Types of multiset systems

A set of sets is often called a family. Therefore, a multiset of sets can be called a multifamily. The idea of a set of multisets has rarely been considered, owing to the rarity of studies of multisets. There are a couple of publications that call them "macrosets" but that is a bit verbose and then adding multi to that wouldn't be right. I developed the idea of calling them clans instead, so we could have a better understanding of the theory.

• Family: a set of sets
• Multifamily: a multisets of sets
• Clan: a set of multisets
• Multiclan: a multiset of multisets

As I discussed earlier the fundamental nature of sets and multisets emerges from commutative operations and particularly the fact that the simplest arithmetic operations like addition and multiplication are commutative. In particular, multisets directly emerge from the prime factorizations of any number. Then prime distributions are the relative frequencies of the factorization, prime signatures are the signatures of these multisets, etc. All the different types of multiset systems have corresponding types of multiplicative partitions in number theory.

• Families: sets of squarefree numbers
• Multifamily: multisets of squarefree numbers
• Clan: any set of numbers
• Multifamily: any multiset of numbers

So for example {2,6,30} is a set of square free numbers which corresponds to a set system, actually a progression family. Well {2,2,3,3} corresponds to a unary multifamily because it is a multiset of prime numbers. Then {1,2,3,4} is a set of numbers well anything like {12,12} is going to be none of these types of structures. Here is an interesting tidbit {2,6} is a set of square free numbers which corresponds to a kuratowski pair set system well their product 12 corresponds to a kuratowski pair multiset because multiplication corresponds to multiset addition due to the freeness of the multiplication operation.

References:
Toward a Formal Macroset Theory:
https://dl.acm.org/citation.cfm?id=721844