Difference hierarchies over lattices 1 C´ elia Borlido (based on joint work with Gerhke, Krebs, and Straubing) LJAD, CNRS, Universit´ e Cˆ ote d’Azur Workshop on Algebra, Logic and Topology in honour of Aleˇ s Pultr, in the occasion of his 80th birthday September 28, 2018 1The research discussed has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.670624)
1 Introduction 2 Difference chains of closed upsets 3 The point-free approach and an application to Logic on Words
Motivation D = bounded distributive lattice Booleanization of D : unique (up to isomorphism) Boolean algebra D − , ι → D − satisfying the together with a bounded lattice embedding D ֒ − − − − following universal property: ι D − D h − h B C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 3
Motivation D = bounded distributive lattice Booleanization of D : unique (up to isomorphism) Boolean algebra D − , ι → D − satisfying the together with a bounded lattice embedding D ֒ − − − − following universal property: ι D − D h − h B D − is the unique (up to isomorphism) Boolean algebra containing D as a bounded sublattice and generated as a Boolean algebra by D . C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 3
Motivation D = bounded distributive lattice Booleanization of D : unique (up to isomorphism) Boolean algebra D − , ι → D − satisfying the together with a bounded lattice embedding D ֒ − − − − following universal property: ι D − D h − h B D − is the unique (up to isomorphism) Boolean algebra containing D as a bounded sublattice and generated as a Boolean algebra by D . Fact: Every element of D − may be written as a difference chain of the form a 1 − ( a 2 − · · · − ( a n − 1 − a n ) . . . ) , for some a 1 , . . . , a n ∈ D . C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 3
Priestley duality Priestley spaces 1 Bounded distributive lattices � X = Priestley space UpClopen( X ) � ( X D , τ, ≤ ), where D = bounded distributive lattice � ◦ X D = { prime filters of D } a ) c | a ∈ D } , with � ◦ τ has basis of (cl)opens { � a , ( � a = { x ∈ X D | a ∈ x } ◦ ≤ is inclusion of prime filters D ∼ X ∼ = UpClopen( X D ) and = X UpClopen( X ) 1 Compact and totally order disconnected topological space C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 4
Priestley duality Priestley spaces 1 Bounded distributive lattices � X = Priestley space UpClopen( X ) � ( X D , τ, ≤ ), where D = bounded distributive lattice � ◦ X D = { prime filters of D } a ) c | a ∈ D } , with � ◦ τ has basis of (cl)opens { � a , ( � a = { x ∈ X D | a ∈ x } ◦ ≤ is inclusion of prime filters D ∼ X ∼ = UpClopen( X D ) and = X UpClopen( X ) In particular, D − ∼ = Clopen( X D ). 1 Compact and totally order disconnected topological space C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 4
The topological formulation X = Priestley space, V ⊆ X = clopen subset. Then, there are clopen upsets W 1 , . . . , W n of X such that V = W 1 − ( W 2 − ( · · · − ( W n − 1 − W n )) . . . ) . Our question: Is there a “canonical form” for such a writing? C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 5
An example y . . . X = 1 2 x C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
An example y . . . X = 1 2 x C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
An example y . . . X = 1 2 x UpClopen( X ) = P fin ( N ) ∪ { W | W ⊆ X is cofinite and y ∈ W } C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
An example y . . . X = 1 2 x UpClopen( X ) = P fin ( N ) ∪ { W | W ⊆ X is cofinite and y ∈ W } V = { x } is clopen C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
An example y . . . X = 1 2 x UpClopen( X ) = P fin ( N ) ∪ { W | W ⊆ X is cofinite and y ∈ W } V = W − W ′ V = { x } is clopen, = ⇒ ↑ V = { x , y } ⊆ W is not open! C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
An example y . . . X = 1 2 x UpClopen( X ) = P fin ( N ) ∪ { W | W ⊆ X is cofinite and y ∈ W } V = W − W ′ V = { x } is clopen, = ⇒ ↑ V = { x , y } ⊆ W is not open! There is no smallest clopen upset containing V : those are precisely the sets of the form W = S ∪ { x , y } , with S ⊆ N cofinite. Moreover, W ′ = W − { x } = ↑ ( W − V ) is also a clopen upset and V = W − W ′ . C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
