Set theory in a multiset theoretic context

Set theory is the basic foundation of mathematics, but we can consider certain aspects of set theory from a multiset theoretic perspective. Sets can be considered to be special types of multisets and set systems can be considered to be special types of multiset partitions. In order to get the multiset corresponding to a set system, you can get the combinational multiset of the set system by combining each set counting multiplicity. The multiplicity of each element in this multiset is the degree of the element in the set system.

• Sets : multisets with repetitiveness zero
• Set systems : distinct multiset partitions with distinct parts

Set systems come in different forms based upon the multisets that they partition. General set systems have various different combinational multisets that come in various degrees of repetitiveness. If the combinational repetitiveness of the set system is zero, then the set system is a set partition. If the combinational repetitiveness is one, then it is pseudoindependent. Well general systems partition multisets, set partitions partition sets. As the set systems with the smallest combinational repetitiveness, set partitions will play a fundamental role in our ontology.

• Set partitions: set systems whose combinational multisets are also sets
• Pseudoindependent families : set systems whose combinational multisets have max repetitiveness one

The isomorphism types of set partitions are equivalent to the signatures of multisets, as multisets are essentially sets that have a set partition associated with them. The set partition associated with a multiset tells which elements are repetitions of one another, well for a set this is simply trivial. This demonstrates why set partitions and multisets must be two of the most basic elements of the ontology. With this realized, there are two types of partial orders that are immediately realised on set partitions and set systems in general:

• Degree reduction ordering
• Refinement and coarsification ordering

The importance of the degree reducing ordering of set systems was already described as it demonstrates the simplification of set systems, and the operation of taking the combinational multiset is monotone with respect to the degree reduction ordering. This makes multisets a fundamental part of set theory. The refinement and coarsification ordering determines the degree to which a set system is partitioned, rather then the element it is partitioning. The refinement ordering on set partitions is especially important as it describes the extent to which a set is partitioned. Multisets can also be coarsified, for example, given a set system which has the finest partition we can get the multiset of cardinalities of that set system which can coarsify the collection to some extent based upon cardinality equality. This demonstrates the relation between multisets and set partitions.