Canonical extensions of lattices are more than perfect
aa r X i v : . [ m a t h . L O ] J a n Canonical extensions of lattices are more thanperfect
Andrew P. K. Craig, Maria J. Gouveia and Miroslav Haviar
In memory of Bjarni J´onsson
Abstract.
In [2] we introduced TiRS graphs and TiRS frames to createa new natural setting for duals of canonical extensions of lattices. Inthis continuation of [2] we answer Problem 2 from there by characteris-ing the perfect lattices that are dual to TiRS frames (and hence TiRSgraphs). We introduce a new subclass of perfect lattices called PTi lat-tices and show that the canonical extensions of lattices are PTi lattices,and so are ‘more’ than just perfect lattices. We introduce morphisms ofTiRS structures and put our correspondence between TiRS graphs andTiRS frames from [2] into a full categorical framework. We illustrate ourcorrespondences between classes of perfects lattices and classes of TiRSgraphs by examples.
Mathematics Subject Classification.
Keywords. bounded lattice, canonical extension, perfect lattice, RS frame,TiRS frame, TiRS graph, PTi lattice.
1. Introduction
An important aspect of the study of lattice-based algebras in recent decadeshas been the theory of canonical extensions. This has its origins in the 1951–52papers of J´onsson and Tarski [13]. We refer to Gehrke and Vosmaer [10] for asurvey of the theory of canonical extensions for lattice-based algebras and, forfurther background, to recent papers by Gehrke [8] and Goldblatt [11] and thereferences there, in particular to the first section of [11] called “A biographyof canonical extension”.The canonical extensions of general (bounded) lattices were first intro-duced by Gehrke and Harding [9] as the complete lattices of Galois-closed setsassociated with a polarity between the filter lattice and the ideal lattice of thegiven lattice. (The same polarity was also used in the lattice representationof Hartonas and Dunn [12].) A new construction of the canonical extension A.P.K. Craig, M.J. Gouveia and M. Haviarof a general lattice was provided in [3] where it was based on a topologicalrepresentation of lattices by Ploˇsˇcica [16]. The Ploˇsˇcica representation pre-sented a well-known representation of general lattices due to Urquhart [18]in the spirit of the theory of natural dualities of Clark and Davey [1]. It usedmaximal partial maps into the two-element set to represent elements of thefirst and second duals of a given lattice.An another construction of the canonical extensions of general latticeswas presented in [2] where Ploˇsˇcica’s topological representation was used intandem with Gehrke’s representation of perfect lattices via RS frames. (Forthe latter we refer to papers [6] by Dunn, Gehrke and Palmigiano and [7] byGehrke.) In [2] we also demonstrated a one-to-one correspondence betweenTiRS frames forming a subclass of the RS frames and TiRS graphs which weintroduced as an abstraction of the duals of general lattices in the Ploˇsˇcicarepresentation. This has led to a new dual representation of the class ofall finite lattices via finite TiRS frames, or equivalently finite TiRS graphs,which generalises the well-known Birkhoff dual representation between finitedistributive lattices and finite posets from the 1930s. (Here we remark thatevery poset is a TiRS graph.) We use a common concept of
TiRS structures when we refer to both TiRS graphs and TiRS frames without distinguishingbetween the two classes.This paper has two goals:(1) To describe the additional properties that perfect lattices dual to TiRSstructures possess. This was listed as “Problem 2” in [2].(2) To describe the appropriate morphisms of TiRS structures and henceto extend the one-to-one correspondence between the TiRS structuresfrom [2] into a full categorical framework.We also show that the canonical extensions of lattices are PTi lattices, whichfollows from their construction in [2] using Ploˇsˇcica’s and Gehrke’s represen-tations in tandem. We present an example of a perfect but not PTi latticetogether with its dual TiRS graph and an example of a PTi lattice that isnot the canonical extension of any lattice together with its dual TiRS frame.
