## 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 related 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 adjacencies family. Both of these families are nullfree and the preorder containment family is also union free. The adjacencies family is also subunique free as it is subsingleton free in addition to nullfree.