Motivation for point-free bitopology Why think point-free? Pointed spaces versus point-free spaces The functors pt and Ω The dual adjunction between frames and spaces is mediated by the two element frame 2 = • nski space space S = • • and Sierpi´ • : pt ( L ) = [ L → • with topology induced from ( • • ) | L | • ] Ω( X ) = [ X → • with frame order induced from ( • • ) | X | • ] Moshier (Chapman) BLAST 2018 14 / 73
Motivation for point-free bitopology Why think point-free? Pointed spaces versus point-free spaces The functors pt and Ω The dual adjunction between frames and spaces is mediated by the two element frame 2 = • nski space space S = • • and Sierpi´ • : pt ( L ) = [ L → • with topology induced from ( • • ) | L | • ] Ω( X ) = [ X → • with frame order induced from ( • • ) | X | • ] Notes: Ω( X ) is essentially sending the space ( X , τ ) to τ — so it “forgets” the points, but retains the topology. pt ( L ) is essentially sending L to its principle prime ideals. Moshier (Chapman) BLAST 2018 14 / 73
Motivation for point-free bitopology Why think point-free? Pointed spaces and point-free spaces The pointed space/point-free space adjunction pt Sp op ⊤ Frm Ω Moshier (Chapman) BLAST 2018 15 / 73
Motivation for point-free bitopology Why think point-free? Pointed spaces and point-free spaces The pointed space/point-free space adjunction pt Sp op ⊤ Frm Ω Remarks: When regarding frames as point-free spaces, it is reasonable to consider the category of locales: Loc = Frm op . In particular, “subframe” and “sublocale” are very different. Sublocale is the right notion of point-free subspace. Convention: We write ⊑ , ⊓ , � , ⊥ , ⊤ for frames. Moshier (Chapman) BLAST 2018 15 / 73
Motivation for point-free bitopology The general idea Toward point-free bispaces What we want A point-free bispace ought to look like this: ( L − , L + , something ) where L − and L + are frames (point-free topologies); “something” is data describing how L − and L + are actually talking about the same underlying point-free space. Continuity Evidently, the morphisms will be pairs of frame homomorphisms ( f − , f + ) that preserve “something”. Moshier (Chapman) BLAST 2018 16 / 73
Motivation for point-free bitopology The general idea Biframes: Maximal something Definition (Biframes) A biframe is a triple ( L − , L + , L ) so that L is also a frame — intended to encode the joint topology of L − and L + ; hence L − , L + are subframes of L ; L is generated as a frame from L − and L + . Morphisms are frame homomorphims f : L → M that restrict to the positive and negative parts in the obvious way. Moshier (Chapman) BLAST 2018 17 / 73
Motivation for point-free bitopology The general idea Biframes: Maximal something Definition (Biframes) A biframe is a triple ( L − , L + , L ) so that L is also a frame — intended to encode the joint topology of L − and L + ; hence L − , L + are subframes of L ; L is generated as a frame from L − and L + . Morphisms are frame homomorphims f : L → M that restrict to the positive and negative parts in the obvious way. The biframe adjunction pt b BiSp op BiFrm ⊤ Ω b Moshier (Chapman) BLAST 2018 17 / 73
Motivation for point-free bitopology The general idea What’s wrong with biframes Pointed and point-free (uni)spaces Recall the adjunction between frames and spaces: Ω( X ) = [ X → • • ] pt ( L ) = [ L → • • ] Moshier (Chapman) BLAST 2018 18 / 73
