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I am a second-year graduate student of pure mathematics. Like most graduate students, I have been exposed to many types of algebraic structures, and it seems standard for the major emphasis, if not all the emphasis, to be on groups, rings, R-modules, and categories. These are very rich structures with very interesting properties, but, when I zoom out to see the big picture of algebraic structures, I have always wondered why a particular set of defining properties make for such a rich structure, while another set of properties gives less "interesting" structures, or nothing worth teaching at all.

For a motivating example, at least one operation over a set (or class, whatever) that is closed under the operation definitely seems necessary to talk about anything meaningful; however, why is the particular combination of 1.having inverse elements, 2.having an identity element, and 3.associativity so much more rich (a group) than simply replacing associativity with commutativity (a structure I don't even know a name for)? In fact, a good question I have always had is why associativity is so much more prevalent and necessary than commutativity? As another motivating example, we are taught much about groups and rings, but why not loops, monoids, semilattices, and near-rings? What makes the former set either richer in structure or more pedagogically sound to teach about?

Finally, along the same path, I predict many answers will mention how the definitions for certain objects are better stated in category-theoretic terms (a group is a groupoid with one object); however, this simply moves the problem, because I can ask what makes the specific combination of defining properties of a category so great?—why associativity and not commutativity?—why categories and not semi categories? Indeed, categories are extremely powerful objects, I guess just don't know why the particular combination of its defining properties is much more "powerful", deep, and pervasive than another combination of properties.

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Welcome to math.SE Alex –  magma Jan 28 at 23:25
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While I don't have a full answer, I would like to point out that associativity is what allows you to move your focus around when solving equations. Commutative but non-associative operations only let you switch two things around, not several in a row, so you barely gain any freedom. Associativity without commutativity is much less restricting. Also function composition is associative, (regardless of formalism; any sensible definition would be) so any algebra that can be interpreted as a collection of transformations (which is a large number of them) has to be associative. –  Robert Mastragostino Jan 28 at 23:30
    
Thank you. Inspired by your line of thinking, I think there is another reason to favor associativity over nonassociativity. In nonassociative operations, there is a prescribed or intended order, but any prescribed or intended order can be imposed on associative operations using parentheses. In this particular sense, it seems prescribing an operation to be associative, ironically, makes for a more general object than not prescribing such a condition. –  Just Some Old Man Jan 29 at 1:59
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You may be interested in the notion of a commutative algebraic theory and its generalizations. Basically, if we have an algebraic theory with a set of function symbols $\mathcal{F}$ (not necessarily unary; possibly nullary), then a schema of axioms asserting that some $f \in \mathcal{F}$ "commutes with" every element of $\mathcal{F}$, including itself, is interesting. Abelian groups (think $\mathcal{F} = \{0,\-,+\}$) fall into this category bigtime; every $f \in \mathcal{F}$ commutes with all three function symbols. –  goblin Jan 29 at 11:04
    
For a simple example of a commutative algebraic theory, see medial magma. –  goblin Jan 29 at 11:07

4 Answers 4

up vote 5 down vote accepted

Examples!

Remember that most (if not all) abstract structures are motivated by specific examples. And it took a long time for the mathematicians to abstract from these examples and develop an axiomatic framework. Since you have been asking for groups: Permutation groups, Symmetry groups, Lie groups (aka transformation groups) and ideal class groups appeared naturally in the 19th century, even before the general notion of a group was born (Cayley, Galois, Klein, Kronecker, Lie, and many others). There is nothing interesting about the group axioms in themselves, but rather in the fact that they subsume what happens in so many examples, and that we can study many phenomena in specific examples for arbitrary groups. The same remarks apply - even more - to the notion of a category.

Monoids also appear very naturally in many examples. They have a rich theory, quite different from the theory of groups. But in general I would say monoids are harder to understand than groups. For example, whereas finitely generated commutative groups are classified, this is not the case for finitely generated commutative monoids. For this reason one often makes a monoid to a group by formally introducing inverses - this is called the Grothendieck construction, which is especially important in K-theory.

Monoids even play a more important rule when we internalize them into arbitrary monoidal categories - this leads to the notion of a monoid object. Monoid objects in $\mathsf{Set}$ are monoids in the usual sense, but monoid objects in $\mathsf{Ab}$ are rings in the usual sense! This offers a considerable overlap between monoid theory and ring theory. In the commutative case, we can even go further and develop algebraic geometry for commutative monoid objects (Toen-Vaquié, Florian Marty).

I haven't worked with loops or near-rings, but I am pretty sure that these aren't covered in most lectures because there are not as many interesting examples as for groups and rings.

The conclusion is very simple: Abstract structures are motivated by specific examples. And this is not restricted to algebra. You could also go ahead and ask "why the union axiom in the definition of a topology?". The answer is the same: Because examples (especially the class of metric spaces) have motivated this axiom. Given a random system of operations and rules between them, you cannot really tell if this is interesting, natural, or not.

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I think that by the Grothendieck construction you mean the Grothendieck group. –  goblin Jan 29 at 10:01

Well, the basic binary operations that we tend to understand more deeply are addition, multiplication of numbers or matrices, composition of functions, concatenation of finite sequences (strings) and intersection, union of sets, say. All these are associative.

