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Number of semigroups of a given order[edit]

Question: How many semigroups are there of a given finite order? Is there a formula?

I doubt that there's a known formula.

Minimal Ideals[edit]

"The intersection of two ideals is also an ideal, so a semigroup can have at most one minimal ideal." Is this really true? What if the ideals are disjoint? Is there a guarantee that any two ideals will have a nonempty intersection? - Gauge 07:36, 8 May 2005 (UTC)

If s and t lie in ideals I and J respectively then, from the definition of an ideal, the product st lies in both I and J, that is, in their intersection. Best wishes, Cambyses 10:04, 10 May 2005 (UTC)

Oops. I should have thought that one through a bit more.  :-) - Gauge (talk) 04:15, 28 October 2008 (UTC)

Empty Semigroups?[edit]

The current page allows "empty semigroups". TTBOMK, it is the universal convention these days to insist that a semigroup be non-empty. Does anyone think differently, or should I change it? It would require a few minor changes further down the page. Cambyses 21:24, 8 Mar 2004 (UTC)

This convention is certainly used, so it should be mentioned in the article. However, it can't be universal - the two links Axel gives above provide a counterexample. To adopt this convention would require changes to a number of pages that talk about semigroups, not just this page, so it's not something to be undertaken lightly. (Also, it's an ugly convention, IMHO. The set of subsemigroups of a semigroup ought to be a lattice.) --Zundark 22:07, 8 Mar 2004 (UTC)

Okay - good point well made! I've added a paragraph at the top on the issue, also taking the opportunity to note that some people (esp. Russians) use semigroup as a synonym for monoid. (With regard to ugliness, I guess it depends if you care more about subsemigroups or homomorphic images - you could also argue that there should be a trivial semigroup which is an image of every semigroup. Or perhaps the group theorists are right, and semigroups are inherently ugly.... ;-) Cambyses 22:48, 8 Mar 2004 (UTC)

Semigroup Applications[edit]

I liked seeing the example of applying Semigroups to computer science. Greater reader interest could be generated by listing more examples of Semigroups used in communications theory, partical physics, and other areas of applied mathematics.

iirc, algebraic automata theory makes heavy use of semigroups. far as i can see, there's currently no page on WP covering that stuff. perhaps folks knowledgable in that area would being willing to take that up. Mct mht 05:50, 13 December 2006 (UTC)
I would like to add: which applications specifically require semigroups as opposed to monoids? It would seem that applications where semigroup elements are some kind of transformations would not suffer much from throwing in the identity transoformation (if it is not already present) making the structure a monoid. Maybe I'm too naive, but in any case the question comes to mind easily and does not seem to be addressed currently. Marc van Leeuwen (talk) 13:54, 21 September 2013 (UTC)


I added a line about the fact that semigroup theory is relatively recent in abstract algebra. Well I know that's a pretty safe statement given how vague it is but does anyone know about the actual history? I seem to remember that the original motivations came from functional analysis but it would be nice if someone had a reference for that. Certainly someone must be credited for coining the term and that's the kind of information I think would help to make this article a bit more than a reference for the mathematically enclined. Pascal.Tesson 22:52, 4 September 2006 (UTC)

