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
the importance of semigroupsthe importance of endomorphisms=the importance of groupsthe 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.
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 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 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 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.