Motivation for point-free bitopology The general idea What’s wrong with biframes Pointed and point-free (uni)spaces Recall the adjunction between frames and spaces: Ω( X ) = [ X → • • ] pt ( L ) = [ L → • • ] Theorem No set D can be equipped with a biframe and bispace structure so that Ω b ( X ) = [ X → D BiSp ] pt b ( L ) = [ L → D BiFrm ] Moshier (Chapman) BLAST 2018 18 / 73
Motivation for point-free bitopology The general idea What’s wrong with biframes Pointed and point-free (uni)spaces Recall the adjunction between frames and spaces: Ω( X ) = [ X → • • ] pt ( L ) = [ L → • • ] Theorem No set D can be equipped with a biframe and bispace structure so that Ω b ( X ) = [ X → D BiSp ] pt b ( L ) = [ L → D BiFrm ] Moral of the story: Biframes are unnatural (technically speaking). Moshier (Chapman) BLAST 2018 18 / 73
Some background Outline Motivation for point-free bitopology 1 Why think bitopologically? Why think point-free? The general idea Some background 2 The natural adjunction of bispaces and d -frames 3 Alternative formulation: skew frames with auxiliary relation 4 Bitopological separation and compactness 5 Regularity, complete regularity and normality Compactness Examples and applications 6 Moshier (Chapman) BLAST 2018 19 / 73
Some background Double lattices and disjointed lattices Definition (The categories DLat 2 and DLat 2 ) The category DLat 2 consists of double distributive lattices: pairs of distributive lattices and pairs of lattice homomorphisms with no additional structure. Moshier (Chapman) BLAST 2018 20 / 73
Some background Double lattices and disjointed lattices Definition (The categories DLat 2 and DLat 2 ) The category DLat 2 consists of double distributive lattices: pairs of distributive lattices and pairs of lattice homomorphisms with no additional structure. The category DLat 2 consists of disjointed distributive lattices: Objects ( L , f , t ) where L is a distributive lattice and f , t ∈ L are complements of each other. Morphisms Lattice homomorphisms that preserve f and t . Moshier (Chapman) BLAST 2018 20 / 73
Some background Double lattices and disjointed lattices Definition (The categories DLat 2 and DLat 2 ) The category DLat 2 consists of double distributive lattices: pairs of distributive lattices and pairs of lattice homomorphisms with no additional structure. The category DLat 2 consists of disjointed distributive lattices: Objects ( L , f , t ) where L is a distributive lattice and f , t ∈ L are complements of each other. Morphisms Lattice homomorphisms that preserve f and t . Theorem The categories DLat 2 and DLat 2 are equivalent. Moshier (Chapman) BLAST 2018 20 / 73
Some background Double lattices and disjointed lattices Definition (The categories DLat 2 and DLat 2 ) The category DLat 2 consists of double distributive lattices: pairs of distributive lattices and pairs of lattice homomorphisms with no additional structure. The category DLat 2 consists of disjointed distributive lattices: Objects ( L , f , t ) where L is a distributive lattice and f , t ∈ L are complements of each other. Morphisms Lattice homomorphisms that preserve f and t . Theorem The categories DLat 2 and DLat 2 are equivalent. Proof: ( L − , L + ) �→ ( L − × L + , ( 1 , 0 ) , ( 0 , 1 )) ( L , f , t ) �→ ( ↑ t , ↑ f ) Moshier (Chapman) BLAST 2018 20 / 73
