Dear Peter, I think what I said amounts to the requirement that a kappa-indexed sum of cardinals smaller than kappa should again be smaller than kappa, i.e., regularity. For a counterexample, consider k = aleph_omega (a non-regular cardinal), and let M_k denote the functor for multisets of cardinality < k. There's an element b of M_k(M_k({*})) given as follows. For each natural n, let a_n be the multiset consisting of aleph_n copies of *, and let b be the multiset [a_1, a_2, ...]. Then the multiset multiplication of b is given by alpha_omega copies of *, and so doesn't live inside M_k({*}). Best wishes, Richard --On 23 June 2009 13:27 Peter Selinger wrote:

I don't think so. As you correctly write, all you need is that k x k is no bigger than k. This is true for any infinite cardinal. -- Peter

Richard Garner wrote:

...

...

In the case where k=omega, T is the well-known finite multiset monad, which associates to each S the free commutative monoid generated by S (whose elements are also known as finite multisets in S).

For other k, I would call this the "monad of multisets of size less than k". I think this works for any infinite small cardinal, not just regular ones.

You really do need the regularity. Otherwise removing brackets from a k-small multiset of k-small multisets might yield something bigger than a k-small multiset, and then one cannot define a multiplication for the monad.

Richard

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]