2. Preliminaries
For a bounded lattice L , a completion of L is defined to be a pair ( e, C ) where C is a complete lattice and e : L ֒ → C is an embedding. By a filter element ( ideal element ) of a completion ( e, C ) of a bounded lattice L we mean anelement of C which is a meet (join) of elements from e ( L ). By F ( C ) and I ( C ) are denoted the sets of all filter and ideal elements of C , respectively.(We remark that in the older literature the filter (ideal) elements had beencalled closed (open) elements.) A completion ( e, C ) of a bounded lattice L iscalled dense if every element of C can be expressed as both a join of meetsand a meet of joins of elements from e ( L ). A completion ( e, C ) of L is called compact if, for any sets A ⊆ F ( C ) and B ⊆ I ( C ) with V A W B , thereexist finite subsets A ′ ⊆ A and B ′ ⊆ B such that V A ′ W B ′ . (We remarkanonical extensions of lattices 3that the sets A, B in the definition of compactness above can alternativelybe taken as arbitrary subsets of L .)Gehrke and Harding [9] defined abstractly the canonical extension L δ ofa general bounded lattice L as a dense and compact completion of L . Theyproved that every bounded lattice L has a canonical extension and that it isunique up to an isomorphism that fixes the elements of L . Concretely, theyconstructed L δ as the complete lattice of Galois-stable sets of the polarity R between the filter lattice Filt( L ) and the ideal lattice Idl( L ) of L where thepolarity is given by ( F, I ) ∈ R if F ∩ I = ∅ .A filter-ideal pair ( F, I ) will be called maximal if F and I are maximalwith respect to being disjoint from one another. In our final section we shalluse the following result from [9]: Lemma 2.1 ( [9, Lemma 3.4] ). Let ( e, C ) be a canonical extension of L . (1) x ∈ J ∞ ( C ) if and only if x = V e [ F ] for some maximal pair ( F, I ) of L ; (2) x ∈ M ∞ ( C ) if and only if x = W e [ I ] for some maximal pair ( F, I ) of L .Further, each element of C is a join of completely join irreducibles and ameet of completely meet irreducibles. Ploˇsˇcica’s dual [16, Section 1] of a bounded lattice L is a graph withtopology, D( L ) = ( L mp ( L , ) , E, T ), where L mp ( L , ) is the set of maximalpartial homomorphisms from L into . The graph relation E is defined by( f, g ) ∈ E if ( ∀ a ∈ dom f ∩ dom g ) f ( a ) g ( a ) , or equivalently, ( f, g ) ∈ E if f − (1) ∩ g − (0) = ∅ . The topology T has as a subbasis of closed sets the set { V a , W a | a ∈ L } , with V a = { f ∈ L mp ( L , ) | f ( a ) = 0 } and W a = { f ∈ L mp ( L , ) | f ( a ) = 1 } .TiRS graphs were defined by the present authors in [2] as an abstrac-tion of the graphs D ♭ ( L ) = ( L mp ( L , ) , E ) obtained from Ploˇsˇcica’s duals ofbounded lattices L by forgetting the topology.For a graph X = ( X, E ) and x ∈ X , the sets { y ∈ X | ( x, y ) ∈ E } and { y ∈ X | ( y, x ) ∈ E } were denoted in [2] by xE and Ex respectively. Wedefined the conditions (S), (R) and (Ti) for any graph X = ( X, E ) as follows:(S) for every x, y ∈ X , if x = y then xE = yE or Ex = Ey ;(R) (i) for all x, z ∈ X , if zE ( xE then ( z, x ) / ∈ E ;(ii) for all y, z ∈ X , if Ez ( Ey then ( y, z ) / ∈ E ;(Ti) for all x, y ∈ X , if ( x, y ) ∈ E , then there exists z ∈ X such that zE ⊆ xE and Ez ⊆ Ey .A TiRS graph was in [2] defined as a graph X = ( X, E ) with a reflexiverelation E and satisfying the conditions (R), (S) and (Ti). For any boundedlattice L , its dual graph X = D ♭ ( L ) is a TiRS graph [2, Proposition 2.3]. A.P.K. Craig, M.J. Gouveia and M. HaviarWe further recall that a frame is a structure ( X , X , R ), where X and X are non-empty sets and R ⊆ X × X . For an arbitrary frame F =( X , X , R ) the conditions (S) and (R) are defined as follows:(S) for all x , x ∈ X and y , y ∈ X ,(i) x = x implies x R = x R ;(ii) y = y implies Ry = Ry .(R) (i) for every x ∈ X there exists y ∈ X such that ¬ ( xRy ) and ∀ w ∈ X (( w = x & xR ⊆ wR ) ⇒ wRy );(ii) for every y ∈ X there exists x ∈ X such that ¬ ( xRy ) and ∀ z ∈ X (( z = y & Ry ⊆ Rz ) ⇒ xRz ).The frames that satisfy the conditions (R) and (S) are called reducedseparated frames, or RS frames for short, and were introduced by Gehrke[7] as a two-sorted generalisation of Kripke frames to be used for relationalsemantics of substructural logics.The (Ti) condition introduced in [2] for frames ( X , X , R ) was moti-vated by the (Ti) condition on graphs:(Ti) for every x ∈ X and for every y ∈ X , if ¬ ( xRy ) then there exist w ∈ X and z ∈ X such that(i) ¬ ( wRz );(ii) xR ⊆ wR and Ry ⊆ Rz ;(iii) for every u ∈ X , if u = w and wR ⊆ uR then uRz ;(iv) for every v ∈ X , if v = z and Rz ⊆ Rv then wRv .A TiRS frame was in [2] defined as a frame ( X , X , R ) that satisfiesconditions (R), (S) and (Ti), i.e. it is an RS frame that satisfies condition(Ti). A one-to-one correspondence between TiRS graphs and TiRS frameswas then shown in [2]. We recall here some facts of this correspondence thatwill be needed in the next section. Definition 2.2 ( [2, Definition 2.5] ). Let X = ( X, E ) be a graph. The associatedframe ρ ( X ) is the frame ( X , X , R ρ ( X ) ) where(i) X = X/ ∼ for the equivalence relation ∼ on X given by x ∼ y if xE = yE ;(ii) X = X/ ∼ for the equivalence relation ∼ on X given by x ∼ y if Ex = Ey ;(iii) R ρ ( X ) is the relation given by[ x ] R ρ ( X ) [ y ] ⇐⇒ ( x, y ) / ∈ E, where [ x ] and [ y ] are, respectively, the ∼ -equivalence class of x andthe ∼ -equivalence class of y .We omit the subscript ρ ( X ) in R ρ ( X ) whenever it is clear to which relation R refers.If X = ( X, E ) is a TiRS graph, then the associated frame ρ ( X ) =( X , X , R ρ ( X ) ) is a TiRS frame [2, Proposition 2.6]. Then it follows that ifanonical extensions of lattices 5 L is a bounded lattice, X = D ♭ ( L ) is its dual TiRS graph and ρ (D ♭ ( L )) isthe associated frame, then ρ (D ♭ ( L )) is a TiRS frame (cf. [2, Corollary 2.7]). Definition 2.3 ( [2, Definition 2.8] ). Let F = ( X , X , R ) be a TiRS frame. The associated graph gr( F ) is ( H F , K F ) where the vertex set H F is the subset of X × X of all pairs ( x, y ) that satisfy the following conditions:(i) ¬ ( xRy ),(ii) for every u ∈ X , if u = x and xR ⊆ uR then uRy ,(iii) for every v ∈ X , if v = y and Ry ⊆ Rv then xRv .and the edge set K F is formed by the pairs (( x, y ) , ( w, z )) such that ¬ ( xRz ).We omit the subscript F in H F and in K F whenever it is clear whichvertex set and edge set we refer to.In [2, Proposition 2.10] we showed that if F = ( X , X , R ) is a TiRSframe, then its associated graph gr( F ) is a TiRS graph. Definition 2.4 ( [2, Definition 2.11] ). Two graphs X = ( X, E X ) and Y =( Y, E Y ) are isomorphic (denoted X ≃ Y ) if there exists a bijective map α : X → Y such that ∀ x , x ∈ X ( x , x ) ∈ E X ⇐⇒ ( α ( x ) , α ( x )) ∈ E Y and we refer to such a map as the graph-isomorphism α : X → Y .Two frames F = ( X , X , R F ) and G = ( Y , Y , R G ) are isomorphic (denoted F ≃ G ) if there exists a pair ( β , β ) of bijective maps β i : X i → Y i ( i = 1 ,
2) with ∀ x ∈ X ∀ x ∈ X (cid:0) x R F x ⇐⇒ β ( x ) R G β ( x ) (cid:1) and we refer to such a pair as the frame-isomorphism ( β , β ) : F → G .Now for a TiRS graph X = ( X, E ), a map α X : X → gr ( ρ ( X )) isdefined by α X ( x ) = ([ x ] , [ x ] ). The next result shows that α X is a graphisomorphism and that the correspondence between TiRS graphs and TiRSframes is one-to-one. Theorem 2.5 ( [2, Theorem 2.13] ). Let X = ( X, E ) be a TiRS graph and F = ( X , X , R ) be a TiRS frame. Then (a) the graphs X and gr( ρ ( X )) are isomorphic; (b) the frames F and ρ (gr( F )) are isomorphic.
3. TiRS graph and TiRS frame morphisms
In this section we extend the one-to-one correspondence between TiRS graphsand TiRS frames from [2] into the full categorical framework. We start bydefining the concepts of TiRS graph and TiRS frame morphisms.
Definition 3.1.
Let X = ( X, E X ) and Y = ( Y, E Y ) be TiRS graphs. A TiRSgraph morphism is a map ϕ : X → Y that satisfies the following conditions:(i) for x , x ∈ X , if ( x , x ) ∈ E X then ( ϕ ( x ) , ϕ ( x )) ∈ E Y ; A.P.K. Craig, M.J. Gouveia and M. Haviar(ii) for x , x ∈ X , if x E X ⊆ x E X then ϕ ( x ) E Y ⊆ ϕ ( x ) E Y ;(iii) for x , x ∈ X , if E X x ⊆ E X x then E Y ϕ ( x ) ⊆ E Y ϕ ( x ).We note that every graph isomorphism and its inverse are TiRS graphmorphisms. Definition 3.2.
Let F = ( X , X , R F ) and G = ( Y , Y , R G ) be TiRS frames.A TiRS frame morphism ψ : F → G is a a pair ψ = ( ψ , ψ ) of maps ψ : X → Y and ψ : X → Y that satisfies the following conditions:(i) for x ∈ X and y ∈ X , if ψ ( x ) R G ψ ( y ) then xR F y ;(ii) for x, w ∈ X , if xR F ⊆ wR F then ψ ( x ) R G ⊆ ψ ( w ) R G ;(iii) for y, z ∈ X , if R F y ⊆ R F z then R G ψ ( y ) ⊆ R G ψ ( z );(iv) for x ∈ X and y ∈ X , if ( x, y ) ∈ H F then ( ψ ( x ) , ψ ( y )) ∈ H G .We note that a frame isomorphism is a TiRS morphism.Henceforth we shall refer to TiRS graph morphisms and to TiRS framemorphisms simply as graph morphisms and frame morphisms respectively.Our main result in this section puts our one-to-one correspondence be-tween TiRS graphs and TiRS frames into a full categorical framework. Thelast two statements are illustrated by the diagrams in Fig. 1. Theorem 3.3.
