Thursday, June 30, 2016

Families of kuratowski pairs

Families of kuratowski pairs which are set systems that define a pair between two elements fully characterize correspondences between sets. It is by using families of kuratowski pairs that we can deal with questions of relation algebra. Each family of kuratowski pairs necessarily has rank at most two because each kuratowski pair has a maximum cardinality of two. This means that these are rank limited set systems.

Wednesday, June 15, 2016

The rank of set systems

Given a set of sets then the rank of that set system is the maximum cardinality of the sets it contains as members. We can determine certain characteristics of a set system based upon their rank. Set systems of rank at most one are rank complete which means they are like a set which may or may not contain the empty set. Set systems of rank at most two include as a particular case set systems that define comparability. This process can be extended further to determine properties of the rank of higher set systems.

The rank property can also be used to understand sets of sets of sets. Given a sets of sets of sets then the rank is the maximum cardinality of a set of sets contained in the set of sets of sets. We can deduce that families of kuratowski pairs necessarily have a rank of at most two because the pairs never have more then two members. The families of kuratowski pairs effectively generalize correspondences between sets. Kuratowski pairs also have the property that their union has at most two elements so the rank of the set of unions is at most two which makes the ranking property even stronger for them.