The earliest use of the term I have found in English is in
Hilton, Harold, Theory of Groups of Finite Order, Oxford: Clarendon Press, 1908.
The book can be downloaded for free at [1]. The use of the term "semi-group" is on p. 51. Interestingly, he doesn't mention the associative law, but I think he implicitly assumes it. At the website Earliest Known Uses of Some of the Words of Mathematics (S), we find this tidbit:
"The term SEMIGROUP apparently was introduced in French as semi-groupe by J.-A. de Séguier in Élem. de la Théorie des Groupes Abstraits (1904)."
Assuming this is correct, semigroups are nearly as old as groups. Michael Kinyon 02:11, 5 September 2006 (UTC)
Oops, that's not what I meant to say at all. I meant that semigroups are not as old as groups (which date back to the early/middle 19th century), but are perhaps less recent than the article suggests. Michael Kinyon 02:15, 5 September 2006 (UTC)
Great stuff. Thanks. Pascal.Tesson 04:29, 5 September 2006 (UTC)
I've added that info. But since you seem to be a good source for history, let me ask you (and anyone else reading this page) a couple of more questions! Was there ever a journal prior to the semigroup forum devoted exclusively to semigroups? What more can we say about how the field has evolved? I am mostly aware of the development of finite semigroup theory because of its links with automata but I wouldn't want to write the history section with too much of a theoretical computer science slant. Pascal.Tesson 05:20, 5 September 2006 (UTC)
I doubt there was a journal prior to Semigroup Forum devoted to the field. I gathered the information above from trying some searches on MathSciNet for early papers and then backtracking using bibliographies, etc. It was just luck, really; I don't know much else about the history of the field. The following might be useful, but I haven't seen it:
Preston, GB, Personal Reminiscences of the Early History of Semigroups, Proceedings. of the Monash Conference on Semigroup Theory 1990, pp. 16-30.
I hope that helps a bit. Michael Kinyon 05:37, 5 September 2006 (UTC)
Interesting. While I could not find that paper, I found one that referenced it and credits Anton Suschkewitsch with "the first major paper on semigroups".[2] Will add that too when I get the time.Pascal.Tesson 05:51, 5 September 2006 (UTC)
You might also ask at the Historia Matematica mailing list or the Semigroups mailing list. Michael Kinyon 05:54, 5 September 2006 (UTC)

Semigroups are so natural and ubiquitous that I guess it is imposible to pinpoint where they first appeared in the literature. The trouble is that the axioms are so weak as to make any kind of "general" study more or less impossible, so the subject has always tended to be a loose affiliation of different areas which each restrict attention to a different "well-behaved" class of semigroups. (Of course, groups are the archetypal example of such a class and so, in a sense, group theory is a "typical" branch of semigroup theory, in which case semigroup theory certainly began with group theory, even if not before!) That said, there are a few themes which tend to recur whenever one studies semigroups. Although Suschkewitz's 1928 paper (referenced above) is formally concerned only with finite semigroups, it is widely recognised as the first major contribution to "general" semigroup theory, because it was the first to introduce one of these themes. Specifically, it contains all the essential ideas for the Rees matrix construction and the Rees theorem. Not sure if that helps at all.... :-) Cambyses 11:16, 7 September 2006 (UTC)

another note[edit]

Is it just me or does the sentence

All subsets of a group that contain the identity form a semigroup with elementwise multiplication.

feel funny to everyone else? I'm tempted to fix it but I'm not even sure what the editor meant. My best guess is that he meant something like the set of subsets forms a group with multiplication AB = {ab: a in A and b in B} If that's the case then why should we care that the thing is a group? A monoid would do the trick so this addition in the list perhaps refers to some well-known applications in gropu theory. In any case, one should probably mention that the power set of any semigroup S is a semigroup for that same reason. Pascal.Tesson 05:31, 5 September 2006 (UTC)

Yes, I've been bothered by that sentence for a while. Please do fix it. Michael Kinyon 05:57, 5 September 2006 (UTC)


This page has no mention of applications. My functional analysis text (AMS109) claims (p. 298) "The notion of a semigroup is the most important notion for describing time-dependent processes in nature in terms of functional analysis." I'm not entirely sure yet what they mean by this. Could someone add more about applications? —Ben FrantzDale 00:04, 13 December 2006 (UTC)

solutions to the abstract Cauchy problem, e.g. a functional differential equation, are formulated in terms of one-parametersemigroups (sometimes one gets more than a semigroup). perhaps that's what they meant. Mct mht 05:43, 13 December 2006 (UTC)
see also C0-semigroup. Mct mht 05:46, 13 December 2006 (UTC)

Inaccurate claim[edit]

I removed the claim that "every abelian semigroup can be extended to a group", which was simply false. Any non-trivial semilattice, for example, is commutative (which I presume is what is meant by abelian) but has multiple idempotents and so clearly cannot be extended to a group. Best wishes, Cambyses 15:43, 28 March 2007 (UTC)

