## 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.