Wednesday, April 30, 2014

Embedding partial orders in complete lattices

Given a partially ordered set we can always embed that partial order within a complete lattice through Dedekind completion so that partial order becomes a suborder of a complete lattice. For finite unique extrema partial orders this simply means adding bounds to the partial order to make it into a lattice. Otherwise there may need to be elements placed between others such as in the case of the [2 2] weak order.

If we represent partial orders as set systems then the process of embedding the partial order within a complete lattice is equivalent to the process of embedding the set system within a Moore family. We can always acquire such a set system by taking the Moore completion of the family of sets.

The general applicability of lattice theory to problems of order theory through the process of embedding partial orders in complete lattices makes me think that lattice theory should play a fundamental role in our understanding of order theory. With an understanding of complete lattices we can better understand partial orders in general.

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.