This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me for quite some time and I'm not entirely certain that it's completely worthless, which is why I've decided to ask it here.

Why are groups so immensely more important in mathematics and its applications than semigroups are? I know little of group theory, mathematics and its applications, so I cannot understand how much more important they are. But I know they are because I see how many more people are interested in groups than in semigroups.

I wouldn't ask this question if this fact didn't seem a little strange to me. I know that groups are associated with symmetries, or with automorphisms of structures. Cayley's Theorem tells me that every group can be seen as a set (closed with respect to taking compositions and inverses) of automorphisms of a structure with no constants, functions or relations, i.e. a plain set. I know that the automorphisms of a vector space, a module or a group form a group. Obviously, automorphisms are important.

Then we have inverse semigroups. These are less popular but this I can understand. They are associated with partial symmetries, by the Wagner-Preston Theorem. Partial functions do seem much less used than functions.

But then come semigroups. Just like in the two previous cases, there is an "embedding theorem", which says that every semigroup can be embedded in a semigroup of maps from a certain set into itself. And, analogously to the case of groups, the endomorphisms of a vector space, a module or a group form a semigroup.

This seems to say that semigroups are to endomorphisms what groups are to automorphisms. The "conclusion" would be that

$\frac{\mbox{the importance of semigroups}}{\mbox{the importance of endomorphisms}}=\frac{\mbox{the importance of groups}}{\mbox{the importance of automorphisms}}$

Assuming that endomorphisms are about as important as automorphisms, we get that semigroups are about as important as groups. My feeling is that the assumption is correct.

I understand that I must be oversimplifying something at some point. But what am I oversimplifying and where?

I realize that this is possibly a very dumb question, but my confusion is genuine.

Edit: I have realized, after reading your answers and comments, that I made the mistake of using a very vague term without even attempting to define it. The problem seems to be what importance is. Is it popularity or usefulness, or being used a lot, or interesting or being interesting, or something else, or a mixture of many traits? I'm not going to force my understanding of the word now. But if someone still wants to answer this question, perhaps it would be good idea if the answer contained an attempt at clearing this up. (Or maybe not. I'm not sure!)

Edit: A question similar to mine is dealt with here.

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    $\begingroup$ +1 Please don't hesitate to ask questions like this. Because mathematicians do not usually speak on such topics in more formal settings, it is only natural that they arise in forums like this one. Such questions can certainly lead to very valuable contributions. $\endgroup$ – Bill Dubuque Jan 23 '12 at 1:36
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    $\begingroup$ My immediate reaction is that I hear more about monoids than about semigroups. That would even fit your analogy better, because when you're looking at endomorphisms of some structure, the identity map surely is included. Furthermore, a monoid is just a category with one object, and categories are hugely popular and important. It might be that most of the interesting things there are to say about monoids can be generalized to categories and therefore are. $\endgroup$ – hmakholm left over Monica Jan 23 '12 at 1:45
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    $\begingroup$ Dear ymar, This question is somewhat related to this earlier one, and my answer there provides a partial answer to your question. Regards, $\endgroup$ – Matt E Jan 23 '12 at 4:09
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    $\begingroup$ #MattE Thank you. I have read it and upvoted! $\endgroup$ – user23211 Jan 23 '12 at 12:00
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    $\begingroup$ @user1729 If you consider white-black move-pairs as transformations between board states, then you have a monoid of transformations. Some move pairs can be undone, some are permanent. This reflects the type of irreversibility you get when talking about left ideals in a monoid of transformations. After you make pawn transformations, you are forever "trapped" in a subset of the whole space of transformations (the ideal, which absorbs further operations on the left). If all moves were reversible, then you could freely travel between board states. $\endgroup$ – rschwieb Jan 7 '16 at 17:55

To add a remark related to Jim Belk's answer and the OP's comments on that answer:

In many naturally occurring situations, including some of those where group theory is particularly useful, endomorphisms are automatically automorphisms.

