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.

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

The exceptional small commutative aperiodic semigroup

Commutative aperiodic semigroups can be studied in order to consider generalizations of semilattices, and to determine their lattice like properties. Ultimately, we can define order-preserving commutative aperiodic semigroups by the joins of distinct irreducible elements are preserved under the algebraic preordering. It is clear that this is the case for all commutative aperiodic semigroups of size three or less, but in the case of aperiodic semigroups of size four it is true for 30/31 of them, all except for a single exceptional case. This semigroup is identified as the GAP small semigroup 4, 34 and it can be considered the simplest counter-case to the order-preserving property.

[ [ 1, 1, 1, 1 ], 
  [ 1, 1, 1, 1 ], 
  [ 1, 1, 2, 1 ], 
  [ 1, 1, 1, 2 ] ]

Every single semigroup is associated with a preordering relation, and for commutative aperiodic semigroups this is a partial ordering relation. In the case of this commutative aperiodic semigroup, its algebraic ordering is the weak order [2 1 1], which is shown below.



The main issue with this small commutative aperiodic semigroup is that the product of 3 and 4 goes to 1 rather then to 2 which is the least upper bound, or the join, of the algebraic preordering. As a result, it does not preserve the ordering of its elements in its operation. It is clear that this is the unique smallest commutative aperiodic semigroup that has this property because it must have the algebraic preordering [2 1 1] so that distinct minimal elements can go to an upper bound which is different from the least one and this is the only such semigroup that does. This principle allows us to compare commutative aperiodic semigroups to semilattices in order to better generalize the properties of lattice theory to other commutative structures.

Generalizing semilattices

The generalizations of semilattices should be commutative and aperiodic (or group-free) because these semigroups are most similar to semilattices. Together, these two conditions ensure that every finite semigroup will have an algebraic preorder that is antisymmetric, that is that the elements of the semigroup are partially ordered. This isn't true for non-aperiodic semigroups, for example, abelian groups are obviously symmetrically ordered as every element can be acquired from every other one due to reversibility. It also isn't true for even the simplest non-commutative semigroup which is also a band as well as aperiodic.

A brief examination of commutative aperiodic semigroups reveals that not all of them tend tend to have the most semilattice-like properties. The first aperiodic commutative semigroup that I noticed did not look like a semilattice is the GAP small semigroup 4, 34 which doesn't preserve order joins over distinct elements of its minimal generating set. It is weak ordered by [2 1 1] and the distinct minimal elements go to the maximal element, rather then the middle which is also the least upper bound with respect to that algebraic ordering. This leads to the following definition.

Definition. a commutative aperiodic semigroup is order-preserving if it has a presentation over a set of generators such that the product of two elements a*b is equal to the join (least upper bound) of the algebraic preorder of the semigroup.

This leads to a broad generalization of semilattices within the category of commutative aperiodic semigroups. Then to consider the semilatticeness of a commutative aperiodic semigroup one could consider the number of relations that need to be added to the presentation in order to define the semilattice besides the order ones which are simply assumed.

This leads to the notion of commutative aperiodic semigroups which can be determined by multiset systems, rather then by set systems as semilattices are. Semilattices are defined by their join representations, which produce join representation families in the theory of set systems, for example, finite power sets correspond to the join representations of boolean algebras.

Commutative operations

Set theory, the theory of set systems, etc provides the best foundation for mathematics. Sets are embedded within a distributive lattice so they have commutative operations of union and intersection. Set systems can therefore be considered in terms of the union operation. Noticeably, the union operation produces redundant elements so in the set system {{0 1} {2 3} {0 1 2 3}} the elements {0 1} and {2 3} can be eliminated. Likewise, in {{0 1} {0 2} {1 2}} any element can be eliminated as redundant without effecting the union.

The missing element is the free commutative monoid, because then there are no identities in the presentation which can be used to eliminate elements except for the identity element. The identity element is always a redundant element that doesn't effect an operation so there is nothing we can do about that. Thats why small set systems of a certain size avoid considerations about the inclusion of the identity element. So for {{0 1} {0 2} {1 2}} the sum in the free commutative monoid is {0 0 1 1 2 2} and the elimination of any element strictly reduces the result of the operation. So the most interesting element is free commutative monoids which of course produce multisets.

Sets are directly linked to semilattices because they forbid repetition, and semilattices make any repetition redundant to the result of the operation. This is why there is an underlying set (the support) of any multiset produced by the idempotent-reduction of the multiset in the free commutative monoid. A natural question is how to generalize semilattices. The first that comes to mind in generalizing semilattices is that the commutative semigroup generalization should be aperiodic, because that preserves the antisymmetry of the algebraic preordering. The closest thing to a semilattice is an aperidoic commutative max index two unique non-idempotent semigroup, because semilattices require every element to be idempotent and these structures only have one element that is not. The max index of the aperidoic commutative semigroup is the max multiplicity of irredundant representations.