Some background Some conventions Double lattice elements In a double lattice ( L − , L + ) , we write a , b , . . . for elements of L + ; and ϕ, ψ, . . . for elements of L − . Disjointed lattice elements In a disjointed lattice ( L , f , t ) , we write α, β, γ ∈ L α − = α ⊔ t α + = α ⊔ f α ∨ β = ( t ⊓ α ) ⊔ ( α ⊓ β ) ⊔ ( β ⊓ t ) α ∧ β = ( f ⊓ α ) ⊔ ( α ⊓ β ) ⊔ ( β ⊓ f ) Moshier (Chapman) BLAST 2018 21 / 73
Some background Some conventions Double lattice elements In a double lattice ( L − , L + ) , we write a , b , . . . for elements of L + ; and ϕ, ψ, . . . for elements of L − . Disjointed lattice elements In a disjointed lattice ( L , f , t ) , we write α, β, γ ∈ L α − = α ⊔ t — projection onto L − α + = α ⊔ f — projection onto L + α ∨ β = ( t ⊓ α ) ⊔ ( α ⊓ β ) ⊔ ( β ⊓ t ) α ∧ β = ( f ⊓ α ) ⊔ ( α ⊓ β ) ⊔ ( β ⊓ f ) Moshier (Chapman) BLAST 2018 21 / 73
Some background Some conventions Double lattice elements In a double lattice ( L − , L + ) , we write a , b , . . . for elements of L + ; and ϕ, ψ, . . . for elements of L − . Disjointed lattice elements In a disjointed lattice ( L , f , t ) , we write α, β, γ ∈ L α − = α ⊔ t — projection onto L − α + = α ⊔ f — projection onto L + α ∨ β = ( t ⊓ α ) ⊔ ( α ⊓ β ) ⊔ ( β ⊓ t ) α ∧ β = ( f ⊓ α ) ⊔ ( α ⊓ β ) ⊔ ( β ⊓ f ) — explained by the following Moshier (Chapman) BLAST 2018 21 / 73
Some background The 45 ◦ lemma (two orders for the price of one) Lemma Suppose ( L , ⊔ , ⊓ , ⊥ , ⊤ ; f , t ) is a bounded distributive lattice with complementary elements. Define x ∧ y = ( f ⊓ x ) ⊔ ( x ⊓ y ) ⊔ ( y ⊓ f ) x ∨ y = ( t ⊓ x ) ⊔ ( x ⊓ y ) ⊔ ( y ⊓ t ) Then ( L , ∨ , ∧ , f , t ; ⊥ , ⊤ ) is also a bounded distributive lattice with complementary elements. Moreover, this construction is involutive. Hence, objects of DLat 2 have two orders: the information order — the given order; the logical order — defined by ∧ and/or ∨ . Moshier (Chapman) BLAST 2018 22 / 73
Some background The 45 ◦ lemma ⊤ t f t ⊥ ⊤ ⊥ f Useful because we can speak about both orders. The information order: ⊑ ; The logical order: ≤ . Moshier (Chapman) BLAST 2018 23 / 73
Some background Auxiliary relations An idea that has come up in several talks (e.g., George Metcalfe’s tutorials) . . . Definition A lattice relation is a relation R ⊆ L × M between lattices satisfying: Weakening x ≤ L x ′ R y ′ ≤ M y implies x R y ; Logical R is a sub-lattice of L × M : Moshier (Chapman) BLAST 2018 24 / 73
Some background Auxiliary relations An idea that has come up in several talks (e.g., George Metcalfe’s tutorials) . . . Definition A lattice relation is a relation R ⊆ L × M between lattices satisfying: Weakening x ≤ L x ′ R y ′ ≤ M y implies x R y ; Logical R is a sub-lattice of L × M : 0 R z ; x R z and y R z implies x ∨ y R z ; x R 1; x R y and x R z implies x R y ∧ z ; A relation ≺ is interpolative if x ≺ z implies x ≺ y ≺ z for some y . Moshier (Chapman) BLAST 2018 24 / 73
Some background Auxiliary relations An idea that has come up in several talks (e.g., George Metcalfe’s tutorials) . . . Definition A lattice relation is a relation R ⊆ L × M between lattices satisfying: Weakening x ≤ L x ′ R y ′ ≤ M y implies x R y ; Logical R is a sub-lattice of L × M : 0 R z ; x R z and y R z implies x ∨ y R z ; x R 1; x R y and x R z implies x R y ∧ z ; An auxiliary relation on L is a lattice relation ≺⊆ L × L contained in the order relation. [It is thus transitive.] A relation ≺ is interpolative if x ≺ z implies x ≺ y ≺ z for some y . Moshier (Chapman) BLAST 2018 24 / 73