For example, if $E/F$ is a finite extension of fields, any endomorphism of $E$ which is the identity on $F$ is automatically an automorphism of $E$.

As another example, if $C$ is a Riemann surface of genus at least $2$, then any (nonconstant) endomorphism of $C$ is necessarily an automorphism.

Any endomorphism of a Euclidean space which preserves lengths is necessarily an automorphism.

Another point to bear in mind is that the groups that arise in practice in geometry are often Lie groups (i.e. have a compatible topological, even smooth manifold, structure). One can define a more general notion of Lie semigroup, but if your Lie semigroup has an identity (so is a Lie monoid) and the semigroup structure is non-degenerate in some n.h. of the identity, then Lie semigroup will automatically be a Lie group (at least in a n.h. of the identity). A related remark: in the definition of a formal group, there is no need to include an explicit axiom about the existence of inverses.

To make a point related to Qiaochu Yuan's answer: in some contexts semigroups do appear naturally.

For example, studying the rings of endomorphisms of an object is a very common technique in lots of areas of mathematics. (E.g., just to make a connection to my first point, for genus $1$ Riemann surfaces, there can be endomorphisms that aren't automorphisms, but then genus $1$ Riemann surfaces can also be naturally made into abelian groups --- so-called elliptic curves --- and there is a whole theory, the theory of complex multiplication, devoted to studying their endomorphisms rings.)

As another example, any ring of char. $p > 0$ has a Frobenius endomorphism, which is not an automorphism in general; but the semigroup of endomorphisms that it generates is typically an important thing to consider in char. $p$ algebra and geometry. (Of course, this semigroup is just a quotient of $\mathbb N$.)

One thing to bear in mind is what you hope to achieve by considering the group/semigroup of automorphisms/endomorphisms.

A typical advantage of groups is that they admit a surprisingly rigid theory (e.g. semisimple Lie groups can be completely classified; finite simple groups can be completely classified), and so if you discover a group lurking in your particular mathematical context, it might be an already well-known object, or at least there might be a lot of known theory that you can apply to it to obtain greater insight into your particular situation.

Semigroups are much less rigid, and there is often correspondingly less that can be leveraged out of discovering a semigroup lurking in your particular context. But this is not always true; rings are certainly well-studied, and the appearance of a given ring in some context can often be leveraged to much advantage.

A dynamical system involving just one process can be thought of as an action of the semigroup $\mathbb N$. Here there is not that much to be obtained from the general theory of semigroups, but this is a frequently studied context. (Just to give a perhaps non-standard example, the Frobenius endomorphism of a char. $p$ ring is such a dynamical system.) But, in such contexts, precisely because general semigroup theory doesn't help much, the tools used will be different.

E.g. in topology, the Lefschetz fixed point theorem is a typical tool that is used to study an endomorphism of (i.e. discrete dynamical system on) a topological space. Interestingly, the same formula is used to study the action of Frobenius in char. $p$ geometry (see the Weil conjectures). So even in contexts such as action of the semigroup $\mathbb N$, there is some coherent philosophy that can be discerned --- it is just that it is supplied by topology rather than algebra, since the algebra doesn't have all that much to say.

I think the conclusion to be drawn is not to be too doctrinaire, and to be sensitive to the actual mathematical contexts in which and from which the various notions of group, semigroup, automorphism, and endomorphism arise and have arisen.

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    $\begingroup$ This is a great answer, thank you. I can't say I understand every word you've written but it did give me a lot of food for thought. $\endgroup$ – user23211 Jan 24 '12 at 0:05

It is true that groups are much more important in mathematics than semigroups. There are two basic reasons for this:

  1. Groups are the mathematical embodiment the concept of symmetry. In the examples you give, you seem to equate the idea of symmetry with automorphisms of algebraic objects. However, most symmetries that mathematicians care about are geometric or analytic, such as the rigid symmetries of a polyhedron, the deck transformations of a covering space, the automorphisms of a graph, the continuous symmetries of a system of differental equations, the isometries of a metric space, and so forth. In these cases, there isn't an obvious analogue of "endomorphism" that could be used to form a semigroup. Even if there were, the semigroup wouldn't be as useful, because the most interesting endomorphisms are the automorphisms.