My fault for adding that claim, I was thinking of the universal property of the free abelian group. However, I'm feeling confused today as to where things like a free lattice would fit into the scheme of things. Bonus points on figuring out why I'm confused. linas 22:41, 30 March 2007 (UTC)
I had to look at my notes to unconfuse myself. Here is a definition from Atiyah; it is nearly identical to the standard definition for a vector product. Let S be any abelian semigroup. Let F(S) be the free abelian group which takes S as its basis. There is a subgroup of F(S), lets call it E(S) that consists of elements . Here, + and − are addition and subtraction in F(S), while is the commutative addition in S. You see where this is going ... E(S) is an equivalence relation, its the kernel of some map of a short exact sequence. Thus G(S)=F(S)/E(S) is the universal group that is the group extension of S. This is a universal property, in that the homomorphism is unique, and, for any homomorphism to any abelian group H, there exists a homomorphism such that .
If S is a lattice or semilattice, so that is "join", for example, then, when I plug into the above, I get G(S)={0} is the trivial group. Because of the universal property, this tells me that every homomorphism of a lattice to an abelian group is always the trivail homomorphism. None the less, G(S) is still validly called the universal cover of S. Right? Ergo, "every abelian semigroup can be extended to an abelian group". More generally, if S has idempotents, then will carry all idempotents to 0 in G(S). linas 14:24, 31 March 2007 (UTC)
I thought of writing an article on this, but appearantly we already have one, called the Grothendieck group. linas 19:38, 1 April 2007 (UTC)

The universal group construction is well-known to semigroup theorists (and if phrased appropriately, makes sense for arbitrary, possibly non-commutative semigroups) but as you say yourself, what you get is a morphism from S to G(S). To say that G(S) is an extension of S, you want a morphism the other way, from G(S) to S. Best wishes, Cambyses 12:35, 3 April 2007 (UTC)

And what's wrong with the forgetful functor? ...semigroups are one of those categories (I think they're called "finitary categories", where the forgetful functor is part of a monad) which has functors going both ways. No matter; the category stuff is all highly abstract nonesense; we are not writing for semigroup theorists but to newcomers to the subject (of which I'm one). What I really wanted to do was to have the article make a clear pointer to the construction given in the planetmath Grothendieck group article, which doesn't actually use the word "category", but simply gives a concrete, approachable construction that can actually be useful to newcomers.
In the same vein, if there's a concrete construction for certain non-commutative semigroups, I'm not aware of it myself, and would like to know more. I'm trying to read about lattices in Johnstone's "Stone spaces" book, but its not a page turner; much of the material is new to me. linas 13:00, 3 April 2007 (UTC)

Assuming you mean the forgetful functor from the category of groups to the category of semigroups, I can't immediately see how this helps. It will take G(S) to G(S) considered as as a semigroup, so I can't see how it gives us anything at all about S. In answer to your question, the universal group of a semigroup is just defined to be the (homomorphically) maximal group generated by the elements of the semigroup and subject to all relations which hold in the semigroup. Equivalently, if < A | R > is a presentation for the semigroup then G(S) is the group with presentation < A | R >. The identity map on the generators induces a morphism from S to G(S), which is injective exactly if S is group-embeddable. Best wishes, Cambyses 13:17, 3 April 2007 (UTC)

I'm not trying to provoke an argument, nor am I trying to study semigroups in all generality; merely that I have a research project in which a certain semigroup plays a role, and I'm trying to scrape up all of the relevant properties it may have, and the implications thereof. In my case, the "free" construction <A|R> doesn't seem to have any legs, since I'm trying to preserve the action of the semigroup on a set. I dunno, I have to think about it more. linas 14:35, 3 April 2007 (UTC)

Sorry, didn't mean to sound argumentative! I can quite believe the universal group is not useful for your application (although I think it coincides with the construction you give in the abelian case). How does your semigroup acton your set? If it acts by permutations then you can always take an image in which the action is faithful (by identifying two elements if they act the same everywhere) and the result can of course be embedded in a group (just by throwing in the inverses of the permutations). If it acts by non-injective or non-surjective functions (eg. as = bs for some distinct a and b in the set and s in the semigroup) then you'll probably have a hard time capturing the essence of the action in a group action. Sorry if that doesn't help either! :-) Best wishes, Cambyses 15:39, 3 April 2007 (UTC)

"Total" in the formal definition?[edit]

Perhaps the formal definition ought to state explicitly that the operation is total? (talk) 12:24, 2 January 2008 (UTC)

