couple questions involving monoids/isomorphisms

calvino

I'm having trouble with these two questions.

Problem 1: Prove that the multiplicative monoid N of natural numbers 1, 2,... is a free commutative monoid.

Problem 2: Is the multiplicative monoid N isomorphic to the additive monoid N_0 x N_0 x ...x N_0 (n times), for any n = 1, 2,...? Prove your claim.

If i can get any tips on where to begin with them, that would be great. from there i will work on an show where i have progressed. right now, all i can think of is using induction in 2 to prove that there is an isomorphism for all n. i do this, by looking at neutral elements in each monoid, and constructing the isomorphism for n=1. that's all i can think up.

calvino

i actually figured out 1. it was fairly simple. 2 is a little trickier though. haven't made any progress on it, im afraid.

calvino

heres what i've been thinking:
every value in N can be written as a product of primes, which can be directly related to N_0 x N_0 x ...x N_0. for instance 45= 9 * 5= (3^2)*5. this can be written as (2,1,0,0,...). where the values in that "vector" are the powers of increasing powers of primes. my problem comes when i start thinking of when n=1. what occurs then? i know the concept is there, and that i might end up using properties of commutative free monoids, but what?

matt grime

Can something that is not finitely generated be isomorphic to something that is?

calvino

no, which is exactly the case with n an integer. so is that already my solution? that i should prove there is no isomorphism? if so, then does it suffice to use that as a proof? that (N_0,+) which itself has rank 1 (and further more any monoid with rank n) cannot be isomorphic to a monoid with countably infinte generators? i believe that's a theorem in itself - two monoids are isomorphic if their ranks are equal

calvino

one thing that troubles me, however, is that i have the following equivalent statements in my notes. perhaps i haven't full understood its meaning?

"the following are equivalent for a monoid S:

i) S =~ F_Abmon (X) for some X

ii) there exists an X, submonoid of S such that for all a in S, a not equal to e_S (netural in S), there exist a unique representation a= * x_i ^k_i for i=1 to r (r,k_i are greater or equal to 1, and no x_i are similar."

here, * is the concatination of elements. ii) resembles problem 2 in the sense that every element in N can be written as a product of primes, the generators in N. this would mean that the isomorphism would exist between (N_O,+) and (N,x)....according to ii), anyway. this is exactly the opposite of what you just said, i believe, since their ranks are totally different.

edit: i should note, just in case, that F-Abmon (X) is just the free commutative monoid of X, and =~ means isomorphic.

matt grime

Why would you think that ii) implies an isomorphism of monoids between N_0 underr addition and N under multiplication? Since * in the former is addition and * in the latter is multiplication there is no isomorphism: there is no uniqueness about writing an integer as the *sum* of primes (5=5=2+3).

calvino

that cleared some things up. thanks.

that being said, does it suffice to say that since N_0 X N_0 X ... X N_0 (n times) equipped with addition , for any n=1,2,3,... is generated by n values, and (N,*) has countably infinite generators that they cannot be isomorphic? or is there a more extensive proof of this?

matt grime

Of course there is a more extensive proof - you could write out all the details. What level of detail you need to furnish depends on for whom you are writing.