  2. Algebraic objects tend to have many more endomorphisms than automorphisms, which makes it much easier to understand the automorphism group than the endomorphism semigroup. Moreover, the definition of a group is "rigid" enough that it leads to a rich structure theory, which makes it much easier to investigate the structure of a given group than the structure of a given semigroup. Indeed, one of the first things you would want to understand for a semigroup would be the structure of the group of units.

Of these two reasons, I would say that the first explains why groups are so much more important to mathematics in general, while the second helps to explain why groups are more important even within the context of abstract algebra.

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  • $\begingroup$ Ad 1. I was determined, when I wrote my question, to make sure I'm not saying "automorphisms of algebraic structures". I did use examples from algebra because I know too little about other examples to be sure I'm not talking nonsense. What I did mean by "automorphisms" is something that I believe category theory deals with. I know nothing about category theory so I didn't use the term in my question either. $\endgroup$ – user23211 Jan 23 '12 at 11:46
  • $\begingroup$ And to add to my previous comment, I know that category theory has the concept of endomorphism, so I would guess it should be appliable outside algebra. I can't know exactly where of course because I only have a vague understanding of what categories and morphisms are. $\endgroup$ – user23211 Jan 23 '12 at 11:59
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    $\begingroup$ This seems more like an argument/explanation for the ubiquity or convenience of groups, rather than importance per se. As for part (1), which I agree with to som extent, what about the inverse semigroups that arise from groupies of partial symmetries? $\endgroup$ – user16299 Jan 23 '12 at 16:30
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    $\begingroup$ that should have been "groupoids" - curse this gutter-minded autocorrect ... $\endgroup$ – user16299 Jan 23 '12 at 16:50

I disagree that groups are more important than semigroups. For example, the multiplication on a ring without identity turns it into a semigroup, and rings are incredibly important in mathematics. In fact a ring without identity is nothing more than a semigroup internal to the category of abelian groups with tensor product.

What I do believe to be the case is that groups were the first to be historically studied because it is more natural to think about isomorphisms than endomorphisms (it is not at all obvious from the perspective of our mathematical ancestors that a non-isomorphism is a useful thing to think about) and that groups are easier to study since they have more structure (e.g. the representation theory of finite groups is much easier than that of monoids or semigroups; see this question).

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    $\begingroup$ @ymar: the larger point is that when we study rings we acknowledge that endomorphisms are important; if we like monoids because we like endomorphisms of sets, then we like rings because we like endomorphisms of abelian groups, and there is a Cayley's theorem for rings to this effect. $\endgroup$ – Qiaochu Yuan Jan 23 '12 at 1:47
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    $\begingroup$ @ymar: it is not particularly hard to work out. Any ring embeds into the endomorphism ring of its underlying abelian group by left multiplication. $\endgroup$ – Qiaochu Yuan Jan 23 '12 at 1:49
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    $\begingroup$ @Mariano: sure. For me "ring" means "ring with unit" by default and I use "rng" to denote "rings without unit" because I think it's cute. $\endgroup$ – Qiaochu Yuan Jan 23 '12 at 2:14
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    $\begingroup$ I don't agree that just because "the multiplication on a ring without identity turns it into a semigroup, and rings are incredibly important in mathematics" then that automatically makes semigroups important. Just because semigroups arise in some context that is important doesn't automatically make semigroups (as an independent subject of study) important. It seems sort of similar to saying "the integers form a group under addition, and the integers are incredibly important in mathematics" so that makes groups important. $\endgroup$ – Ted Jan 23 '12 at 2:48
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    $\begingroup$ @Ted: again, the larger point is not only that you can form a semigroup (in fact a monoid for rings with identity, so I'll just say that) from a ring, but that a ring is the same thing as a monoid internal to abelian groups: there is a two-way implication here. Any monoidal category (en.wikipedia.org/wiki/Monoidal_category) carries an internal notion of monoid, and it's a funny fact about mathematics that historically the most important groups have been the ones in $\text{Set}$ while the most important monoids have been the ones in $\text{Ab}$. $\endgroup$ – Qiaochu Yuan Jan 23 '12 at 2:50

I had thought about this for a long time, this is what I got.