Let X = ( X, E X ) and Y = ( Y, E Y ) be TiRS graphs and let F = ( X , X , R F ) and G = ( Y , Y , R G ) be TiRS frames. (1) If ϕ : X → Y is a TiRS graph morphism, then, for ρ ( ϕ ) : X/ ∼ → Y / ∼ and ρ ( ϕ ) : X/ ∼ → Y / ∼ the maps defined by ρ ( ϕ ) ([ x ] ) = [ ϕ ( x )] and ρ ( ϕ ) ([ x ] ) = [ ϕ ( x )] , for all x ∈ X , the pair ρ ( ϕ ) = ( ρ ( ϕ ) , ρ ( ϕ ) ) is aTiRS frame morphism from ρ ( X ) to ρ ( Y ) . (2) If the pair ψ = ( ψ , ψ ) : F → G is a TiRS frame morphism, then themap gr( ψ ) : gr( F ) → gr( G ) defined by gr( ψ )( x, y ) = ( ψ ( x ) , ψ ( y )) , for ( x, y ) ∈ H F , is a TiRS graph morphism. (3) If ϕ : X → Y is a TiRS graph morphism, then gr( ρ ( ϕ )) ◦ α X = α Y ◦ ϕ . (4) If ψ : F → G is a TiRS frame morphism, then ρ (gr( ψ )) ◦ β F = β G ◦ ψ .Proof. (1) First we show that ρ ( ϕ ) is well defined. Let x, y ∈ X . If [ x ] = [ y ] then xE X = yE X which implies ϕ ( x ) E Y = ϕ ( y ) E Y and so [ ϕ ( x )] = [ ϕ ( y )] ,by the definition of a TiRS graph morphism. Similarly we prove that [ ϕ ( x )] =[ ϕ ( y )] whenever [ x ] = [ y ] . Next we prove that conditions (i) to (iv) ofthe definition of a TiRS frame morphism are satisfied by ρ ( ϕ ). For (i), let x, y ∈ X and assume ρ ( ϕ ) ([ x ] ) R ρ ( Y ) ρ ( ϕ ) ([ y ] ). Then ( ϕ ( x ) , ϕ ( y )) / ∈ E Y yielding that ( x, y ) / ∈ E X and so [ x ] R ρ ( X ) [ y ] . For (ii), let x, w ∈ X . Thenthe following holds:[ x ] R ρ ( X ) ⊆ [ w ] R ρ ( X ) ⇐⇒ wE X ⊆ xE X ⇒ ϕ ( w ) E Y ⊆ ϕ ( x ) E Y ⇐⇒ [ ϕ ( x )] R ρ ( Y ) ⊆ [ ϕ ( w )] R ρ ( Y ) . Hence (ii) is satisfied. Similarly we conclude that (iii) holds. Finally (iv)follows from [2, Lemma 2.12] where we showed that for a TiRS graph X =( X, E ), the elements of H ρ ( X ) are exactly the pairs ([ x ] , [ x ] ), with x ∈ X .anonical extensions of lattices 7 X Y F G gr( ρ ( X )) gr( ρ ( Y )) ρ (gr( F )) ρ (gr( G )) ϕα X α Y gr( ρ ( ϕ )) ψρ (gr( ψ )) β G β F Figure 1.
TiRS graph morphisms and TiRS frame mor-phisms(2) First we note that condition (iv) of the definition of a TiRS framemorphism satisfied by ψ guarantees that the map gr( ψ ) is well defined. Nextwe prove that conditions (i) to (iii) of the definition of a TiRS graph morphismare satisfied by gr( ψ ). Let ( x, y ) , ( w, z ) ∈ H F . If (( x, y ) , ( w, z )) ∈ K F then ¬ ( xR F z ) which implies ¬ ( ψ ( x ) R G ψ ( z )) and consequently(gr( ψ )( x, y ) , gr( ψ )( w, z )) = (( ψ ( x ) , ψ ( y )) , ( ψ ( w ) , ψ ( z ))) ∈ K G . Hence ( x, y ) , ( w, z ) satisfies (i). For (ii), we observe that,( x, y ) K F ⊆ ( w, z ) K F ⇐⇒ wR F ⊆ xR F and gr( ψ )( x, y ) K G ⊆ gr( ψ )( w, z ) K G ⇐⇒ ψ ( w ) R G ⊆ ψ ( x ) R G , which follows from (iii) of [2, Lemma 3.9]. As ψ is a TiRS morphism, we alsohave wR F ⊆ xR F ⇒ ψ ( w ) R G ⊆ ψ ( x ) R G . Hence ( x, y ) , ( w, z ) satisfies (ii). Similarly we conclude that (iii) also holds.(3) Let x ∈ X . We have that(gr( ρ ( ϕ )) ◦ α X )( x ) = gr( ρ ( ϕ ))([ x ] , [ x ] )= ( ρ ( ϕ ) ([ x ] ) , ρ ( ϕ ) ([ x ] ))= ([ ϕ ( x )] , [ ϕ ( x )] )= ( α Y ◦ ϕ )( x ) . (4) Let x ∈ X . There exist y ∈ X such that ( x, y ) ∈ H F . We have that( ρ (gr( ψ )) ◦ β F )( x ) = ρ (gr( ψ )([( x, y )] )= [gr( ψ )( x, y )] = [( ψ ( x ) , ψ ( y ))] , where ψ : X → Y and ψ : X → Y satisfy ψ = ( ψ , ψ ). Since ( x, y ) ∈ H F and ψ is a TiRS morphism, we also have ( ψ ( x ) , ψ ( y )) ∈ H G and so[( ψ ( x ) , ψ ( y ))] = β G ( ψ ( x )) = ( β G ◦ ψ )( x ) . A.P.K. Craig, M.J. Gouveia and M. Haviar (cid:3)
Corollary 3.4.