An example y . . . X = 1 2 x UpClopen( X ) = P fin ( N ) ∪ { W | W ⊆ X is cofinite and y ∈ W } V = W − W ′ V = { x } is clopen, = ⇒ ↑ V = { x , y } ⊆ W is not open! There is no smallest clopen upset containing V : those are precisely the sets of the form W = S ∪ { x , y } , with S ⊆ N cofinite. Moreover, W ′ = W − { x } = ↑ ( W − V ) is also a clopen upset and V = W − W ′ . However, ↑ V is closed and V = ↑ V − ↑ ( ↑ V − V ). C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 6
We will see: 1. Every clopen subset of a Priestley space may be canonically written as a difference chain of closed upsets of the space. Such writing has a nice topological interpretation. C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 7
We will see: 1. Every clopen subset of a Priestley space may be canonically written as a difference chain of closed upsets of the space. Such writing has a nice topological interpretation. 2. This provides a canonical writing as a difference chain for the elements in the Booleanization of a co-Heyting algebra. C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 7
We will see: 1. Every clopen subset of a Priestley space may be canonically written as a difference chain of closed upsets of the space. Such writing has a nice topological interpretation. 2. This provides a canonical writing as a difference chain for the elements in the Booleanization of a co-Heyting algebra. 3. This provides a topological proof of the fact that every element in the Booleanization of a bounded distributive lattice D may be written as a difference chain a 1 − ( a 2 − ( · · · − ( a n − 1 − a n ) . . . )) , with a 1 , . . . , a n ∈ D . C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 7
We will see: 1. Every clopen subset of a Priestley space may be canonically written as a difference chain of closed upsets of the space. Such writing has a nice topological interpretation. 2. This provides a canonical writing as a difference chain for the elements in the Booleanization of a co-Heyting algebra. 3. This provides a topological proof of the fact that every element in the Booleanization of a bounded distributive lattice D may be written as a difference chain a 1 − ( a 2 − ( · · · − ( a n − 1 − a n ) . . . )) , with a 1 , . . . , a n ∈ D . 4. The point-free version of 1. allows for an elegant generalization having an application to Logic on Words. C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 7
The degree of an element of a poset P = poset, S ⊆ P , p ∈ P p 1 < p 2 < · · · < p n in P is an alternating sequence of length n for p (with respect to S ) provided p n = p and p i ∈ S if and only if i is odd. C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 8
The degree of an element of a poset p = p 4 = p ′ 2 p 3 P = poset, S ⊆ P , p ∈ P p ′ p 2 1 p 1 p 1 < p 2 < · · · < p n in P is an alternating sequence of length n for p (with respect to S ) provided p n = p and p i ∈ S if and only if i is odd. C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 8
The degree of an element of a poset p = p 4 = p ′ 2 p 3 P = poset, S ⊆ P , p ∈ P p ′ p 2 1 p 1 p 1 < p 2 < · · · < p n in P is an alternating sequence of length n for p (with respect to S ) provided p n = p and p i ∈ S if and only if i is odd. The degree of p (wrt S ), deg S ( p ), is the largest k for which there is an alternating sequence of length k for p , and p has degree 0 if there is no alternating sequence for p (wrt S ). Example: p has degree 4. C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 8
The degree of an element of a poset p 1 < p 2 < · · · < p n in P is an alternating sequence of length n for p (with respect to S ) provided p i ∈ S if and only if i is odd. p n = p and The degree of p (wrt S ), deg S ( p ), is the largest k for which there is an alternating sequence of length k for p , and p has degree 0 if there is no alternating sequence for p (wrt S ). Remarks: ◦ The elements of degree 0 are precisely those of ( P − ↑ S ). 1 S is convex if x ≤ y ≤ z with x , z ∈ S implies y ∈ S . C. Borlido (LJAD) Difference hierarchies over lattices September 28, 2018 9
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