Finite commutative groups and multiplicative partitions

There are two fundamental types of semigroups: groups and the group-free semigroups which we call aperiodic (Krohn-Rhodes theory decomposes semigroups in terms of these). In particular, these are the two most basic types of commutative semigroups. The theory of finite commutative groups is completely solved, by their fundamental theorem. Finite commutative aperiodic semigroups have a more interesting theory, so they need to be considered more later. Nonetheless, the fact that the isomorphism types of finite commutative aperiodic semigroups correspond to multiplicative partitions is worth considering.

{}
{2}
{3}
{4} 
{2,2}
{5}
{2,3}
{7}
{2,2,2}
{2,4}
{8}
{3,3}
{9}
{2,5}
{11}
{2,2,3}
{4,3}

All multiplicative partitions correspond to multisets of multisets, which further can be classified into four types based upon the distinctiveness of their members as previously mentioned. Finite order-rank set systems correspond to a special type of multiplicative partition. An alternative is to classify by the equality of members rather then their distictiveness.

• Multiclan: a multiset of multisets
• Equal multiclan: a multiset of multisets such that each multiset is equal to one another
• Max order one multiclan: a multiset of equal multisets
• Equal max order one multiclan: a multiset of equal multisets all of which are equal to one another

The prime power multiplicative partitions of a number correspond to nullfree max order one multiclans, so the interesting thing is that these finite abelian groups have a multiset systems theory associated with them.

Theorem. a prime power multiplicative partition forms a disjoint union of total preorders under divisibility.

Proof. let $S$ be the support of the product of the elements of the prime power partition, in other words, all the distinct prime factors of all the elements of the multiplicative partition. Then for each element of $x$ of $S$ we can get the supermembers of the singleton set ${x}$ of the multiset of prime factorization multisets of the system, or in other words, all the numbers divided by that prime number $x$. Then by the fact that this a prime power multiplicative partition, we know that these are all disjoint from one another, as no number contains two distinct primes as divisors. Then in order to see that these are total preorders, we will prove that the support of these elements are total orders. A set of distinct powers of a prime is totally ordered under divisibility because the set of distinct prime powers can be embedded in the total order of the natural numbers by mapping the exponent to a natural number and then using that to determine divisibility. Therefore the whole collection is a disjoint union of total preorders.

A basic example is the distinct multiplicative partition of 216 which is {2,4,3,9} which being a distinct partition forms a disjoint union of total orders rather then preorders. This is equal to the partial order 2+2 formed by combining the two total orders with two elements. This is just a small tidbit about these commutative groups. Next we will further explore finite commutative aperiodic semigroups.