Completions of Pseudo Ordered Sets Maria D Cruz BLAST 2018 August - - PowerPoint PPT Presentation

Completions of Pseudo Ordered Sets

Maria D Cruz

BLAST 2018

August 10,2018

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Basics for Pseudo Ordered sets

Definition

A relation ≤ on a set A is a pseudo order if ≤ is reflexive and

• antisymmetric. We call (A, ≤) a pseudo-ordered set.

Definition

A trellis is a pseudo-ordered set (A, ≤) in which any two elements have a least upper bound and a greatest lower bound.

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Definition

A trellis X is a complete trellis if every subset of X has a least upper bound and a greater lower bound. In lattice theory, every finite lattice is complete. In trellises, there are finite trellises which are not complete such as

Example

the three element cycle Z = ({0, 1, 2} , ≤) in which 0 < 1 < 2 < 0

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Completion

Definition.

A completion of a pseudo ordered set X is a pair (E, f ) where E is a complete trellis and the map f : X → E is an order embedding.

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Skala’s Completion

Theorem (Skala 1971)

Every trellis has a completion.

Properties of Skala’s completion f : X → E

Preserves existing joins Preserves existing meets Join dense

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Skala’s Completion Misbehaves

The Skala’s completion of a lattice may not even be a poset. X

v1 v2 v3 v4 u3 u2 u1 u0

X is a bounded lattice X is not complete

Skala’s Completion of X

X union all infinite subsets of botton that contain 0.

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Idea: To make a better behaved completion than Skala’s

Definition

A subset P ⊆ A such that P = LUP is called a normal ideal.

Note!

For a ∈ A, both LU{a} and L{a} are normal ideals with largest element a. Unlike posets, need not be equal LU{a} is the smallest normal ideal containing a. L{a} is the largest normal ideal containing a.

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Normal Ideals of (A, ≤)

Definition

Let N(A) be the set of all normal ideals of A partially ordered by ⊆.

Definition

Let Θ be a relation on N(A) given by PΘQ iff P = Q or LU{a} ⊆ P, Q ⊆ L{a} for some a

Proposition

Θ is a c-bounded relation on N(A) meaning Θ is an equivalence relation. The equivalence classes of Θ are convex. I/Θ has a largest and a smallest element I + and I − respectively. If I − ⊆ J+ and J− ⊆ I + imply IΘJ.

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Proposition

If P is a poset and Θ is a c-bounded relation on it, there is a pseudo order

• n P/Θ where

x/Θ y/Θ iff x− ≤ y+. Further, If P is a complete lattice, then (P/Θ, ) is a complete trellis

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Pseudo MacNeille Completion

Definition

For a pseudo ordered set A, let (S(A), ) be N(A)/Θ where Θ is the relation defined before.

Theorem

For a pseudo ordered set A, The pair (S(A), ) is a complete trellis. The pair (S(A), f ) where f : A → S(A) defined by f (a) = LU{a}/Θ is a completion of A.

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The completion (S(A), f ) satisfies the following:

1 Preserves joins 2 Preserves meets 3 Join dense 4 Meet dense 5 When (S(A), f ) is applied to a poset A it is the MacNeille completion 6 When (S(A), f ) is applied to a complete trellis A it does nothing

We call to (S(A), f ) the Pseudo MacNeille completion of A.

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Another Feature of the Pseudo MacNeille completion (S(A), f )

Definition

Let (A, ≤1) and (C, ≤2) be pseudo ordered sets and let f : A → C be an

• rder embedding. The pair (C, f ) is a strict extension if: For each a, b ∈ A

and for any c ∈ C, if f (a) ≤2 c ≤2 f (b), then either c is in the image of f or a ≤1 b.

Proposition

(S(A), f ) is a strict completion.

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Characterization of the Pseudo MacNeille Completion

Theorem

If (E, h) is a completion of a pseudo ordered set (A, ≤) satisfying:

1 (E, h) is a strict extesion. 2 h is join dense. 3 h is meet dense.

Then there is a unique isomorphism g : S(A) → E making h = g ◦ f (i.e the diagram commutes). S(A) A E f ∃!g h

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We extract some ideas from the pseudo MacNeille completion to continue

• ur study of pseudo orders.

Idea

For A a pseudo ordered set make a poset Γ(A) and Γ(A) → A. Call this the covering poset.

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Costructing a Covering Poset Γ(A)

Definition

Let A be a pseudo ordered set. For an element a ∈ A we say that a is a transitive element if b ≤ a ≤ c implies b ≤ c. Let T be the set of all transitive elements in A. And let N be the set of elements of A that are not transitive.

Definition

Consider Γ(A) to be the set N × {0, 1} ∪ T in which for each a ∈ A we define a+, a− ∈ Γ(A) as follows: If a ∈ N then a+ = (a, 1) and a− = (a, 0). If a ∈ T then a+ = a = a−. In other words, Γ(A) = {a−, a+ : a ∈ A}.

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Covering Poset Γ(A)

Proposition

Let ⊑ be a partial order on Γ(A) defined as follows:

1 a+ ⊑ b+ iff l ≤ a implies l ≤ b. 2 a+ ⊑ b− iff l ≤ a and b ≤ u implies l ≤ u. 3 a− ⊑ b+ iff a ≤ b. 4 a− ⊑ b− iff b ≤ u implies a ≤ u. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 16 / 24

The Covering Poset Γ(A) mod Θ

Definition

Let Θ be a c-bounded relation on Γ(A) given by xΘy iff x = y or x, y ∈ {a−, a+} for some a ∈ A Let cA : Γ(A) → A be given by cA(a+) = a = cA(a−).

Theorem

For a pseudo ordered set A Θ is c-bounded. The map cA : Γ(A) → A with Ker(cA) = Θ. Γ(A)/Θ is isomorphic to A.

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Other Completions

Given the following data Γ(A) A P cA f where P is a poset and f is order embedding.

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Other Completions

Given Γ(A) P/Θf A P cA f k gf Let xΘf y iff x = y or f (a−) ≤ x, y ≤ f (a+) for some a ∈ A. And gf (a) = f (a+)/Θ.

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Other Completions

Theorem

The square commutes Γ(A) P/Θf A P cA f k gf Further, if P is a complete lattice, then P/Θf is complete and (P/Θf , gf ) is a completion of A.

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Pushouts

Theorem

The square Γ(A) P/Θf A P Q cA f k gf u h v Is a pushout w.r.t. the pseudo ordered set Q and the order preserving u, v.

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Other Completions (P/Θf , gf ) of A

Note

This allows various types of completions of A by applying poset completions to Γ(A).

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Final Comments/Questions

What properties do the completions (P/Θf , gf ) have? Is the pseudo MacNeille completion given by applying the MacNeille completion to Γ(A)? Is the poset cover cA : Γ(A) → A some kind of coreflector? (Not in an

• bvious way)

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Thank you!!!

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