Some background Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element ( L , ∧ , ∨ , 0 , 1 , ⊥ ) . In a based distributive lattice, x ⊓ y = ( ⊥ ∧ x ) ∨ ( x ∧ y ) ∨ ( y ∧ ⊥ ) defines a semilattice operation that distributes over ∧ and ∨ , and ⊥ is least element with respect to ⊑ . Moshier (Chapman) BLAST 2018 25 / 73
Some background Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element ( L , ∧ , ∨ , 0 , 1 , ⊥ ) . In a based distributive lattice, x ⊓ y = ( ⊥ ∧ x ) ∨ ( x ∧ y ) ∨ ( y ∧ ⊥ ) defines a semilattice operation that distributes over ∧ and ∨ , and ⊥ is least element with respect to ⊑ . Example 1 • • • • • • 0 Moshier (Chapman) BLAST 2018 25 / 73
Some background Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element ( L , ∧ , ∨ , 0 , 1 , ⊥ ) . In a based distributive lattice, x ⊓ y = ( ⊥ ∧ x ) ∨ ( x ∧ y ) ∨ ( y ∧ ⊥ ) defines a semilattice operation that distributes over ∧ and ∨ , and ⊥ is least element with respect to ⊑ . Example 1 • • • • • ⊥ • 0 Moshier (Chapman) BLAST 2018 25 / 73
Some background Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element ( L , ∧ , ∨ , 0 , 1 , ⊥ ) . In a based distributive lattice, x ⊓ y = ( ⊥ ∧ x ) ∨ ( x ∧ y ) ∨ ( y ∧ ⊥ ) defines a semilattice operation that distributes over ∧ and ∨ , and ⊥ is least element with respect to ⊑ . Example 1 • 1 • • �− → • • • • • • • ⊥ 0 • • 0 ⊥ Moshier (Chapman) BLAST 2018 25 / 73
Some background Skew frames Definition A skew frame is a based distributive lattice ( L , ∧ , ∨ , 0 , 1 , ⊥ ) for which the induced order ⊑ is a dcpo; Moshier (Chapman) BLAST 2018 26 / 73
Some background Skew frames Definition A skew frame is a based distributive lattice ( L , ∧ , ∨ , 0 , 1 , ⊥ ) for which the induced order ⊑ is a dcpo; ∧ and ∨ preserve � ↑ . Moshier (Chapman) BLAST 2018 26 / 73
Some background Skew frames Definition A skew frame is a based distributive lattice ( L , ∧ , ∨ , 0 , 1 , ⊥ ) for which the induced order ⊑ is a dcpo; ∧ and ∨ preserve � ↑ . Remarks: In any based distributive lattice, if x and y are bounded in ⊑ , they have a least upper bound. Hence a skew frame is conditionally complete in ⊑ . A frame is precisely a skew frame in which 0 = ⊥ . Moshier (Chapman) BLAST 2018 26 / 73
Some background Double frames Definition The category Frm 2 (double frames) is the subcategory of DLat 2 consisting of frames and pairs of frame homomorphisms. The category Frm 2 (disjointed frames) is the subcategory of DLat 2 consisting of frames ( L , f , t ) with complements and frame homomorphisms that preserve f and t . Moshier (Chapman) BLAST 2018 27 / 73
Some background Double frames Definition The category Frm 2 (double frames) is the subcategory of DLat 2 consisting of frames and pairs of frame homomorphisms. The category Frm 2 (disjointed frames) is the subcategory of DLat 2 consisting of frames ( L , f , t ) with complements and frame homomorphisms that preserve f and t . Remarks: The 45 ◦ lemma still obtains, so that a disjointed frame has the given frame order ( ⊑ ) and its derived logical order ( ≤ ). A disjointed frame is a complete (distributive) lattice order in the logical order, not necessarily a frame or a coframe. Auxiliary relations must be specified as pertaining to ⊑ or ≤ . Both kinds will appear later. Moshier (Chapman) BLAST 2018 27 / 73
The natural adjunction of bispaces and d -frames Outline Motivation for point-free bitopology 1 Why think bitopologically? Why think point-free? The general idea Some background 2 The natural adjunction of bispaces and d -frames 3 Alternative formulation: skew frames with auxiliary relation 4 Bitopological separation and compactness 5 Regularity, complete regularity and normality Compactness Examples and applications 6 Moshier (Chapman) BLAST 2018 28 / 73
