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.

[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)

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