8 March 2012 (UTC)[reply]
Modified to remove ordered set and graph. I'll see if I can figure out how to implement this in the template. Rschwieb (talk) 13:45, 9 March 2012 (UTC)[reply]
As I was looking at the calculus template, I didn't like it as much, and started shopping around. I like the Lie algebra template. Struggling to come up with suitable header names, I just changed the sketch below. As you can see the headers have been dubbed "(structure)-like". It's just an idea, we can change it all back if there is a lot of disagreement. I am trying to avoid heading categories with names that are too general (for example, a person unfamiliar with monoids would not expect groups to be under "monoid") and on the other hand I'm trying to avoid using headers that are too specific, because it's problematic to list something more general underneath it. The (structure)-like scheme is a possible solution which I hope you are able to comment on. Rschwieb (talk) 14:12, 9 March 2012 (UTC)[reply]
Rings
A ring without unity is called pseudo-ring in this Wikipedia. Just ring (mathematics) defaults to rings with unity. Possibly, we should explicitly specify "ring without unity" and "ring with unity" in display forms for corresponding links. Incnis Mrsi (talk) 14:53, 9 March 2012 (UTC)[reply]
IMO, the differences in theory between rings with/without unity are not enough to warrant mentioning rngs independently. (In contrast, commutative ring theory has flavor completely different from noncommutative, sometimes.) I'm interested in seeing more opinions from others on this point, though. Rschwieb (talk) 15:01, 9 March 2012 (UTC)[reply]
I agree with Rschwieb. This is enforced by Pseudo-ring#Adjoining an identity element, which, essentially, shows that every result on pseudo-rings may restated mechanically as a result on rings and ideals. In other words, pseudo-ring theory is a trivial theory without any specific interest. D.Lazard (talk) 15:59, 9 March 2012 (UTC)[reply]
(structure)-like headers
I'm still looking for feedback for whether or not the idea of using "grouplike" "ringlike" is a good one or a bad one. Rschwieb (talk) 13:44, 11 March 2012 (UTC)[reply]
Another thing: does anyone think that the template might look good uncollapsed? Rschwieb (talk) 13:19, 12 March 2012 (UTC)[reply]
I think that loops and quasigroups are rather rarely talked about, but I'm interested in having at least one up there. Monoids and semigroups are the same sort of situation, except they are both more common. It seems silly though to have things on different lines just because they have identity, though. I'd like to put these pairs in the same line. I'm also fixing the link to "loop" because "loop group" was not the intended target. "Loop" is supposed to be a quasigroup with identity. Rschwieb (talk) 19:31, 10 April 2012 (UTC)[reply]
Presently, the template does not include the algebraic structures that are widely used outside pure abstract algebra. An incomplete list appears in the heading. I propose to add the three first ones in the template and will do it. I'll not include "Boolean algebra", because "Boolean ring" may suffice, and it is not clear if it should be included in "algebra like", "ring like" or "lattice like" section. There are other algebraic structures of interest, such as Jordan algebras and Clifford algebras. My opinion is not to insert them, as they are of a much more specialized usage. D.Lazard (talk) 13:18, 25 May 2014 (UTC)[reply]
Of course. I have added it. D.Lazard (talk) 15:36, 30 April 2016 (UTC)[reply]
D.LazardJimw338 I don't know, I feel like division rings aren't a good candidate. A crude way to express it is that they are "a little different than fields but way less prominent." I just don't want it to turn into "scope creep" of the template. This case is border-line, so I don't intend to do anything about it. I just want to ward off bursts of over-specific entries. Thanks. Rschwieb (talk) 18:53, 2 August 2016 (UTC)[reply]