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.

Information theory

Information theory studies the ways in which we can encode information. This includes the study of the lossless compression of information. Information theory is related to thermodynamics by Landauer's principle which is always related to reversible computing.