Young's lattice

There are many different lattices, so it is important to know which ones are important and why. This is especially true of integer partitions where are there many different types of orderings. The importance of the Young's lattice is that it describes the ordering of the signatures of multisets, for example, for one number to divide another, its prime signature must precede the other number's in the Young's lattice. A multiset can only be included in another if its signature is in the Young's lattice.



The signatures play a similar role in multiset theory to cardinal numbers in set theory, as they describe the ordering properties of multisets. The multiset ordering forms a distributive lattice. The amazing thing is that the Young's lattice forms a distributive lattice as well. As a distributive lattice, the signatures of the Young's lattice are sets of multisets of maximum order two whose two atoms are the types of partitions conjugate to one another.