Foundational distributive lattices

An unordered collection can be either a set or a multiset. Unordered collections are the foundational core of mathematics, as well as the basis of all ontologies. All unordered collections form distributive lattices. Sets and multisets also have an isomorphism types associated with them, for the sets it is the cardinal numbers, and for multisets it is the multiplicities signatures. In order for a set to be included in another its cardinality must be less then the cardinality of the other set. In order for a multiset to be included in another its signature must be less then the signature of the other multiset.

• Sets : cardinal numbers
• Multisets : multiplicities signature

The cardinals form a total order which means they are also distributive. The multiplicities signatures are partially ordered by the Young's lattice, which also forms a distributive lattice. Both the cardinals and the multiplicities signatures can therefore be included among the set of distributive lattice ordered structures. As a consequence of this, the Young's lattice is one of the most fundamental distributive lattices in lattice theory. I will talk about the Young's lattice in more detail next.