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.