Semigroups with the cancellation property[edit]

Is it the case that any (not necessarily commutative) semigroup with the cancellation property can be embedded in a group? If not, is there a characterization of those semigroups that are isomorphic to subsemigroups of groups? (talk) 07:19, 14 November 2008 (UTC)

There are cancellative monoids that don't embed in any group. The first example was given by Mal'cev in 1937. I think there is a characterization (also due to Mal'cev) of semigroups that embed in a group, but I don't have the details. --Zundark (talk) 10:24, 14 November 2008 (UTC)
Thanks. (talk) 03:38, 16 November 2008 (UTC)

Point-Set Topologist, your reversion is unjustified.[edit]

The definition given here should be the same format as all of the other definitions for magma, semigroup, abelian group, group, monoid, etc. Negi(afk) (talk) 02:45, 25 April 2009 (UTC)

This was discussed at Talk:Group (mathematics). --PST 02:18, 26 April 2009 (UTC)
No, it wasn't I just searched the page. This new definition isn't even right.Negi(afk) (talk) 05:37, 26 April 2009 (UTC)

Removed the "more references needed" tag[edit]

This article is still very, very basic, even though I've done a little work on it. It's so basic (heck it still lacks the Cayley theorem for semigroups) that practically everything in it except the C0-stuff is covered in all introductory books on semigroups. Taking the FA Group (mathematics) as example, basic facts are need not be given with an inline citation; only historical info, applications or results not commonly found in an introductory book on the topic (should) have a direct citation. Pcap ping 20:32, 10 August 2009 (UTC)

I'm the one that added the tag, so I suppose I should explain myself. First of all, I'm not a huge fan of the style in which the references are given: with chatty little notes. General references should be fine for (most of) an article like this, but the silly notes render the references list very unprofessional. Also, if we are to go down the general referencing path, then some care should be given only to list references that are used for the article. For instance, the reference to the "Semigroup forum" is to an entire periodical, and so clearly not acceptable as a "reference" for an article. I'll take the liberty of partially fixing some of these issues. Sławomir Biały (talk) 13:55, 22 August 2009 (UTC)
About Semigroup forum: you could put it in a Further reading section, but it's rather unusual to do that with a journal too. Probably just best mentioned in text, e.g. in the history section. I think it's sufficiently non-controversial to do that w/o a citation. It's true however that more significant (recent?) results in this field got published in more general venues, e.g. [3], and follow-ups usually appear in Semigroup Forum-only but that pattern is probably the same in any field. (Mental note: add Mitsch's order to the article.) Pcap ping 15:26, 22 August 2009 (UTC)


The article currently claims that the study of semigroups started later than that of groups and rings in the mid 19th century. Should this be 20th century? The article also claims that the study of finite semigroups is more developed than that of infinite semigroups, but there is an entire field of functional analysis and partial differential equations that deals only with infinite semigroups, so this statement also seems quite dubious (or at least written from a particular point of view). Sławomir Biały (talk) 14:11, 22 August 2009 (UTC)

The first dubious fact is almost certainly false -- it probably wanted to say that in Clifford and Preston's era a number of important results were obtained (Green's relations etc.); anyway the statement is too vague to convey much to the reader. I'll remove it. As for the 2nd, I don't enough about functional analysis, but there aren't that many special results for finite semigroups that I know of. So it's rather overreaching as a conclusion even from, say, a computer science point of view. Springer's EOM was not of much help here. Pcap ping 15:34, 22 August 2009 (UTC)
Your solution solves my earlier objection. Sławomir Biały (talk) 22:02, 22 August 2009 (UTC)

Braindead citation template[edit]

Does anyone know how to make it link to authors via redirects, e.g. link to John Mackintosh Howie. The other semigroup theorist have wiki pages too... Pcap ping 14:23, 22 August 2009 (UTC)

