MacNeille completion
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:
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the rationals can be embedded in its completion (the reals)
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every subset with an upper bound has a least upper bound
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every subset with a lower bound has a greatest lower 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
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can be embedded in as a poset (so that is compatible with ), and
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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.
References
- 1 H. M. MacNeille, Partially Ordered Sets. Trans. Amer. Math. Soc. 42 (1937), pp 416-460
- 2 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd edition, Cambridge (2003)
Title | MacNeille completion |
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Canonical name | MacNeilleCompletion |
Date of creation | 2013-03-22 16:05:27 |
Last modified on | 2013-03-22 16:05:27 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06B23 |
Synonym | Dedekind-MacNeille completion |
Synonym | normal completion |
Related topic | DedekindCuts |