The natural adjunction of bispaces and d -frames d -lattices Definition A d -lattice is a double distributive lattice ( L − , L + , con , tot ) equipped with relations con ⊆ L + × L − — intended to mean ϕ and a are disjoint tot ⊆ L − × L + — intended to mean a and ϕ cover satisfying con - ↓ con is a lower set con is a sublattice of L − × L ∂ con -logic + tot - ↑ tot is an upper set tot is a sublattice of L ∂ + × L − tot -logic con - tot a con ; tot b implies a ⊑ + b ψ tot ; con ϕ implies ψ ⊒ ϕ Moshier (Chapman) BLAST 2018 29 / 73
The natural adjunction of bispaces and d -frames d -lattices Definition A d -lattice is a double distributive lattice ( L − , L + , con , tot ) equipped with relations con ⊆ L + × L − — intended to mean ϕ and a are disjoint tot ⊆ L − × L + — intended to mean a and ϕ cover satisfying con - ↓ con is a lower set con is a sublattice of L − × L ∂ con -logic a lattice relation + tot - ↑ tot is an upper set tot is a sublattice of L ∂ + × L − tot -logic a lattice relation con - tot a con ; tot b implies a ⊑ + b ψ tot ; con ϕ implies ψ ⊒ ϕ Moshier (Chapman) BLAST 2018 29 / 73
The natural adjunction of bispaces and d -frames con - ↓ and tot - ↑ ϕ a Moshier (Chapman) BLAST 2018 30 / 73
The natural adjunction of bispaces and d -frames con - ↓ and tot - ↑ ϕ a ψ b Moshier (Chapman) BLAST 2018 30 / 73
The natural adjunction of bispaces and d -frames con - ↓ and tot - ↑ ϕ a ψ b ϕ a Moshier (Chapman) BLAST 2018 30 / 73
The natural adjunction of bispaces and d -frames con - ↓ and tot - ↑ ϕ a ψ b ϕ a ψ b Moshier (Chapman) BLAST 2018 30 / 73
The natural adjunction of bispaces and d -frames ”Logic” ϕ a Moshier (Chapman) BLAST 2018 31 / 73
The natural adjunction of bispaces and d -frames ”Logic” ψ b Moshier (Chapman) BLAST 2018 31 / 73
The natural adjunction of bispaces and d -frames ”Logic” ϕ ∨ ψ a ∧ b Moshier (Chapman) BLAST 2018 31 / 73
The natural adjunction of bispaces and d -frames ”Logic” ϕ ∨ ψ a ∧ b Likewise for tot . Moshier (Chapman) BLAST 2018 31 / 73
The natural adjunction of bispaces and d -frames con - tot ϕ a a con ϕ Moshier (Chapman) BLAST 2018 32 / 73
The natural adjunction of bispaces and d -frames con - tot ϕ a b a con ϕ ϕ tot b Moshier (Chapman) BLAST 2018 32 / 73
The natural adjunction of bispaces and d -frames con - tot ϕ a b a con ϕ ϕ tot b So a ⊑ b Moshier (Chapman) BLAST 2018 32 / 73
The natural adjunction of bispaces and d -frames The order dual of a d -lattice Definition For d -lattice L = ( L − , L + , con , tot ) , define L ∂ = ( L ∂ + , L ∂ − , tot , con ) . Moshier (Chapman) BLAST 2018 33 / 73
The natural adjunction of bispaces and d -frames The order dual of a d -lattice Definition For d -lattice L = ( L − , L + , con , tot ) , define L ∂ = ( L ∂ + , L ∂ − , tot , con ) . Because the axioms for con and tot are symmetric, this is clearly a d -lattice. ( − ) ∂ extends to a functor by swapping component homomorphisms. Moshier (Chapman) BLAST 2018 33 / 73
The natural adjunction of bispaces and d -frames The order dual of a d -lattice Definition For d -lattice L = ( L − , L + , con , tot ) , define L ∂ = ( L ∂ + , L ∂ − , tot , con ) . Because the axioms for con and tot are symmetric, this is clearly a d -lattice. ( − ) ∂ extends to a functor by swapping component homomorphisms. Our next axiom defining d -frames breaks the symmetry, so L ∂ of a d -frame is not a d -frame. Moshier (Chapman) BLAST 2018 33 / 73
