# Why semigroups could be important?

There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in Mathematics? I understand that it is a research question, but may be somebody can hint me the direction to look on so that I would see sensibility of semigroups, if you see what I mean (so some replies like look for wikipedia are not working as they are anti-answers).

• This is an extremely general question, given that whole books have been written about semigroups or monoids. Feb 18 '11 at 18:21
• @Jim: Yes, perhaps this question asks for a direction where to begin ... Feb 18 '11 at 18:24
• @Victor: The wikipedia article shows that semigroups are important in applied mathematics, PDE, theoretial computer science and probability theory ... perhaps you should explain why you think that this is not evidence enough for their importance. Check also related articles such as en.wikipedia.org/wiki/C0-semigroup Feb 18 '11 at 18:30
• @Martin: ok, in PDE's, in computer science (say, particularly, semi-Thue systems), the so-called Cuntz-algebras in C*-algebras etc. -- all this is where semigroups appear is like there appears something with associative operation and that's it, no further miracle of that we then go looking what is going on with those semigroups with further yielding nice results about our initial problem. And, C_0-semigroups aren't really semigroups :-) I'm not saying that semigroups are not important, I just wonder if somebody knows where semigroups can do some tricks (hence it's a good research question) Feb 18 '11 at 18:41
• Shouldn't this be titled "Why are semigroups important?" :) Feb 18 '11 at 18:55

Am slightly surprised no one has mentioned the Galvin-Glazer proof of Hindman's theorem via the existence of semigroup structure on $\beta{\mathbb N}$, the Stone-Cech compactification of the positive integers (see, for instance, part of this note by Hindman.

The relevance to the original question is that knowing that compact right topological semigroups have idempotents'' may sound recondite, but it is just what was needed to answer Galvin's original question about translation-invariant ultrafilters, which was itself motivated by a "concrete" question in additive combinatorics.

On a related note, while it is in general not possible to embed a locally compact group as a dense subgroup of something compact (the map from a group to its Bohr compactification need not be injective), you can always embed it densely into various semigroups equipped with topological structure that interacts with the semigroup action: there are various of these, perhaps the most common being the WAP-compactification and the LUC-compactification. Unfortunately this often says more about the complicated behaviour of compact semitopological semigroups (and their one-sided versions) than about anything true for all locally compact groups, but the compactifications are a useful resource in some problems in analysis, and the semigroup structure gives one some extra grip on how points in this compactification behave. (Disclaimer: this is rather off my own fields of core competence.)

• +1: I was thinking of remarking on this. Furstenberg and Katznelson's paper on idempotents in compact semigroups (math.stanford.edu/~katznel/colorla.pdf) is amazing. Feb 26 '11 at 1:38
• The link in the post is now dead. I suppose it is supposed to be link to the paper Neil Hindman: Algebra in the Stone-Cech compactification and its applications to Ramsey theory, Scientiae Mathematicae Japonicae 62 (2005), 321-329. Here is Wayback Machine version and als a new link - nhindman.us/research/jams.pdf Jan 3 '18 at 12:11
• Thanks Martin for pointing out the linkrot. It also gives me an excuse to reread Hindman's entertaining and informative survey! Jan 3 '18 at 14:59

Semigroups provide a fundamental, algebraic tool in the analysis of regular languages and finite automata. This book chapter (pdf) by J-E Pin gives a brief overview of this area.

• I overlooked this answer while writing my own; Pin's book is a nice reference, thanks for mentioning it. Feb 18 '11 at 19:06

Victor, I don't understand your claim that $C^0$-semigroups aren't really semigroups. You are not free to decide for all the mathematical community what is a semigroup (I guess that you are interested only on discrete semigroups, aren't you ?).

$C^0$-semigroups are fundamental in PDEs (in probability too as mentioned by Steinhurst). The reason is that a lot of evolution PDEs (basically all parabolic ones, like the heat equation, or Navier-Stokes) can be solved only forward but not backward. In linear PDEs, this is a consequence of the Uniform Boundedness Principle (= Banach-Steinhaus Theorem). There is a nice theory relating operators and semigroups, the former being the generator of the latter. In the linear case, a fundamental result is the Hille-Yosida Theorem. Subsequent tools are Duhamel's principle and Trotter's formula. A part of the theory extends to nonlinear semigroups.

Edit. John B. expresses a doubt on the fundamental aspect of semigroups, compared with the evolution equations from which they arise. Let me say that semi-groups say much more, for the following reason. Evolutionary PDEs have classical solutions only when the initial data $u_0$ is smooth enough, typically when $u_0$ belongs to the so-called domain of the generator. This result can never be used to pass from a linear context to a non-linear one via the Duhamel's principle. In other words, in order to have a well-posed Cauchy-problem in Hadamard's sense, we need to invent a notion of weaker solutions ; this is where the semi-group theory comes into play.