Associativity is very natural: it allows one to apply the operation on more than two terms, that is, an associative binary operation generates a unique $n$-ary operation for all $n$.

However, of course, not every operation that occurs is associative, but all these theories must use a lot of brackets (or convention how to place the brackets if omitted). For example, the exponentiation is not associative: $$(x^y)^z\,\ne\,x^{(y^z)}\,.$$ Or the Cayley numbers over the quaternions. See also Lie algebras.

Commutativity combined with associativity also permits you to permute the terms: $$a_1a_2\dots a_n\ =\ a_{\sigma(1)}a_{\sigma(2)}\dots a_{\sigma(n)}$$ where $\sigma$ is a permutation of indices $\{1,2,\dots,n\}$.

Without associativity, it is much more useless. At least, so to say, it is not deeply studied at all, probably because of lack of real motivation.


Let me show instead one more property of operations that has importance in category theory:

Let a binary relation $*$ satisfy the following interchange property $$(a*b)*(c*d)\ =\ (a*c)*(b*d)\,,$$ then the category $\mathcal V$ of such structures is self-enriched, in the sense that the pointwise $*$ operation will map a pair of homomorphism to a homomorphism (just like in the case of Abelian groups).

(Note also that if $*$ has a unit element, then $*$ is already both commutative and associative.)

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the operation you mention is typical of medial magmas, right? en.wikipedia.org/wiki/Medial_magma –  magma Jan 29 at 1:00
    
Yes, it is just that. –  Berci Jan 29 at 1:14

This is a very soft question in my opinion, hence I added a the tag.

Here are some - soft- considerations.

The simplest algebraic structures with a binary operation are called magmas, and altough they may not be as popular as groups or rings they have their fans and a relatively large theory, considering how simple their axioms are. Some algebraic structures were studied earlier - and are now more popular - simply because they were used to solve problems in other areas of mathematics or geometry or physics. Think at groups. Later came rings and vector spaces. Modules came only recently, followed by categories and now even more general structures exist.

Categorically all algebras are algebras for a monad so, depending on the monad, you may get different properties. Associativity tends to be a desirable property, probably just because people like to iterate their operations and getting rid of parenthesis is a bonus. The interesting thing is that in physics many operations are indeed associative, thus reinforcing the pure mathematical notion.

Ultimately students and researchers tend to gravitate around tools (algebraic structures, in this case) that seem to be the most effective in solving real world problems, and this includes problems in other areas of mathematics

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by the way, magmas have no connection with my nickname :-) –  magma Jan 29 at 1:01
    
Real-world problems—well, I guess pure mathematics may not have been as pure as I thought. Even pure mathematicians have an indirect connection to the practical, for they must be employed by society! :) I do wonder, though it may me harder, could the reverse process be profitable; should one embark on constructing a seemingly useless rich theory, then find real world applications for it to apply to? –  Just Some Old Man Jan 29 at 1:20
    
experience shows that sooner or later every branch of pure mathematics is applied in some - often unexpected - context. Think at G.H. Hardy: en.wikipedia.org/wiki/G._H._Hardy. Number theory, the purest of all mathematics, is now routinely applied to criptography. –  magma Jan 29 at 1:48
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@magma, I think its a bit rich to say that every branch of pure mathematics is applied in some context eventually. Surely there are branches that are never applied, or whose usefulness is far exceeded by the amount of effort that was put in to develop those areas. –  goblin Jan 30 at 6:38

Since you already got 3 deep and pervasive answers, I want to concentrate on a single minor point.

I can ask what makes the specific combination of defining properties of a category so great?

In my humble opinion, almost nothing: you will certainly be interested in the work a friend of mine and I (in minor extent) are laying down where we study multi-object partial magmas and recover a great deal of classical "category" theory: you get these things called plots where your composition is not defined for each pair of consecutive arrows, and when it's defined it is possibly non-associative. Finally, you don't have identities everywhere.

"You fool, nothing good can come out of these poorly behaved, thorny things!"

:) not at all.

You can define "isomorphisms" (yes, without identities), and notice that "being an isomorphism" and "admitting an inverse" are different notions in this world, and that they collapse in the categorical world (a category is an associative plot, where the composition is defined and every object has a 1, in the same vein a monoid is an extremely smooth partial magma). You can then define isoids, i.e. plots where every arrow is an isomorphism.

We're even able to define morphisms of plots (p-unctors), natural transformations (trimmings, if I remember well the name Salvatore and I chose), adjoints, limits, and a chain of free-forgetful adjunctions which connects the category (it is a category) of plots to the category of associative plots, semicategories [in this case we've even two different adjunctions for two different fully faithful embeddings], and categories. Other things in the to-do list: what's a $n$-plot? How can one define localization of a plot with respect to a family of arrows? How about simplicial stuff, how about enrichments (whatever this means)?

"You are only children playing with symbols! Examples, Examples!!"

:) Functional analysis and symplectic geometry provide "natural factories" of examples of such structures. For example, one of our two unitization functors applied to the category of symplectic relations gives precisely the Woodward-Wehrheim category.

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