Just add "|authorlink=John Mackintosh Howie" to the citation template. Sławomir Biały (talk) 14:26, 22 August 2009 (UTC)
Well, that is the braindead part. It should not avoid redirects like John M. Howie. You basically have to write the redirects again in each template instance; it defeats the reason for having the "write-once" redirects. Pcap ping 14:39, 22 August 2009 (UTC)
I don't follow. Adding "|authorlink=John M. Howie" should also work, but having a direct link is usually preferrable because it marginally reduces the server load. Sławomir Biały (talk) 15:50, 22 August 2009 (UTC)
It was my impression that even without an explicit link some citation templates (perhaps not this one?) auto-detect if an article for the author's names exists and links to it. Perhaps I'm confusing it with what I saw on a different wiki... Pcap ping 16:02, 22 August 2009 (UTC)
I just checked the template documentation for {{citation}} and {{cite book}}, and you're right, it must always be done manually on this wiki. Pcap ping 16:07, 22 August 2009 (UTC)

Section on group of fractions[edit]

I have removed the section on group of fractions, because it is not discussed in the main algebra sources and was misleading. In particular in the analytic theory of semigroups as applied to the heat operator (cf the theory of one-parameter semigroups explained in the books of Hille & Phillips, Yoshida and in elsewhere), the section was meaningless. Taking a positive unbounded self-adjoint operator A on a Hilbert space, the operators will not be bounded for t negative. Similarly the heat equation has no solution for negative t. Putting spurious synthesized material into mathematics articles is unhelpful, even disruptive. It gives undue emphasis to what is essentially a minor side remark, even a mathematical dead-end. It is true that, in combinatorial group theory (now usually called geometric group theory, post-Gromov), a discrete semigroup defined by generators and relations has an enveloping semigroup, which could be trivial; but the topic is not represented in the two main textbooks and is in the nature of a side remark, not warranting a discursive and misleading section of its own bordering on WP:OR. Mathsci (talk) 03:24, 29 March 2010 (UTC)

