Eric Forgy
Multiset
Warning: This page represents personal notes and likely contains errors. Feedback is welcome.
See An Overview of the Applications of Multisets
A nice way to denote a multiset is
where is a set and is a multiplicity function.
What is the βbestβ way to both define multisets and to denote multisets?
One thing I would like to do is allow on some elements of , but as Toby pointed out, this leads to thinks like
and
denoting βdifferentβ multisets, but weβd like to think of these as describing the βsameβ multiset .
How can we settle this?
One way is to specify a universal set and then a multiset is just a function .
Then weβd have
In this way, it is easy to verify that
and
This is βokβ, but Iβm not excited about it because we lose the feeling that weβre working with βsetsβ.
Is there another way?
We can define a multiset via a surjection with multiplicity function given by the cardinality of the fiber, i.e.
Then weβd have
and is the multiset with underlying set and multiplicity given by (not sure if this is right)
snip
Oh! Free Abelian monoids
A multiset is a free Abelian monoid on Set.
Oh! Hassler Whitney (one of my all-time favorite mathematicians) has thought about multisets!
- Characteristic Functions and the Algebra of Logic, Annals of Mathematics, Vol 34 (1933), 405-414.
Revised on October 30, 2009 at 21:53:51
by
Eric Forgy