Tuesday, December 16, 2014

Multiset systems

One of the critical problems of set theory is how to represent sequences as sets. Well Kuratowski solved the problem for sequences of size two through the set theoretic definition of the ordered pair there is no obvious solution to the problem of representing sequences of arbitrary size as set systems because sequences may contain repeated elements. I have thought about this problem so much in terms of set systems that I did not consider the possibility of using a multiset system instead.

In order to represent any sequence all we need to do is use a multiset system containing the multiset of elements up to a point in the sequence for each point in the sequence. This solution is so simple I am surprised I did not hear about this before. It is probably because multisets are far too often pushed aside for sets but no more. From now on I am going to make full use of multisets when I think about mathematics.

Wednesday, November 26, 2014

Disjoint union closed families

The disjoint union closed families are precisely those families for which it is the case that the union of any two disjoint sets in the family is contained within the family. The disjoint union closed families generalize both the union closed families which are both disjoint union closed and nondisjoint union closed and the symmetric difference closed families. The symmetric difference of any two disjoint sets is their union so symmetric difference closed families are disjoint union closed.

Besides the union closed families and the symmetric difference closed families the antidisjoint families are disjoint union closed. The antidisjoint families are precisely those families of sets whose every pair of sets is not disjoint. These include the nullfree chain families which also happen to have the property that they are union closed as they are chain families as well as the antidisjoint sperner families which are antidisjoint families that also contain no comparable pairs.

Monday, November 24, 2014

Nondisjoint union closed families

A family of sets is nondisjoint union closed if for all pairs of sets in the family whenever those sets have a nonempty intersection then their union is contained in the family. Such families of sets are order connectivity preserving because each dependent component of the family is upper bounded by its union and it is therefore connected.

The laminar families are of course nondisjoint union closed because the only nondisjoint pairs of sets that appear in a laminar family are chains. This means that independent families which never contain nondisjoint pairs and laminar multichain families are nondisjoint union closed.

The union closed families are of course also nondisjoint union closed as they are both disjoint union closed and nondisjoint union closed at the same time. The connectivity complexes are precisely those families of sets that are subunique closed as well as nondisjoint union closed. The connectivity complexes describe the connected sets of a structure which are nondisjoint union closed because if two nondisjoint sets are connected then so is their union.

Sunday, November 16, 2014

Order connectivity preserving families

The order connectivity preserving families are precisely those families whose comparable connected components and dependent connected components are equal. This implies that order connectivity preserving families generalize preorder containment families and irreducible containment families which are two of the major ways of converting partial orders into set systems. Here are some examples of such order connectivity preserving families:
#{#{0 1} #{2 3}}
#{#{} #{0} #{1} #{0 1}}
#{#{0} #{0 1} #{0 2} #{0 1 2 3}}
An example of a family of sets that is not order connectivity preserving is #{#{0 1} #{1 2}}. The two sets #{0 1} a #{1 2} are dependent despite being incomparable so this is not an order connectivity preserving family. Well it is possible to have a family of sets with such intersecting elements that is order connectivity preserving such as #{#{0 1} #{1 2} #{0 1 2}} considering that all uniquely order connected families preserve order connectivity it is only the laminar families which forbid dependent incomparable sets that are order connectivity preserving for all their subsets. All order connectivity preserving multichain families are laminar as is the case with multichain containment families.

Friday, October 31, 2014

Laminar families

The laminar families are precisely those families of sets in which each pair of sets is either comparable or independent. It is therefore implied that the laminar families include both chain families which are precisely those families in which each pair of sets is comparable and independent families which are precisely those families in which ear pair of sets is independent. Here are some such laminar families:
#{#{0 1 2} #{3 4 5}}
#{#{0} #{0 1} #{2} #{2 3}}
#{#{0} #{0 1} #{0 1 2} #{0 1 2 3}}
By definition the intersection of any pair of incomparable sets in a laminar family is the empty set so nullfree laminar families are intersection free. Laminar multichain families which generalize both chain families and independent sperner families are union free in addition to intersection free so they are all examples of extrema free laminar families.

Neighbourhood families

Given a family of sets and a point in the union of the family of sets the neighbourhood of sets around that point is the family of all sets that contain that point as a member. Given the collection of the neighbourhoods of a given family of sets there are two set systems that we can form from the neighbourhoods using union and intersection.

By taking the intersection of each of the neighbourhoods of the family of sets we can get a preorder containment family and by taking the union of each of the neighbourhoods of the family of sets we can get a neighbourhoods family. Both of these families are nullfree and the preorder containment family is also union free. The neighbourhoods family is also subunique free as it is subsingleton free in addition to nullfree

Tuesday, September 30, 2014

Managing complexity

Complexity is interpreted in a variety of ways. With mereology systems that have more parts tend to be more complex then ones that do not. Within information theory the notion of complexity is determined by Kolmogrov complexity. In computer science there is a notion of complexity called computational complexity which is related to the growth rate of the computation time with respect to argument size.