Talk:Semigroup

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

I doubt that there's a known formula.

Contents

[edit] Minimal Ideals

"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)

[edit] Empty Semigroups?

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)

[edit] Semigroup Applications

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)

[edit] history

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)

[edit] another note

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)

[edit] Applications?

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)

[edit] Inaccurate claim

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 x+y-x\oplus y. Here, + and − are addition and subtraction in F(S), while \oplus 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 \mu:S\to G(S) is unique, and, for any homomorphism \phi:S\to H to any abelian group H, there exists a homomorphism \psi:G(S)\to H such that \phi=\psi\circ\mu.
If S is a lattice or semilattice, so that \oplus 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 μ 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)


[edit] "Total" in the formal definition?

Perhaps the formal definition ought to state explicitly that the operation is total? 70.111.117.246 (talk) 12:24, 2 January 2008 (UTC)