The category of TiRS graphs with TiRS graph morphisms isequivalent to the category of TiRS frames with TiRS frame morphisms viathe functors given by ρ and gr as described above. There are other definitions of morphisms between two frames (or con-texts) F = ( X , X , R F ) and G = ( Y , Y , R G ) that are used in the litera-ture. Deiters and Ern´e [5] use a pair of maps ( α, β ) where α : X → Y and β : X → Y as we do above. Gehrke [7, Section 3] uses a pair of relations( R, S ) where R ⊆ X × Y and S ⊆ X × Y . More recently, Moshier [15](see also Jipsen [14]) defined a context morphism to be a single relation S ⊆ X × Y .
4. Perfect lattices dual to TiRS structures
Consider a complete lattice C and let F ( C ) = ( J ∞ ( C ) , M ∞ ( C ) , ) where J ∞ ( C ) and M ∞ ( C ) denote the sets of completely join-irreducible and com-pletely meet-irreducible elements of C , respectively. We will refer to F ( C ) asthe frame coming from C . For the opposite direction, consider an RS frame F = ( X, Y, R ). For A ⊆ X and B ⊆ Y , let R ⊲ ( A ) = { y ∈ Y | ( ∀ a ∈ A )( aRy ) } and R ⊳ ( B ) = { x ∈ X | ( ∀ b ∈ B )( xRb ) } . Now consider the complete lattice of Galois-closed sets (ordered by inclusion): G ( F ) = { A ⊆ X | A = ( R ⊳ ◦ R ⊲ )( A ) } . By results from Gehrke [7, Section 2] we know that the completely join-irreducible elements and completely meet-irreducible elements of G ( F ) areidentified as follows: J ∞ ( G ( F )) = { ( R ⊳ ◦ R ⊲ )( { x } ) | x ∈ X } and M ∞ ( G ( F )) = { Ry | y ∈ Y } . Below we introduce a condition that refines the class of perfect lattices.At the end of this section we will conclude that every perfect lattice that isthe canonical extension of some bounded lattice will have this property.
Definition 4.1.
A perfect lattice satisfies the condition (PTi) if for all x ∈ J ∞ ( C ) and for all y ∈ M ∞ ( C ), if x (cid:10) y then there exist w ∈ J ∞ ( C ), z ∈ M ∞ ( C ) such that(i) w x and y z (ii) w (cid:10) z (iii) ( ∀ u ∈ J ∞ ( C ))( u < w ⇒ u z )(iv) ( ∀ v ∈ M ∞ ( C ))( y < v ⇒ w v )In Fig. 2 we give a pictorial depiction of the (PTi) condition. We haveindicated the sets ↑ x , ↑ w , ↓ y and ↓ z . We see that the (PTi) condition for C essentially starts with an arbitrary disjoint filter-ideal pair ( ↑ x, ↓ y ) gen-erated by elements x ∈ J ∞ ( C ) and y ∈ M ∞ ( C ). It says that every suchanonical extensions of lattices 910 xy wz Figure 2.
The (PTi) condition illustrated.disjoint filter-ideal pair is contained in a maximal disjoint filter-ideal pair( ↑ w, ↓ z ) where again w ∈ J ∞ ( C ) and z ∈ M ∞ ( C ) (here maximality is un-derstood such that neither of ↑ w and ↓ z can be enlarged without breakingtheir disjointness). Lemma 4.2.
Let C be a perfect lattice. If C satisfies (PTi) then the RS frame F ( C ) = ( J ∞ ( C ) , M ∞ ( C ) , ) satisfies (Ti) .Proof. First observe that when translating the condition (Ti) from a generalRS frame to F ( C ) we have that xR = ↑ x and Ry = ↓ y . The fact that F ( C )satisfies (Ti) follows then from the fact that u < w implies u = w and ↑ w ( ↑ u . (cid:3) We want to characterise the condition (PTi) on the Galois closed setsarising from an RS frame F = ( X, Y, R ). The following lemma will assist usin this task.
Lemma 4.3.
