Tuesday, April 29, 2014

Representing order elements as sets

Given a partially ordered set we can always represent the elements of that partially ordered set as sets of join irreducible elements even if that set is not a distributive lattice. Consider for example the antichain on two elements #{(0 0) (1 1)} which is not a lattice. This antichain can be represented as the set #{#{0} #{1}}. Likewise larger antichains such as #{#{0} #{1} #{2}} can be represented as set systems.

The set systems that correspond to lattices are Moore families which means that they are intersection closed and they have a closure operation. The union of elements of the Moore family is equal to the closure of their union. Non-distributive lattices such as #{#{} #{0} #{1} #{2} #{0 1 2}} and #{#{} #{0} #{1} #{1 2} #{0 1 2}} can also be represented as set systems which demonstrates the applicability of this representation to non-distributive lattices.

The ability to represent any elements of partial orders as sets makes me think that it makes sense to use sets to represent essentially every mathematical entity. Even entities which aren't typically considered to be sets like numbers and booleans can always be described as singleton sets such that they have a cardinality of one. It is worthwhile nonetheless, to consider the different sets involved within a set system so for example a structured set might have a set corresponding to the underlying set and another set corresponding to the frame of the structure. With this there are different notions of sets in the structure but all the while the structure is still a set of some sort.

No comments:

Post a Comment