In a first course on real analysis, one is generally introduced to the concept of a Dedekind cut. It is a way of constructing the set of real numbers from the rationals. This is a process commonly known as the completion of the rationals. Three key features of this completion are:
the rationals can be embedded in its completion (the reals)
every subset with an upper bound has a least upper bound
If we extend the reals by adjoining and and define the appropriate ordering relations on this new extended set (the extended real numbers), then it is a set where every subset has a least upper bound and a greatest lower bound.
When we deal with the rationals and the reals (and extended reals), we are working with linearly ordered sets. So the next question is: can the procedure of a completion be generalized to an arbitrary poset? In other words, if is a poset ordered by , does there exist another poset ordered by such that
can be embedded in as a poset (so that is compatible with ), and
every subset of has both a least upper bound and a greatest lower bound
In 1937, MacNeille answered this question in the affirmative by the following construction:
Given a poset with order , define for every subset of , two subsets of as follows:
Then ordered by the usual set inclusion is a poset satisfying conditions (1) and (2) above.
This is known as the MacNeille completion of a poset . In , since lub and glb exist for any subset, is a complete lattice. So this process can be readily applied to any lattice, if we define a completion of a lattice to follow the two conditions above.
|Date of creation||2013-03-22 16:05:27|
|Last modified on||2013-03-22 16:05:27|
|Last modified by||CWoo (3771)|