Completions of Pseudo Ordered Sets Maria D Cruz BLAST 2018 August 10,2018 Maria D Cruz (NMSU) BLAST 2018 August 10,2018 1 / 24
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. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 2 / 24
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 Maria D Cruz (NMSU) BLAST 2018 August 10,2018 3 / 24
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. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 4 / 24
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 Maria D Cruz (NMSU) BLAST 2018 August 10,2018 5 / 24
Skala’s Completion Misbehaves The Skala’s completion of a lattice may not even be a poset. u 0 u 1 u 2 X is a bounded lattice u 3 X is not complete Skala’s Completion of X v 4 X union all infinite subsets of botton that v 3 contain 0. v 2 v 1 0 X Maria D Cruz (NMSU) BLAST 2018 August 10,2018 6 / 24
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 . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 7 / 24
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 P Θ Q 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 I Θ J . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 8 / 24
Proposition If P is a poset and Θ is a c-bounded relation on it, there is a pseudo order � on P / Θ where x / Θ � y / Θ iff x − ≤ y + . Further, If P is a complete lattice, then ( P / Θ , � ) is a complete trellis Maria D Cruz (NMSU) BLAST 2018 August 10,2018 9 / 24
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 . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 10 / 24
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 . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 11 / 24
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 order 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. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 12 / 24
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). h A E ∃ ! g f S ( A ) Maria D Cruz (NMSU) BLAST 2018 August 10,2018 13 / 24
We extract some ideas from the pseudo MacNeille completion to continue our 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. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 14 / 24
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 } . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 15 / 24
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 x = y or x , y ∈ { a − , a + } for some a ∈ A x Θ y iff Let c A : Γ( A ) → A be given by c A ( a + ) = a = c A ( a − ). Theorem For a pseudo ordered set A Θ is c-bounded. The map c A : Γ( A ) → A with Ker ( c A ) = Θ. Γ( A ) / Θ is isomorphic to A . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 17 / 24
Other Completions Given the following data c A Γ( A ) A f P where P is a poset and f is order embedding. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 18 / 24
Other Completions Given c A Γ( A ) A g f f k P / Θ f P Let x Θ f y iff x = y or f ( a − ) ≤ x , y ≤ f ( a + ) for some a ∈ A . And g f ( a ) = f ( a + ) / Θ. Maria D Cruz (NMSU) BLAST 2018 August 10,2018 19 / 24
Other Completions Theorem The square commutes c A Γ( A ) A g f f k P / Θ f P Further, if P is a complete lattice, then P / Θ f is complete and ( P / Θ f , g f ) is a completion of A . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 20 / 24
Pushouts Theorem The square c A Γ( A ) A g f f v k P / Θ f P u h Q Is a pushout w.r.t. the pseudo ordered set Q and the order preserving u , v . Maria D Cruz (NMSU) BLAST 2018 August 10,2018 21 / 24
Other Completions ( P / Θ f , g f ) of A Note This allows various types of completions of A by applying poset completions to Γ( A ). Maria D Cruz (NMSU) BLAST 2018 August 10,2018 22 / 24
Final Comments/Questions What properties do the completions ( P / Θ f , g f ) have? Is the pseudo MacNeille completion given by applying the MacNeille completion to Γ( A )? Is the poset cover c A : Γ( A ) → A some kind of coreflector? (Not in an obvious way) Maria D Cruz (NMSU) BLAST 2018 August 10,2018 23 / 24
Thank you!!! Maria D Cruz (NMSU) BLAST 2018 August 10,2018 24 / 24
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