Max order one multiset systems

In a previous post I described how the ordering of distinct max order one multiset systems is a disjoint union of total orders. This describes the order type of the support of a max order one multiset system but it does not completely describe the multiset system type of the max order one multiset system.

Therefore, we need to further consider max order one multiset systems, and the multisets of prime powers, which often emerge from commutative groups. Consider as an example {2,2,4,4,3,3,27} then in this case we can partition this by the support of each of these multisets and therefore we will get {2,2,4,4} and {3,3,27}. Then each of these has their own signature defined by the multiset of exponents of each of these multisets of prime factorizations in this case, though the multiplicative concept of a positive integer and a finitary multiset are equivalent. The multiset system type of these is then the multiset of exponent signatures of these components.

So for the example of {2,2,4,4} we will have {1,1,2,2} and for {3,3,27} we will have {1,1,3}. The overall type is then {{1,1,2,2},{1,1,3}. This perhaps demonstrates that multisets of signatures will play a key role in understanding multiset systems. The only other detail is the multiplicity of the empty set, in the not necessarily nullfree case, which is simply a non-negative integer. These multisets of signatures can also be acquired from the membership signatures of each dimember of a multiset system, which generalizes signatures of set systems themselves.

A larger example is {2,2,4,3,3,9,5,125,125,3125,3125,3125,7,343} which produces the multiset system type {{1,1,2},{1,1,2},{1,3,5},{1,3}}. This demonstrates that this multiset of signatures can have repetition as we can see that {2,2,4} is isomorphic to {3,3,9} as a multiset system of prime factorizations. Together this fully defines these multiset systems that emerge from commutative groups. The height of each element of each total order is the support size of each of these multisets, so we can get the order type as well from this full description.