Set systems of small size

In the most general sense, the size of any entity is the number of parts that it has. The parts of any entity in metaphysics rather it is an abstract object like a collection or a number or a concrete entity like a physical object is determined by a partial ordering relation. For general sets, it is rather obvious that the size of a set is the number of elements it has which is the cardinality. This is rather trivial, and it doesn't require further consideration. When considering set systems, not only does the set system have parts so does its members. So a better way of determining the size of a set system is the cardinality sum, which is the sum of all the cardinalities of members in the set system. Size is monotone with respect to complexity.

Set systems of size one or less:

All the set systems of size one or less are trivial like {} and {{0}}. It doesn't matter for the purposes of this classification to consider rather or not the set systems contain the empty set.

Set systems of size two:

All the set systems of size two are pairwise disjoint so we can consider the two set partitions of two elements which are {{0} {1}} and {{0, 1}}.

Set systems of size three:

Signature 1,1,1
There are three set partitions of a set of three elements. These are the set partition {{0}, {1}, {2}}, the set partition {{0}, {1,2}} and the set partition {{0, 1, 2}}. It should be clear that all the set partitions are determined up to isomorphism by their cardinality multisets.

Signature 1,2
The next set system on three elements is {{0},{0,1}} which is the first ordered set system and the simplest set system that is not pairwise disjoint. This is why it is chosen to be the set theoretic definition of the ordered pair. Extending from the theory of set systems and continuing to the theory of sets of sets of sets, one arrives at the theory of sets of kuratowski pairs, also known as binary relations, which forms one of the most basic concepts of pure set theory.

Set systems of size four:

Signature 1,1,1,1
There are five set partitions of four elements:
{{0},{1},{2},{3}}
{{0},{1},{2,3}}
{{0,1},{2,3}}
{{0},{1,2,3}}
{{0,1,2,3}}.

Signature 1,1,2
There are three types of set systems with degree signature 1,1,2. These set systems are pseudo-independent because they are nearly pairwise disjoint as seen by their degree signature. Each degree signature whose multiplicities are consecutive starting with one, are associated with a single chain partition which in the case of this set system looks like {{0},{0,1,2}}. This is a chain because it is totally ordered under inclusion, and it is laminar because all chains are laminar.

Another laminar set system with this degree signature is {{0},{1},{1,2}} which is the unique non-order connected laminar set system with this degree signature. This is equal to a kuratowski pair unioned with a disjoint singleton set. As a result its connectivity cardinality multiset is {1,3}.

The set system {{0,1},{1,2}} is the simplest non-laminar set system. In the same manner that the kuratowski pair represents the simplest set system which isn't pairwise disjoint, this set system represents the simplest set system which isn't laminar. The complementary clique family to the laminar family is the set of all anti-disjoint sperner families, so this is the simplest non-disjoint sperner family. Anti-disjoint sperner families are not order connectivity preserving, because they are intersection connected but not inclusion connected.

Signature 2,2
The power set like set system {{0},{1},{0,1}} is the only set system with its degree signature, as its the maximal degree signature of order two. It is a laminar set system, which corresponds to the join representations of a height two upper tree of order three.

Set systems of size five:

Signature 1,1,1,1
There are seven set partitions of five elements:
{{0} {1} {2} {3} {4}}
{{0} {1} {2} {3 4}}
{{0} {1} {2 3 4}}
{{0} {1 2} {3 4}}
{{0} {1 2 3 4}}
{{0 1} {2 3 4}}
{{0 1 2 3 4}}

Signature 1,1,1,2
Set systems of this degree signature are pseudo-independent because they are nearly pairwise disjoint as removing a single element makes them set partitions. As a result, all of the laminar set systems of this sort are going to be height two interval ordered subsingleton-free multichain families. To begin with there is the chain partition {{0},{0,1,2,3}} constructing by adding a subsingleton to the four element set.

Then we can consider the case of adding a subsingleton to the three element which gets us {{0},{1},{1,2,3}} which also is not order connected as before. There has 1,4 connectivity signature.

Then we can consider the cases where we add a subsingleton to a two element set. This results in {{0},{1},{2},{2,3}} which has 1,1,3 connectivity signature and {{0,1},{2},{2,3}} which has 2,3 connectivity signature. These are both set systems based upon adding a subsingleton to pair but they are distinguished by their connectivity signatures. These can also be obtained by unioning a disjoint set system with a kuratowski pair.

The two non-laminar set systems with these degree signature are the maximal cliques of the paw and the co-paw. The paw cliques family {{0,1,2},{2,3}} is uniquely dependent well the co-paw cliques family {{0,1},{1,2},{3}} is not uniquely dependent. As the co-paws family is triangle free all its cliques are trivial and it could also be considered an independency family.

Signature 1,2,2
There are four types of set systems with signature 1,2,2. To begin with there is the chain partition, which is always associated with a degree signature of this sort which is {{0 1} {0 1 2}}. As a chain, this naturally corresponds to a total preorder under the operation of taking principal ideals.

Then there is the upper tree order containment family {{0},{1},{0,1,2}}. This corresponds to the principal ideals on the height two upper tree of size three, rather then the join representations discussed previously because even though 2 is join irreducible it is an element of this set system.

The set system {{0} {1} {2} {1 2}} corresponds to the join representations of an upper tree as well as a singleton set added to it. It is like the set system on 2,2 with a singleton added to it because it is not uniquely connected.

