Definitions and Propositions pdf
Part A: Order Theory 3
The question now is, how to find, for any arbitrary poset (X,P), the smallest possible complete lattice into which to embed the poset. The Dedekind-MacNeille Completion is the answer to that question. Before we can illustrate how the completion of a poset works, we need to introduce some central notions.
Definition A.13 - Down-Set, Up-Set, AL, AU
(X,P) is a poset, A ⊆ X is a subset. We define:
- ↓A = {x ∈ X : ∃a ∈ A with x ≤ a} (Down-Set of A)
- ↑A = {x ∈ X : ∃a ∈ A mit a ≤ x} (Up-Set of A)
- ↓x = ↓{x}
- ↑x = ↑{x}
- AL = {x ∈ X : x ≤ a ∀a ∈ A} (Set of lower bounds of A)
- AU = {x ∈ X : a ≤ x ∀a ∈ A} (Set of upper bounds of A)
We are looking for the smallest possible lattice into which we can fit our poset. This means that the poset needs to be dense in the lattice. In mathematical terms:
(X, P) is a poset. S ⊆ X is called ∧-dense (or ∨-dense), if and only if for all x ∈ X there exists a T ⊆ S where inf T = x (or sup T = x).
If S is ∧-dense as well as ∨-dense, then S is dense.
(X, P) is a poset. The (closure-)operator ΓUL is defined as follows:
ΓUL: A ⊆ X ↦ (AU)L
Proposition A.16 - Dedekind-MacNeille (cf. also Dedekind-MacNeille Completion)
(X, P) is a poset and DM(X, P) = (im(ΓUL), ⊆). φ: X → 𝒫(X) is given by x ↦ ↓x. Then, the following holds:
- DM(X, P) is a complete lattice
- φ is an order embedding
- If (X, P) is already a complete lattice, then (X, P) ≅ DM(X, P), especially φ is an isomorphism of lattices.
- φ(X) is dense in DM(X, P)
The poset DM(X, P) = (im(ΓUL), ⊆) together with the embedding φx is called Dedekind-MacNeille Completion of the poset (X, P). This is the wanted smallest complete lattice which contains the poset (X, P).
Cf. also the illustration in Part C: Dedekind-MacNeille Completion.