>Where do I find a proof that every subsemigroup of the additive group

>of all natural numbers is finitely generated?

Sorry, my previous reaction was too vague (laziness, common laziness...)

There was a reference to related results (provided by Melvin Henriksen).

The Redei theorem says: congruences on a finitely generated free

commutative semigroup satisfy the ascending chain condition (= are

themselves finitely generated = a finitely generated comm. semigroup

is finitely presentable). Take the free comm. semigroup with a single

generator and...

L. R\'edei, Theorie der endlich erzeugbaren kommutativen Halbgruppen,

Physica Verlag, W\"urzburg, 1963 (English edition

appeared in 1965).

There are quite a few proofs, it's a kind of sport in which people

tried to make the proof as short as possible. There are 1-page proofs

and, by now, probably even shorter proofs. I can give some

references, if needed.

However, this is not exactly Jiri's question.

Concerning Weierstrass, Hurwitz etc. see, for example, J. Komeda's

papers and a big crowd of similarly minded people.

The structure of EVERY subsemigroup S of the additive sgp of

natural numbers is this: It has finitely many numbers followed (in

the natural order sense) by an infinite arithmetic progression with

some difference (=step of the progression) d. This d is the

greatest common divisor of all elements of S. If we divide all the

elements of S by d, we get an isomorphic semigroup S' that has

finitely many initial numbers followed by ALL natural numbers

beginning from a certain number n + 1.

I think this was known to the "ancients". If I remember correctly, it

was Dedekind who was interested in the size of n provided the

finitely many generators (in the irregular initial part of S') are

known. This is a tough problem. There were some papers (and tech

reports) on that published in Norway.

Now, Dedekind published papers and also wrote letters and also gave

talks and also... So if it was him indeed, the historians may know

the reference.

I guess if we look at things in a sufficiently creative way,

we can find this fact mentioned in the Bible.

Boris.