Consider the RS frame F = ( X, Y, R ) . Then (i) w ∈ ( R ⊳ ◦ R ⊲ )( { x } ) if and only if xR ⊆ wR ; (ii) ( R ⊳ ◦ R ⊲ )( { w } ) ⊆ ( R ⊳ ◦ R ⊲ )( { x } ) if and only if xR ⊆ wR ; (iii) ( R ⊳ ◦ R ⊲ )( { x } ) ⊆ Ry if and only if xRy .Proof. For (i) we have w ∈ ( R ⊳ ◦ R ⊲ )( { x } ) ⇔ ( ∀ z ∈ R ⊲ ( { x } ))( wRz ) ⇔ ( ∀ z ∈ Y )( xRz ⇒ wRz ) ⇔ xR ⊆ wR. To assist with the proof of (ii), note that R ⊲ ( { x } ) = Rx and R ⊲ ( { w } ) = Rw .If we assume that xR ⊆ wR then the fact that R ⊳ : ℘ ( Y ) → ℘ ( X ) is order-reversing gives us that ( R ⊳ ◦ R ⊲ )( { w } ) ⊆ ( R ⊳ ◦ R ⊲ )( { x } ). For the converse,if ( R ⊳ ◦ R ⊲ )( { w } ) ⊆ ( R ⊳ ◦ R ⊲ )( { x } ) then since R ⊲ : ℘ ( X ) → ℘ ( Y ) is order-reversing and since ( R ⊲ ◦ R ⊳ ◦ R ⊲ )( { w } ) = R ⊲ ( { w } ), we get xR ⊆ wR . Thestatement (iii) is exactly [7, Proposition 2.6]. (cid:3) F = ( X, Y, R ) satisfies the(Ti) condition, the perfect lattice of Galois-closed sets G ( F ) satisfies (PTi).In order to make the proof easier to follow, it will be useful to translate thecondition (PTi) from the setting of a general perfect lattice to the setting of G ( F ). Lemma 4.4.
Let F = ( X, Y, R ) be an RS frame. Assume that the followingset of conditions is satisfied by G ( F ) :For all x ∈ X and all y ∈ Y , if ¬ ( xRy ) then there exist p ∈ X , q ∈ Y such that (i) xR ⊆ pR and Ry ⊆ Rq (ii) ¬ ( pRq )(iii) ( ∀ u ∈ X )( pR ( uR ⇒ uRq )(iv) ( ∀ v ∈ Y )( Rq ( Rv ⇒ pRv )Then the lattice G ( F ) satisfies (PTi). Proof.
This follows using Lemma 4.3 to translate (PTi) conditions to thecomplete lattice G ( F ). (cid:3) Lemma 4.5.
Let F = ( X, Y, R ) be an RS frame. If F satisfies (Ti) then G ( F ) satisfies (PTi) .Proof. Let F = ( X, Y, R ) be an RS frame satisfying (Ti) (i.e. a TiRS frame).Take arbitrary x ∈ X and y ∈ Y and assume that ¬ ( xRy ). In the perfectlattice G ( F ) coming from F consider the sets A = ( R ⊳ ◦ R ⊲ )( { x } ) and B = Ry .Then A ∈ J ∞ ( G ( F )), B ∈ M ∞ ( G ( F ) and A * B using Lemma 4.3(iii).We have that( R ⊳ ◦ R ⊲ )( { x } ) * Ry ⇒ ( ∃ w ∈ X )( xR ⊆ wR & ¬ ( wRy )) ⇒ ( ∃ w ∈ X ) h xR ⊆ wR &( ∃ p ∈ X )( ∃ q ∈ Y ) (cid:16) ¬ ( pRq ) & wR ⊆ pR & Ry ⊆ Rq & ( ∀ u ∈ X )( pR ( uR ⇒ uRq )& ( ∀ v ∈ Y )( Rq ( Rv ⇒ pRv ) (cid:17)i The only part of the (PTi) condition for G ( F ) that is not now immediate isthe fact that we need xR ⊆ pR . This follows from the xR ⊆ wR ⊆ pR andthe transitivity of set containment. (cid:3) Now we are ready to show that the canonical extensions of lattices arePTi lattices and so they indeed are ‘more’ than just perfect lattices. For thiswe cite our final result from [2]:
Proposition 4.6 ( [2, Corollary 3.11] ). Let L be a bounded lattice and X =D ♭ ( L ) be its dual TiRS graph. Let ρ ( X ) be the frame associated to X and G( ρ ( X )) be its corresponding perfect lattice of Galois-closed sets.The lattice G( ρ ( X )) is the canonical extension of L . anonical extensions of lattices 11The result can be illustrated by the diagram in Fig. 3. The given boundedlattice L is firstly assigned its Ploˇsˇcica dual space D( L ) = ( L mp ( L , ) , E, T ),and then the Ploˇsˇcica dual graph X = D ♭ ( L ) = ( L mp ( L , ) , E ) is obtainedby forgetting the topology. This is a TiRS graph and so the frame ρ ( X ) asso-ciated to X in our one-to-one correspondence developed in [2] between TiRSgraphs and TiRS frames is a TiRS frame. Hence by Lemma 4.5 above, theperfect lattice G( ρ ( X )) of Galois-closed sets corresponding in Gehrke’s rep-resentation to the frame ρ ( X ) is a PTi lattice. By Proposition 4.6, the latticeG( ρ ( X )) is the canonical extension of the given lattice L . Lat
D(Ploˇsˇcica) ✲ PlSpPerLat δ ❄ ✛ G(Gehrke)
TiGr ( Fr ) ♭ ❄ Figure 3.