Deletion of well-sourced material for personal reasons is disruptive. Either explain or strike the word "misleading". When you say "meaningless", what you mean is that a purely algebraic construction to map a semigroup into a group may not be appropriate for an analytic theory: so add a section on the appropriate analytic setting if you like. This is not a reason for deletion, and suggests that you do not understand the concept adequately: the enveloping semigroup in the context of topological group actions is quite a different thing. The topic of the group of fractions is sufficiently central that there is a Wikipedia article Grothendieck group on a special case, namely the group of fractions of a commutative cancellative semigroup, and it has been discussed at this talk page twice already. Quotient group (talk) 20:37, 29 March 2010 (UTC)
It appears very strange to me that you removed this. I thought the times when parts of algebra were rejected because they are too abstract and don't fit certain applications were long past. The case of commutative monoids is of course very well known (it's covered in Cohn, Algebra vol. I and in Bourbaki, Algebra Chapter I). I have found groups of fractions of semigroups mentioned in various places on the web, including a paper by P.T. Johnstone (which I don't have access to) that appears to generalise the construction to categories.
There is also an entire section about The division problem for semigroups and rings in Cohn, Universal Algebra, Chapter VII. [4]
However, I note that it uses different terminology: Cohn refers to the group that always exists as the universal group U(S) of the semigroup S, so there is always a homomorphism SU(S). He writes: "the division problem for semigroups consists in finding practical criteria for a semigroup to be embeddable in a group", i.e. for this homomorphism to be injective. Cohn only calls U(S) the group of fractions of S if the homomorphism is injective. He gives a sufficient criterion for the existence of a group of fractions.
In my opinion the section should be restored, and the terminology it uses verified with the sources. If the sources used by the section differ from Cohn, the diverging terminology needs to be pointed out for the benefit of the readers. Hans Adler 21:30, 29 March 2010 (UTC)
It's also discussed as universal group here:
Grillet, P. A. (1995), Semigroups: An introduction to the structure theory, New York: Dekker, ISBN 978-0-8247-9662-4 
Grillet only speaks of the group of fractions in the commutative case, though.
As a general comment I would like to add that as far as I can tell Hilbert spaces are only one of the areas where semigroups appear in practice. Another, that may be even more significant, is words in the context of automata theory. (The operation is concatenation of two words.) Hans Adler 21:44, 29 March 2010 (UTC)
Section 1.10 in Clifford & Preston vol. I also comes close to the removed section, and has more information in another direction. It states Ore's theorem: Any right reversible (i.e. any two left ideals intersect), cancellative semigroup is embeddable in a group. (I.e. has a group of fractions in the stronger sense, i.e. the homomorphism to its universal group is injective.) The converse is almost true: A semigroup that can be embedded in a group must of course be cancellative. If it can be embedded in a group that consists only of its left quotients, then it must also be right reversible. Hans Adler 22:04, 29 March 2010 (UTC)
It turns out that Clifford & Preston vol. II has an entire chapter about "Embedding a semigroup in a group"; it's the last one. They call the universal group of S the free group on S.
I haven't checked all the details of the removed section and the terminology seems to be slightly eccentric, but something like this really belongs in the article. Hans Adler 22:13, 29 March 2010 (UTC)
As far as analytic semigroups was concerned, the content was completely misleading. I did look at Clifford and Preston. and saw the material about embeddings, which I think I already mentioned somewhere. In that case there should be a section called "embeddings in groups". It is important to distinguish the analytic semigroups from the algebraic theory, something that's not done at all carefully in the article. (There is the Hille-Yosida theorem for example.) The article treats both on an equal footing at the moment (eg the section on the heat equation), so some care in writing is required. I agree with almost your suggestions, but would start from the textbooks when writing this material. I am quite willing to assist you in doing this, if you find that agreeable. Best regards, Mathsci (talk) 00:43, 30 March 2010 (UTC)
That sounds like a nice project, although it's not really my field. But as far as I'm concerned it will have to wait for a few weeks. (I am going to travel, and then my little daughter and my parents are visiting me in Vienna.) Hans Adler 01:07, 30 March 2010 (UTC)
Just a side comment and suggestion. It also irks me that the article treats the algebraic theory of semigroups and the analytic theory are placed on equal footing. It may be better to fork off a separate semigroup (algebra) article, since there is almost no meaningful intersection between the subjects. Sławomir Biały (talk) 12:42, 30 March 2010 (UTC)
It seems to me that the pure algebraic meaning is the primary one and is already the main focus of the article, except for the section Semigroup#Semigroup methods in partial differential equations. We also have articles analytic semigroup and topological semigroup, which, strangely, are not linked from here. (I will put them under "See also" as a quick fix.) Only C0-semigroup is linked at the moment.
This structure is parallel to group (mathematics), which also has the abstract, most general notion, along various other articles such as Lie group, topological group, algebraic group that deal with special cases. It would be a bit absurd to introduce an application-centric view here that puts the main focus on the analytic concept, when we can get along without any disambiguators by just leaving the algebraic notion as the primary concept. Hans Adler 13:11, 30 March 2010 (UTC)
If it is indeed the case that this is the primary topic, then the section under discussion should probably be removed. As MathSci points out, we cannot solve the initial value problem for the heat equation backwards in time. Otherwise, it seems that a different article whose focus is the algebraic treatment of semigroups would be warranted in the interests of maintaining a neutral point of view. Obviously then the discussion of continuous semigroups should be expanded as well. Sławomir Biały (talk) 18:33, 30 March 2010 (UTC)
Are we talking about the same article? I am talking about Semigroup, which is an article about semigroups as monoids that may not have a unit, i.e. in the sense that semigroups are defined by almost every modern algebra book. Continuous semigroups are a special case of that and have additional structure that does not play any role in this article. Why do you think the article is about them? Why do you think this article is not "algebraic"? I have given you links to three other articles that probably discuss things that are much closer to what you have in mind. This article is about the general concept, which covers the following examples (I am sure I am missing many obvious ones):
  • All groups, e.g. all finite groups.
  • All monoids.
  • Strings with concatenation as operation. E.g. the strings over the alphabet { a,b,c } form a semigroup with the concatenation operation *. E.g. "aab" * "caacb" = "aabcaacb".
  • and also but not primarily topological semigroups, analytic semigroups and C0-semigroups.
How is the heat equation a reason for not asking whether we can embed the string semigroup in a group??? Hans Adler 18:47, 30 March 2010 (UTC)
Well, now I'm very confused. Earlier you said that this article should focus on the primary concept, and now you are suggesting that this article should focus on the aspects of semigroups that are relevant only to abstract algebra. The algebraic structure of continuous semigroups is fairly trivial, but there is a sizable plurality of mathematicians for whom "semigroup" means this latter sort of semigroup. Designating the former as the "primary" concept, and the latter as just an auxiliary concept seems to be rather contrary to the spirit of maintaining a neutral point of view. Apart from my suggestion of splitting out a separate article for the notion in abstract algebra, I don't really have any concrete suggestions. But the focus of this article is something that has bothered me for a long time. Sławomir Biały (talk) 20:41, 30 March 2010 (UTC)

