Talk:Semigroup
From Academic Kids
Question: How many semigroups are there of a given finite order? Is there a formula?
- I doubt that there's a known formula.
- http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A023814 for the total number of semigroups on a set with n elements
- http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A027851 for the number of non-isomorphic semigroups with n elements --AxelBoldt
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)
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)
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.