Ploˇsˇcica and Gehrke in tandem.Hence we have our final result of this section:
Theorem 4.7.
The canonical extension of a bounded lattice is a PTi lattice.
Gehrke and Vosmaer [10] showed that the canonical extension of a lat-tice need not be meet-continuous, and hence need not always be algebraic.Theorem 4.7 gives us further information about the structure of canonicalextensions of bounded lattices.
5. Examples
Our goal in this section is to illustrate that the PTi condition adds to thecurrent description of the canonical extension of a bounded lattice. We focuson non-distributive examples. Canonical extensions of distributive lattices areknown to be completely distributive complete lattices. To show that our newcondition does indeed add to the current description, we give an example ofa perfect lattice that is not PTi. Giving an example of a PTi lattice thatis not the canonical extension of a lattice would be the same as giving anexample of a TiRS graph that is not of the form ( L mp ( L , ) , E ) for somebounded lattice L . Hence this is the same as the representable TiRS graph(representable poset) problem.Our goals are:(1) Give an example of a complete non-distributive lattice which is a PTilattice but is not the canonical extension of any bounded lattice.(2) Give an example of a perfect non-distributive lattice that is not a PTilattice.2 A.P.K. Craig, M.J. Gouveia and M. Haviar p p p q q q k z = y xmwA L Figure 4.
A TiRS graph that is not the graph of MPH’s ofany bounded lattice (left) and its dual PTi lattice A L that isnot a canonical extension (right). The double-headed arrowson the graph emphasize that transitivity holds amongst thevertical edges. Example 5.1.
Consider the complete lattice A L depicted on the right in Fig. 4.We will denote by m , the middle element of the infinite chain ω ⊕ ⊕ ω ∂ and y is above the bottom and below the top but incomparable with all otherelements.The TiRS graph dual to A L is X = { p i | i ∈ ω } ∪ { q j | j ∈ ω } ∪ { k } with the relation E given by p < p < p < . . . < p n < p n +1 < . . . < q n +1 < q n < q n − . . . < q < q ∪{ ( k, p i ) | i ∈ ω } ∪ { ( k, q j ) | j > } (it is depicted on the left in Fig. 4). To be clear, the p i ’s and q j ’s form aposet (it is transitive) that is order-isomorphic to ω ⊕ ω ∂ while the element k is related to everything except the top of the chain.Recall that MPE’s are ordered by: ϕ ψ if and only if ϕ − (1) ⊆ ψ − (1).The MPE’s ϕ and ϕ are defined by ϕ ( x ) = 1 and ϕ ( x ) = 0 for all x ∈ X .All but one of the other MPE’s have k ϕ splits the chain by sending the p i ’s to 0 and the q j ’s to1 then you get the limit point in the middle of A L . The interesting MPE isthe map does the following for a ∈ X : ϕ ( a ) = a = k a = q − otherwiseThis interesting MPE is the incomparable point that makes A L non-distributive.anonical extensions of lattices 13 a a a a b b b b y x x x ω ∂ M L Figure 5.
An RS frame that is not Ti (left) and its dualperfect lattice that is not PTi (right).It is quite easy to show that the lattice A L is a PTi lattice; we indicatedon the right in Fig. 4 what the elements w ∈ J ∞ ( A L ) and z ∈ M ∞ ( A L ) arefor the chosen elements x ∈ J ∞ ( A L ) and y ∈ M ∞ ( A L ). The fact that thelattice A L is not the canonical extension of any bounded lattice is harder toshow and it follows from Proposition 5.2 below. Proposition 5.2.
There is no bounded lattice L and lattice embedding e : L → A L such that ( e, A L ) is the canonical extension of L .Proof. Suppose there are no bounded lattice L and an embedding e : L → A L such that ( e, A L ) is the canonical extension of L . Clearly the top and bottomelement of A L are, respectively e (1) and e (0) where 1 and 0 are the top andbottom element of L . Now consider the set of elements ( A L ) \ { e (0) , e (1) , m } .It is easy to see that each of these elements is completely join-irreduciblein A L and hence we have, by Lemma 2.1, that each of these elements isthe meet of the embedding of a filter of L . Hence each of the elements of( A L ) \ { e (0) , e (1) , m } is a filter element. Dually, it is easy to see that eachelement of ( A L ) \ { e (0) , e (1) , m } is completely meet-irreducible and againby Lemma 2.1 they are all the join of the embedding of an ideal of L andhence are all ideal elements. Thus every element of ( A L ) \ { e (0) , e (1) , m } isboth ideal and filter and hence must be of the form e ( a ) for some a ∈ L . Nowconsider the element m . Since m = V ω ∂ , and since every element of ω ∂ isthe image of an element of L under e , we have that m is a filter element of A L . Also, m = W ω and every element of ω is the image of an element of L under e . Therefore m is also an ideal element of A L . Hence m must be of theform e ( b ) for some b ∈ L . Thus we have that L ∼ = A L and that the embedding e is a bijection.Now we show that ( e, A L ) cannot be the canonical extension of L . Ob-serve that since m = V ω ∂ = W ω we have that V ω ∂ W ω . However, for anyfinite subset A ′ ⊆ ω ∂ and any finite subset B ′ ⊆ ω we will have W B ′ < V A ′ .Hence ( e, A L ) is not a compact completion of L . (cid:3) Example 5.3.