(←) I too was baffled by the relevance of the comment "we cannot solve the initial value problem for the heat equation backwards in time". I think that what this is saying is that this is a case of a semigroup of operators which are not invertible as operators. But of course the point about the group of fractions construction is that it posits a formal inverse of each element (and then deals with all the other relations too). So the formal inverses are not expected to be the inverse operators. This is actually a rather fruitful notion in algebra and in analysis (think of the algebraic notion extending the notions of integrable functions to distributions as members of the double dual for example.)

Anyway, this all seems rather disconnected from the question of whether the group of fractions is encyclopedic. I've just been adding a little to Garside element where I'm trying to build up something on recent work about solvability of the conjugacy problem for Artin groups by means of automata, based on Garside's work on the conjugacy problem for braid groups. The point is that these groups arise as groups of fractions of semigroups with certain properties. So it would be rather hard to develop that article without a mention somewhere on WP of the group of fractions concept. It also seems strange to describe the embedding problem as a "dead end". Quotient group (talk) 20:28, 30 March 2010 (UTC)

I am really puzzled now. What's wrong about discussing abstract semigroups without any additional structure under semigroup and putting any more complicated / more specific objects under their respective more precise names?
  • I looked "semigroup" up in the Encyclopedic Dictionary of Mathematics. The index sent me to three places. In one place semigroups are defined as a generalisation of groups. In one they are defined as an example of mathematical structures. Only one place uses the term for C0-semigroups, but also makes it clear that this is actually an abbreviation and that there are several full names for the concept that have "semigroup" as a component.
  • The two-volume book "Algebraic theory of semigroups" by Clifford and Preston has a total of 600 pages and deals only with the purely algebraic aspects: volume 1, volume 2.
  • Similarly for Howie's "Fundamentals of semigroup theory" (350 pages) and Grillet's "Semigroups: An introduction to the structure theory" (400).
  • When skimming through all these works I haven't once seen the exponential function or any other indication that anyone was doing analysis.
  • An Amazon search for "semigroup" suggests that most if not all books that deal with semigroups that are of a special type or have extra structure say so very clearly in the title. Many of them speak about "operator semigroups" or "semigroups of operators", for example. But books that deal with the pure concept, such as algebraically oriented books or computer science books, just talk about "semigroups".
For all these reasons it seems absolutely clear to me that the pure algebraic concept, which in contrast to all the others doesn't have an alternative name, and which is the only one that fits the name that indicates "more general than a group", is the primary one for the name "semigroup". And I am amazed that I have to argue for it. Hans Adler 21:42, 30 March 2010 (UTC)
I can attest that, in certain parts of analysis, the word semigroup is used, by itself, to mean a C0-semigroup. If someone encountered the word "semigroup" for the first time in an analysis context, they could potentially be very confused if they looked up this article and found a whole lot of abstract algebra. I would suggest that a disambiguation message be placed at the top of the article indicating that this article is about the algebraic structure, and that the reader should go to C0-semigroup for information on one-parameter semigroups. Jim (talk) 02:56, 31 March 2010 (UTC)
Jim, thanks for that, it seems to resolve the perplexities that have surrounded this debate. Are we now agreed that the main focus of this article is the algebraic concept, and as a corollary that the group of fractions is encyclopedic? Quotient group (talk) 09:27, 2 April 2010 (UTC)

The deletion was a bad call. As the Springer Encyclopedia shows ([5] and [6]) "semigroup" and "semigroup of operators" can reasonably be treated as different topics; but that is not how things are done here, and semigroup must be written to include various points of view. That point needs to be taken into account, in terms of how matters are expressed. As for the content itself, it may be worth a separate article (adding an identity to get a monoid is trite, mapping a monoid into a group is a problem for which the commutative case is already famous as the Grothendieck group, and the material should be taken more slowly, suggesting a treatment that is less hasty). Charles Matthews (talk) 19:03, 3 April 2010 (UTC)

"Totality" for group-like structures[edit]

The chart of group-like structures has a column for "totality" which links to the idea of a total function. But it's not quite clear how this applies to algebras. The fact that categories are classified as not possessing "totality" only perplexes me. Can someone illuminate this in the article?