The set systems described so far have been laminar. There are two types of set systems with this size and this order which are non-laminar and they are based upon adding singleton elements to the previously-described non-laminar path independency family. The one with this degree signature is {{0,1},{1,2} {2}} which is based upon adding a loop at one of the endpoints of the path independency family rather than in its center.

Signature 1,1,3
There is only one set system with this degree signature and it is the star {{1},{0,1},{1,2}} which has a common member of all of its sets. This is the order containment family on the height two lower tree with three elements. Although this is order connectivity preserving, it is also not laminar because laminar set systems tend to be upper trees under their inclusion ordering. It is based upon adding a loop to the center of the non-laminar path independency family described previously.

Set systems of size six:

Signature 1,1,1,1,1,1
There are eleven set partitions of size elements.
{{0},{1},{2},{3},{4},{5}}
{{0},{1},{2},{3},{4,5}}
{{0},{1},{2,3},{4,5}}
{{0,1},{2,3},{4,5}}
{{0},{1},{2},{3,4,5}}
{{0},{1,2},{3,4,5}}
{{0,1,2},{3,4,5}}
{{0},{1},{2,3,4,5}}
{{0,1},{2,3,4,5}}
{{0},{1,2,3,4,5}}
{{0,1,2,3,4,5}}

Signature 1,1,1,1,2
In order to classify all set systems of this degree signature we will begin with the uniquely dependent set systems. The only laminar uniquely depedent family is the chain family corresponding to this signature which is {{0},{0,1,2,3,4}}.

There are also two uniquely dependent families which are maximal clique families of certain graphs. The maximal cliques of the butterfly {{0,1,2},{2,3,4}} is the first maximal cliques family of this sort and the other is the maximal cliques of the (4,1) lollipop graph {{0,1,2,3},{3,4}}.

Among the non uniquely dependent ones we have the chain plus one element {{0},{1},{1,2,3,4}} which is a laminar family for a multichain preorder.

Then we have the paw plus one element {{0,1},{1,2,3},{4}} which is effectively a maximal cliques family with a singleton set corresponding to an isolated vertex.

The path can be connected to two types of partitions yielding {{0,1},{1,2},{3},{4}} and {{0,1},{1,2},{3,4}} the later of which is a uniform set system.

The 3-chain can also be connected to two types of partitions yielding {{0,1},{2},{2,3,4}} and its refinement {{0},{1},{2},{2,3,4}}.

Then finally there are three types of set partitions that can be added to a kuratowski pair {{0,1,2},{3},{3,4}} its partial refinement {{0},{1,2},{3},{3,4}} and its total refinement {{0},{1},{2},{3},{3,4}}. All of these set systems are laminar.

Signature 1,1,2,2
There are two types of rank four laminar families with this signature the chain {{0,1},{0,1,2,3}} and the lower tree {{0},{1},{0,1,2,3}}.

The maximal cliques of the diamond graph are {{0,1,2},{2,3,4}} which are two sets that intersect at two common points which have each have a degree of two.

The paw graph has two types of peripheral vertices which can have loops added to them to in their maximal cliques family to get two different types of set system {{0,1},{1,2,3},{0}} and {{0,1},{1,2,3},{3}}.

Then the maximal cliques on the path graph of four elements {{0,1},{1,2},{2,3}} also has this signature and it is rank two.

There are two types of laminar families with a singleton added to them {{0},{1},{2},{1,2,3}} and {{0},{1,2},{1,2,3}}. Both of these have connectivity type 1,5.

Then there is the non-laminar family with a singleton added to it {{0},{1,2},{2,3},{3}} which has this signature and connectivity type 1,5.

Finally we can consider the two types of set systems with a 2+2 component and the two other types of disjoint components added to them which are {{0},{1},{2},{3},{2,3}} and {{0,1},{2},{3},{2,3}}. Both of these set systems are laminar like the other types with larger connected components described previously.

Signature 2,2,2
As this a regular degree signature, all set systems of this signature will be regular families. To begin with, there are two laminar families with this signature both of which have rank three, which means they have a triangle. The first is {{0,1,2},{0},{1},{2}} and the second is {{0,1,2},{0},{1,2}}.

The path graph has two peripheral vertices and one central vertices which has a higher degree then the periphery ones. In order to regularize them, we can add loops to the peripheral vertices to get the set system {{0,1},{1,2},{0},{2}} which corresponds to the path independency family with subsingletons added to its endpoints.

Finally there is the triangle independency family {{0,1},{0,2},{1,2}} which is the simplest independency family which is not a maximal cliques family. Actually this corresponds to the two section of the clique {0,1,2}.

Signature 1,1,1,3
The only laminar set system with this degree signature is the order containment family of the height two lower forest constructed by adding a single element to the size three lower tree {{0},{1},{1,2},{1,3}}.

The triangle free independency family {{0,1},{0,2},{0,3}} corresponds to the non-trivial edges of the star graph on four elements. It is also a uniform set system because each set has the same cardinality.

Adding a singleton set to the center of the maximal cliques set system of the paw gets the set system {{1},{0,1},{1,2,3}} which has this degree signature. It is neither a laminar family or a maximal cliques family.

Signature 1,2,3
The chain progression partition corresponding to this degree signature {{0},{0,1},{0,1,2}} is the unique progression family type on three elements.

Well the chain partition is naturally associated with this degree signature, another set system with this degree signature is the join representations of the four element fence partial order {{0},{1},{0,1},{1,2}} which is also an uneven path graph with a loop at one center vertex and one endpoint vertex and one periphery vertex. Both of these set systems are laminar.