The natural adjunction of bispaces and d -frames The order dual of a d -lattice Definition For d -lattice L = ( L − , L + , con , tot ) , define L ∂ = ( L ∂ + , L ∂ − , tot , con ) . Because the axioms for con and tot are symmetric, this is clearly a d -lattice. ( − ) ∂ extends to a functor by swapping component homomorphisms. Our next axiom defining d -frames breaks the symmetry, so L ∂ of a d -frame is not a d -frame. We will use this when we consider ideal and filter completions. Moshier (Chapman) BLAST 2018 33 / 73
The natural adjunction of bispaces and d -frames d -frames Definition A d -frame is a d -lattice ( L − , L + , con , tot ) for which L − and L + are frames, and con - ↓ con is a lower set con is a sublattice of L − × L ∂ con -logic + ; tot - ↑ tot is an upper set tot is a sublattice of L + × L ∂ tot -logic − con - tot a con ; tot b implies a ⊑ + b ψ tot ; con ϕ implies ψ ⊒ − ϕ con - � ↑ con is closed under directed joins Moshier (Chapman) BLAST 2018 34 / 73
The natural adjunction of bispaces and d -frames d -frames Definition A d -frame is a d -lattice ( L − , L + , con , tot ) for which L − and L + are frames, and con - ↓ con is a lower set con is a sublattice of L − × L ∂ con -logic + ; tot - ↑ tot is an upper set tot is a sublattice of L + × L ∂ tot -logic − con - tot a con ; tot b implies a ⊑ + b ψ tot ; con ϕ implies ψ ⊒ − ϕ con - � ↑ con is closed under directed joins We expected to formalize a point-free bispace as ( L − , L + , something ) with morphisms preserving “something.” Moshier (Chapman) BLAST 2018 34 / 73
The natural adjunction of bispaces and d -frames d -frames Definition A d -frame is a d -lattice ( L − , L + , con , tot ) for which L − and L + are frames, and con - ↓ con is a lower set con is a sublattice of L − × L ∂ con -logic + ; tot - ↑ tot is an upper set tot is a sublattice of L + × L ∂ tot -logic − con - tot a con ; tot b implies a ⊑ + b ψ tot ; con ϕ implies ψ ⊒ − ϕ con - � ↑ con is closed under directed joins We expected to formalize a point-free bispace as ( L − , L + , something ) with morphisms preserving “something.” So a d -frame homomorphism is two frame homomorphisms that, together, preserve con and tot . Moshier (Chapman) BLAST 2018 34 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames con - � ↑ Moshier (Chapman) BLAST 2018 35 / 73
The natural adjunction of bispaces and d -frames The dualizing object Definition Define the bispace S . S to be • open in τ − open in τ + • • • Moshier (Chapman) BLAST 2018 36 / 73
The natural adjunction of bispaces and d -frames The dualizing object Definition Define the bispace S . S to be • open in τ − open in τ + • • • Definition Define the d -frame 2 . 2 to be • tot • • con • L − L + Moshier (Chapman) BLAST 2018 36 / 73
The natural adjunction of bispaces and d -frames The natural adjunction between pointed bispaces and pointfree bispaces Definition For bispace ( X , τ − , τ + ) , Ω d ( X ) = all bicontinuous maps from X to S . S For d -frame ( L − , L + , con , tot ) , pt d ( L ) = all d -frame homomorphisms from L to 2 . 2 Moshier (Chapman) BLAST 2018 37 / 73