• While you say that $C_0$ semigroups are fundamental, isn't it true that all can one can do with them can be done by using the evolution families that give rise to them? I would call these evolution families fundamental, but I have difficulties agreeing with the same for the consequent semigroups. Dec 20 '15 at 23:01

An important application of semigroups and monoids is algebraic theory of formal languages, like regular languages of finite and infinite words or trees (one could argue this is more theoretical computer science than mathematics, but essentialy TCS is mathematics).

For example, regular languages can be characterized using finite state automata, but can also be described by homomorphisms into finite monoids. The algebraic approach simplifies many proofs (like determinization of Buchi automata for infinite words or proving that FO = LTL) and gives deeper insight into the structure of languages.

• Interestingly enough, N. Bourbaki describes exactly this use of the theory in his set theory book, in the appendix to chapter 1. Feb 19 '11 at 20:07
• That's interesting - could you provide a reference? Feb 19 '11 at 21:14
• I just gave you a reference... Here, is this better: N. Bourbaki Éléments de Mathématique I: Théorie des Ensembles. Feb 20 '11 at 4:46

Circuit complexity. See

Straubing, Howard, Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Birkhäuser Boston, Inc., Boston, MA, 1994.

If you want a research problem relating circuit complexity with (finite) semigroups, there are many in the book and papers by Straubing and others. See also Eilenberg and Schutzenberger (that is in addition to Pin's book mentioned in another answer) - about connections between finite semigroups and regular languages and automata.

Semigroups of bounded $L^{2}$ operators are very important in probability. They in fact provide one of the main ways show the very close connection between a self-adjoint operator and a `nice' Markov process (nice can be taken to mean strong Markov, cadlag, and quasi-left continuous.) So how does one get this semigroup from a Markov process? If $X_t$ is your process let $\mu_{t}(x,A)$ be measure with mass $\le 1$ with value $P(X_t \in A | X_0=x)$. Then $\int f(y) \mu_t(x,dy) = T_tf(x)$ gives a semigroup of bounded $L^{2}$ operators.

Why are such constructions important and natural? If $f$ is your initial distribution of something (heat for example) and $X_t$ is Brownian motion. Then $T_t$ acts by letting the heat distribution $f$ diffuse the way heat should. This then gives a nice way to connect PDE and probability theory. I'd end by offering that semigroups are important, in part, because they do arise is so many places and can bridge between disciplines. There are other reasons as well.

• When people say "semigroup" in relation to functional analysis they really only mean semigroups isomorphic to the non-negative reals, right? Feb 18 '11 at 20:22
• Not necessarily, although that is the most common usage. But there is a huge literature on non-commutative dynamical systems, motivated by quantum mechanical problems. Feb 18 '11 at 20:34
• Thank you very much -- this is the type of answers I'm looking for!!!!!!!!!!!!!!!!!!!!!! I'll look in that direction. Feb 19 '11 at 18:28

(Commutative) semigroups and their analysis shows up in the theory of misère combinatorial games. The "misère quotient" semigroup construction gives a natural generalization of the normal-play Sprague-Grundy theory to misere play which allows for complete analysis of (many) such games. (See http://miseregames.org/ for various papers and presentations.)

• Also finitely generated commutative semigroups are equivalent to Petri nets.
– user6976
Feb 18 '11 at 20:09

Though you say that $C_0$ semigroups are not really semigroups, the structure of compact semitopological semigroups plays an important role in the investigation of their asymptotic behaviour. For example, Glicksberg-DeLeeuw type decompositions or Tauberian theorems are obtained such a way, see Engel-Nagel: One-Parameter semigroups for Linear evolution Equations, Springer, 2000, Chapter V.2.

Toric varieties in Algebraic Geometry!! Indeed, the category of normal toric varieties is equivalent with the dual of the category of finitely generated, integral semigroups.

Why nobody explicitly mentioned one of the most natural examples: symmetric semigroup of a set? This is the semigroup analogue of a permutation group (https://en.wikipedia.org/wiki/Transformation_semigroup).

Unary (1-variable) functions mapping a set X to itself under composition is a semigroup. Cayley's Theorem (one of them) says that every semigroup is isomorphic to one of this kind.

Given a group $G$, the Block Monoid $B(G)$ consists of sequences of elements in $G$ that sum to zero. So for example, an element of $B(\mathbb{Z})$ is $(-2,-3,1,1,3)$. The monoid operation is concatenation, and the empty block is the identity element.