In fact, it might be good to give the reader a tour of that little chart. The adjacent section says almost nothing directly about the chart. Ezrakilty (talk) 03:23, 16 March 2011 (UTC)

I think it would be better to get rid of the table. In any case, this is not the article in which to explain it. As for "totality", this is intended to mean that the operation can be applied to any ordered pair of elements. In a category, the "elements" are the morphisms, and the operation is composition of morphisms, so totality fails if there is more than one object. --Zundark (talk) 09:01, 16 March 2011 (UTC)


The section on generalizations mentions semigroupoids, but the accompanying table mentions "semicategory" at what would appear to be the corresponding place.

While I'm mentioning apparent inconsistencies, the section on semigroup applications in PDE's seems to have some as well. I think that H2 should be L2 and that A in the final set of equations should be D. Marc van Leeuwen (talk) 13:59, 21 September 2013 (UTC)

Making the article accessible[edit]

I have an engineering degree from MIT, and this article is still pretty much incomprehensible at first read. A visualization or an elementary worked-through example or two would be enormously helpful. Having lots of prose with mathematical symbols makes it very hard to unpack. -- Beland (talk) 19:55, 19 December 2014 (UTC)

Well, we could copy a visual example from Semigroup_with_two_elements. The catch is that methods of semi-group theory are in general rather different from those of group theory. So it's a strange new world the first time you read through it in any presentation. (talk) 07:34, 12 April 2015 (UTC)
Actually since this page contains little more than definitions, can you clarify where you got lost? (talk) 10:08, 12 April 2015 (UTC)

Yes, please help make this article easier to understand for those who are not math specialists! I have a PhD in Computer Science, and I've certainly had reasonable exposure to math and logic over the years, but I too find this article unbearably difficult to unpack. The main problem is the high density of jargon and the domain-specific notation. Obviously the reason for that is precision, which is important also. But the problem is that a definition of an unfamiliar term is often given in terms of 2-3 other unfamiliar terms, which in turn are defined in terms of more unfamiliar terms, etc.. I find myself diving through several layers of dependency definitions -- each with a different wikipedia page -- just to unpack one definition. And the problem with specialized notation is that it is not even clear how to look up the definition of a symbol such as "*" or (which I do know, but I'm just using as an example). Ask yourself: What would a reader type into a google search, to find out what the symbol means?

Examples help a lot. And plain language definitions help enormously, even if they are imprecise, though of course they should be clearly noted as being imprecise. The key point in a plain language definition is to avoid domain-specific jargon. For example: "Roughly speaking, a foo is . . . ". And later: "More precisely, a foo is . . . " (with full rigor and jargon).

As a case in point, I was just looking up the definition of "inverse relation" on wikipedia, and the explanation talked about a "semigroup with involution", so I had to look up that, which was defined in terms of a "semigroup", so I had to look up that page (which brought me here), which says that "A semigroup generalizes a monoid", so I had to look up "monoid" . . . except that I gave up at that point. (Stack overflow?)

There are two main use cases that a page like this should address, and they are different: (a) someone runs across an unfamiliar term and reads the page to get a rough idea of what that term means; and (b) someone wants to dig deeply and precisely into the meaning of the term. The (possibly imprecise) plain language definition should be given first, free of jargon. The gory details and jargon should come later. I do think it is important to introduce the jargon that is used in the field, but it should be fairly clearly separated from first providing a layman's (approximate) definition, so that readers can get the gist of what the term is about before they face the prospect of a deeply nested recursive traversal through many pages of jargon-filled definitions.

I hope the above suggestions are helpful and don't just sound like complaints. I know it is hard to write such things in widely understandable ways, and I very much appreciate the efforts of all editors who contribute. Thanks! -- DBooth (talk) 16:29, 17 April 2015 (UTC)

I rewrote the lead. Let me know what you think. (talk) 10:22, 18 April 2015 (UTC)