Ah yes I see. Multisets <= k will always form a monad because they are the same as multisets < k^+ and successor cardinals are always regular! Best wishes, Richard --On 23 June 2009 15:55 Peter Selinger wrote:

Hi Richard,

thanks for correcting me on this. I was confused by what was meant by a k-set. I momentarily took this to mean "a set of cardinality at most k", and not "a set of cardinality strictly less than k". With the first definition, the construction would work for any infinite cardinal, but then of course the class of finite sets would not be an example.

-- Peter

Richard Garner wrote:

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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:

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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:

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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

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