Many more aperiodic commutative semigroups share properties with semilattices though, without being the most similar to them. In particular, they can tend to preserve the monotonicity of presentations which is true until the weak ordered [2 1 1] commutative aperiodic semigroup of order four which doesn't preserve joins of its minimal elements. Commutative semigroups are clearly the closest to commutative semigroups, which is why we can compare the combination of elements to their joins to see similarity to semilattices and the semilattice decompositions feature in their decomposition theory. Its also worth mentioning max order two abelian groups like the xor operation which generalize the symmetric difference and therefore operate on sets, but they are not aperiodic and as groups their preorder is actually symmetric rather then antisymmetric which is what we wanted.



Commutative semigroups can naturally be described by lattice theory by considering presentations in terms of the distributive lattice of multisets. The questions of commutative semigroups then are related to their presentation in this lattice, and its operations like interiors and closures. In the other direction considerations of sets of multisets (as subsets of the multiset inclusion lattice) can be used to better understand commutative semigroups. In this sense, commutative semigroups relate the most to lattices and semilattices.

We considered commutative semigroups and their relation to lattices. However, there are two main objects of study in basic abstract algebra: structures with a single binary operation and structures with two binary operations related to one another. We can generalize from semilattices to abandon idempotence which gets us commutative semigroups, but something different happens when we abandon idempotence in lattices. A commutative semigroup pair (S,+,*) contains two commutative semigroups related by distributivity.

• The operations + and * are semigroups
• Commutativity : the operations are both commutative
• Distributivity : the operations distribute over one another

By this definition all distributive lattices and commutative hemirings are special cases and of course this means that commutative rings are as well as they are a special case of commutative hemirings. Commutative rings are of course a common common area of study then commutative hemirings because the relation between congruence and ideals in them. Importance is lended to commutative hemirings because of the fact that the natural numbers (N,+,*) are one of them which means all arithmetic can be described by hemiring theory. On the other hand, logical operations like boolean algebra and set theory can be described by distributive lattices. It is fascinating to notice this relation between fundamental arithmetic and logic.

The combined conditions of commutativity and distributivity mean that any multivariate polynomial can be described as a multiset of multisets, or a multiset system in some normal form. This relates the theory of multiset systems we are familiar with to the theories of commutative algebra and algebraic geometry. In boolean algebra, this is the relationship between set systems and boolean operations which plays a key role in the set theoretic ontology. This property noticeably only works for distributive lattices though, so for other lattices polynomials can take different forms. Lattice theory can be divided into two halves (1) the theory of distributive lattices and (2) the theory of non-distributive lattices. So a great many important lattices don't have the distributive property.

The commutative hemiring (N,+,*) is actually described by the free commutative monoid on one generator and by the fundamental theorem of arithmetic the free commutative monoid on an infinite generating set adjoined with a zero element. Which means multiplication is free except when dealing with the zero element, which absorbs or annihilates everything else as redundant when it appears in an expression. This means that fundamentally arithmetic is just the combination of multisets keeping repetition. Which demonstrates why free commutative monoids are the most fundamental concept we introduced when studying commutative semigroup theory.

I want the initial guiding principle of commutative hemiring theory to be the actually be the study of iteration, which can then be used in any semigroup because all monogenic semigroups are commutative (as seen in the ontology of commutative semigroups above). Addition is the composition of iterations, and multiplication is repeated iteration. In this way, modular arithmetic forms a commutative ring which describes the iteration properties of automorphisms. Eventually modular arithmetic could be used to describe the iteration of any monogenic semigroup element. Natural numbers describe iteration of functions without congruences, integers exactly describe iteration of functions and elements with inverses, and the rational and real numbers describe fractional iterations. Selecting structures with a specific purpose limits the selection of arbitrary structures.

Besides the theory of arithmetic and monogenic semigroups, it is noted that the subsemigroups of a semigroup form a lattice which means lattice theory can also be used in understanding general semigroups. Monogenic semigroups include the smallest possible semigroups because if a semigroup is not monogenic, a smaller monogenic semigroup can always be produced by determining the subsemigroup generated by a single element. Monogenic semigroups are also commutative, which means that certain commutative semigroups build up all semigroups. In this sense, commutative semigroups can be used to understand all semigroups. In finite group theory, the join irreducibles are actually precisely the cyclic groups of prime power order.

Commutative operations are strictly simpler then general binary operations, because they use less information then them. The output of a commutative operation is entirely determined by the underlying multiset. This property is also shared by left and right invariant binary operations, which only use the first or second argument to determine their output. These types of binary operations aren't that interesting though, because they are entirely determined by ordinary functions. This leaves us with the key of commutative operations.