We consider the complete lattice M L depicted on the rightin Fig. 5. The order is given by the poset ⊕ ω with an additional element y incomparable to all elements except the top and the bottom. It can easily beseen that M L is a perfect lattice ( J ∞ ( M L ) = M ∞ ( M L ) = { y }∪{ x i | i > } ).It is not PTi since there are no w and z for the pair x j (cid:10) y ( j > X = { a i } i ∈ ω , X = { b i } i ∈ ω and let R = { ( a , b ) , ( a , b ) } ∪ { ( a i , b j ) | i, j i } . By considering ¬ ( a Rb ) it is rather straightforward to show that ( X , X , R )does not satisfy (Ti).CEs of BDL’sCEs of BL’sPTi lattices RepresentableposetsRepresentable graphsTiRS graphs Figure 6.
The correspondences between classes of PTi lat-tices and classes of TiRS graphs.Our final picture Fig. 6 describes the correspondence between PTi lat-tices and TiRS graphs and between their important subclasses: (i) the canon-ical extensions of bounded lattices inside the PTi lattices and representablegraphs (as dual graphs of bounded lattices) inside the TiRS graphs; (ii)the canonical extensions of bounded distributive lattices inside the canon-ical extensions of bounded lattices and representable posets (as dual graphsof bounded distributive lattices) inside the representable graphs.A natural question that we asked already in [2, pages 126–127] waswhich TiRS graphs arise as duals of bounded lattices. In the case of boundeddistributive lattices (denoted as BDL’s in Fig. 6) this question reduces tothe question of which posets are representable posets which seems to be ex-tremely hard. Examples of non-representable posets are also examples ofnon-representable graphs as any poset is automatically a TiRS graph. Wemention an example of a non-representable poset due to Tan [17] from the1970s: T := ω ⊕ ω δ . The perfect lattice corresponding to this TiRS graph isthe PTi lattice T L := ω ⊕ ⊕ ω δ .anonical extensions of lattices 15 Acknowledgements
The first author gratefully acknowledges the hospitality of Matej Bel Univer-sity during his research visit in September 2017. The third author acknowl-edges support from Slovak grant VEGA 1/0337/16 and the hospitality of theUniversity of Lisbon during his visit in September 2019.
References [1] Clark D.M., Davey, B.A.: Natural Dualities for the Working Algebraist, Cam-bridge University Press (1998)[2] Craig, A.P.K, Gouveia, M.J., Haviar, M.: TiRS graphs and TiRS frames: anew setting for duals of canonical extensions, Algebra Universalis , 123–138(2015)[3] Craig, A.P.K., Haviar, M., Priestley, H.A.: A fresh perspective on canonicalextensions for bounded lattices, Appl. Categ. Structures , 725–749 (2013)[4] Craig, A.P.K., Haviar, M.: Reconciliation of approaches to the construction ofcanonical extensions of bounded lattices, Math. Slovaca , 1335–1356 (2014)[5] Deiters, K., Ern´e, M.: Sums, products and negations of contexts and completelattices, Algebra Universalis , 469–496 (2009)[6] Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relationalcompleteness of some substructural logics, J. Symbolic Logic , 713–740 (2005)[7] Gehrke, M.: Generalized Kripke frames, Studia Logica , 241–275 (2006)[8] Gehrke, M.: Canonical extensions: an algebraic approach to Stone duality, Al-gebra Universalis : 63 (2018)[9] Gehrke, M., Harding, J.: Bounded lattice expansions, J. Algebra , 345–371(2001)[10] Gehrke, M., Vosmaer, J.: A view of canonical extension. In: Proceedings ofthe Eighth International Tbilisi Symposium, TbiLLC, 2009. Lecture Notes inComputer Science vol. 6618, pp. 77–100. Logic, Language and Computation(Tbilisi, 2011)[11] Goldblatt, R.: Canonical extensions and ultraproducts of polarities, AlgebraUniversalis : 80 (2018)[12] Hartonas, C., Dunn, J.M.: Stone duality for lattices, Algebra Universalis ,391–401 (1997)[13] J´onsson, B., Tarski, A.: Boolean algebras with operators, I & II, Amer. J.Math. , 891–939 (1951) & , 127–162 (1952)[14] Jipsen, P:, Categories of algebraic contexts equivalent to idempotent semiringsand domain semirings, RAMiCS 2012, LNCS 7560, 195–206 (2012)[15] Moshier, M.A.,: A relational category of formal contexts, Pre-print.[16] Ploˇsˇcica, M.: A natural representation of bounded lattices, Tatra MountainsMath. Publ. , 75–88 (1995)[17] Tan, T.: On representable posets. PhD Thesis, University of Manitoba (1974)[18] Urquhart, A.: A topological representation theory for lattices, Algebra Univer-salis , 45–58 (1978) Andrew P. K. CraigDepartment of Mathematics and Applied MathematicsUniversity of JohannesburgPO Box 524, Auckland Park, 2006South Africae-mail: [email protected]
Maria J. Gouveiae-mail: [email protected]