I don't think that semigroups are less important than groups, mostly because groups are semigroups too. I think that many people tend to disregard semigroups because we cannot prove lot of staff with semigroups axioms, by the way the same applies to groups axioms: every equation in a group has always a unique solution, group homomorphisms preserve identities and inverses, and few more. Many other theorems can be proven if we add some hypothesis, for instance if we add that a group is also finite we can prove a lot of stuff (Lagrange, Cauchy anb Sylow theorem for instance), other things can be proven if we require that our group has also a topological structure compatible with the group-structure. The point is that to prove significant results in all mathematics we have to add more properties then the axioms of groups (but same holds for other structures as topological spaces). So to produce significant results about semigroups we have to consider semigroups with more structure then those of semigroups: one way is to add properties to the operation (for instance adding identity or inverses), another way is to give an action (or representation) of our semigroup (for example if we made act our monoid on an abelian group (with the same underlying set) we get a rings [with or without identities, depending if our semigroups are monoids too]).

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    $\begingroup$ To build on what you've said... Group theory is a subtheory of the theory of semigroups, so groups can only end more important than semigroups if a subtheory S of a theory T can come as equally important to the theory T. This will only happen, I think, if the complementary subtheory of S, T-S has no importance whatsoever. In this case, the parts of the theory of semigroups which don't concern groups, do have importance, so the complementary subtheory has importance. Thus, one might reason that the theory of semigroups has more importance. $\endgroup$ – Doug Spoonwood Feb 4 '12 at 20:58
  • $\begingroup$ @ymar: For the same reason, time zone, I did not see this question earlier. I agree with what ineff said. I believe there are more studies on groups than semigroups because semigroups lack of inverses. That's why inverse semigroups play a significant role in semigroup theory. I myself pay attention to Lallement's book because it contains applications so I have things to play with. To answer your question, all groups are semigroups. Groups are special cases of semigroups. Group theory is a branch of semigroup theory. $\endgroup$ – scaaahu May 1 '12 at 4:25

I would take a step back and talk about magmas. A magma $M$ is a set of objects and a "product" $*\colon M\times M \rightarrow M$. We can think of a magma as a collection of (binary) trees for some set. In particular, a magma is basically a notion that describes "syntax" or "position relations" for some class of objects. But the magma axioms are very weak. A magma includes all the "position relations", and none of the elements are (necessarily) special in any way. In fact, just about the only thing you can prove about all magmas at once is that $x = x$.

Note that a semigroup is an associative magma. In particular, that means that some trees are to be treated as equal to other trees. Super, but none of the elements are (necessarily) special in anyway.

In the magma $\rightarrow$ semigroup $\rightarrow$ group hierarchy, the group is the first class which imposes special conditions on its objects, and not merely the product. It captures some notion of "semantics" for the elements beyond merely saying that "these two paths go to the same object". (Arguably, the elements themselves are "path-like" objects, and group theory is what we get when these two distinct notions of "path-ness" interact)

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An arbitrary semigroup may lack an identity (or left or right element) element and/or inverse elemments for every element in the semigroup. This prevents such a rich theory without other constraints on the semigroup.

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    $\begingroup$ This doesn't really suffice as an answer. For example I could just as easily argue that because a group lacks the second operation and the extra properties a ring has, that group theory is prevented from being as rich as ring theory (which is clearly not the case). $\endgroup$ – Zev Chonoles Jan 15 '13 at 17:58

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