The natural adjunction of bispaces and d -frames The natural adjunction between pointed bispaces and pointfree bispaces Definition For bispace ( X , τ − , τ + ) , Ω d ( X ) = all bicontinuous maps from X to S . S For d -frame ( L − , L + , con , tot ) , pt d ( L ) = all d -frame homomorphisms from L to 2 . 2 Theorem pt d d-Frm op BiSp ⊤ Ω d Moshier (Chapman) BLAST 2018 37 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? • • • • Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ • • • • Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ • • • false true • Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ • • • information order ⊑ false true • Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ • • • information order ⊑ false true No information • Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ • • • true information order ⊑ false No information • IDK Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ What? True/False? • • • true information order ⊑ false • IDK Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames So 2 . 2 ( S . S ) is the dualizer — what about truth values? logical order ≤ WTF What? True/False? • • • true information order ⊑ false • IDK Moshier (Chapman) BLAST 2018 38 / 73
The natural adjunction of bispaces and d -frames Sub- d -frames Lemma For any d-frame L = ( L − , L + , con , tot ) and subframes, M − ⊆ L − and M + ⊆ L + , ( M − , M + , con ∩ ( M + × M − ) , tot ∩ ( M − × M + )) is a d-frame, and the inclusion maps constitute a d-frame homomorphism. Moshier (Chapman) BLAST 2018 39 / 73
The natural adjunction of bispaces and d -frames Sub- d -frames Lemma For any d-frame L = ( L − , L + , con , tot ) and subframes, M − ⊆ L − and M + ⊆ L + , ( M − , M + , con ∩ ( M + × M − ) , tot ∩ ( M − × M + )) is a d-frame, and the inclusion maps constitute a d-frame homomorphism. Proof. The result is clearly a sub- d -lattice. If { ( a i , ϕ i ) } i ⊆ con ∩ ( M + × M − ) is directed, then � ↑ a i con � ↑ ϕ i . But these two joins are in M + and M − , respectively. Moshier (Chapman) BLAST 2018 39 / 73
The natural adjunction of bispaces and d -frames Ideal completion of a d -lattice Let Idl ( L ) ( Filt ( L ) ) denote ideals (filters) of a distributive lattice — this is always a frame (the free frame over L , resp, L ∂ ). Moshier (Chapman) BLAST 2018 40 / 73
The natural adjunction of bispaces and d -frames Ideal completion of a d -lattice Let Idl ( L ) ( Filt ( L ) ) denote ideals (filters) of a distributive lattice — this is always a frame (the free frame over L , resp, L ∂ ). Theorem Let L = ( L − , L + , con , tot ) be a d-lattice. For I ∈ Idl ( L − ) , J ∈ Idl ( L + ) , define I con ∗ J ⇔ I × J ⊆ con J tot ∗ I ⇔ ( J × I ) ∩ tot � = ∅ Then L ∗ = ( Idl ( L − ) , Idl ( L + ) , con ∗ , tot ∗ ) is the free d-frame over L . Moshier (Chapman) BLAST 2018 40 / 73
The natural adjunction of bispaces and d -frames Ideal completion of a d -lattice Let Idl ( L ) ( Filt ( L ) ) denote ideals (filters) of a distributive lattice — this is always a frame (the free frame over L , resp, L ∂ ). Theorem Let L = ( L − , L + , con , tot ) be a d-lattice. For I ∈ Idl ( L − ) , J ∈ Idl ( L + ) , define I con ∗ J ⇔ I × J ⊆ con J tot ∗ I ⇔ ( J × I ) ∩ tot � = ∅ Then L ∗ = ( Idl ( L − ) , Idl ( L + ) , con ∗ , tot ∗ ) is the free d-frame over L . Also, write Filt ( L ) for L ∂ ∗ . Moshier (Chapman) BLAST 2018 40 / 73
The natural adjunction of bispaces and d -frames Ideal completion of a d -lattice Proof. Suppose M = ( M − , M + , con ′ , tot ′ ) is a d -frame and ( h − , h + ): L → M is a d -lattice homomorphism. Moshier (Chapman) BLAST 2018 41 / 73
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