Dualizing sup-preserving endomaps of a complete lattice
DDavid I. Spivak and Jamie Vicary (Eds.):Applied Category Theory 2020 (ACT2020)EPTCS 333, 2021, pp. 335–346, doi:10.4204/EPTCS.333.23
Dualizing sup-preserving endomaps of a complete lattice
Luigi Santocanale
LIS, CNRS UMR 7020, Aix-Marseille Universit´e, France [email protected]
It is argued in [5] that the quantale [ L , L ] ∨ of sup-preserving endomaps of a complete lattice L is aGirard quantale exactly when L is completely distributive. We have argued in [16] that this Girardquantale structure arises from the dual quantale of inf-preserving endomaps of L via Raney’s trans-forms and extends to a Girard quantaloid structure on the full subcategory of SLatt (the category ofcomplete lattices and sup-preserving maps) whose objects are the completely distributive lattices.It is the goal of this talk to illustrate further this connection between the quantale structure,Raney’s transforms, and complete distributivity. Raney’s transforms are indeed mix maps in theisomix category
SLatt and most of the theory can be developed relying on naturality of these maps.We complete then the remarks on cyclic elements of [ L , L ] ∨ developed in [16] by investigating itsdualizing elements. We argue that if [ L , L ] ∨ has the structure a Frobenius quantale, that is, if ithas a dualizing element, not necessarily a cyclic one, then L is once more completely distributive.It follows then from a general statement on involutive residuated lattices that there is a bijectionbetween dualizing elements of [ L , L ] ∨ and automorphisms of L . Finally, we also argue that if L isfinite and [ L , L ] ∨ is autodual, then L is distributive. SLatt
The homset [ X , Y ] ∨ in SLatt . The category
SLatt of complete lattices and sup-preserving functionsis a well-known ∗ -autonomous category [1, 9, 8, 5]. For complete lattices X , Y , we denote by [ X , Y ] ∨ the homset in this category. The two-element Boolean algebra is a dualizing element and [ X , ] ∨ is,as a lattice, isomorphic to the dual lattice X op . More generally, the functor ( · ) ∗ = [ · , ] ∨ is naturallyisomorphic to the functor ( · ) op where, for f : Y −→ X , f op : X op −→ Y op is the right adjoint of f , notedhere by ρ ( f ) (the left adjoint of an inf-preserving function g : Y −→ X shall be denoted by ℓ ( g ) : X −→ Y ).Let us describe the internal structure of the homset [ X , Y ] ∨ as a complete lattice. For x ∈ X and y ∈ Y ,we define the following elements of [ X , Y ] ∨ : c y ( t ) : = ( y , t = ⊥ , ⊥ , t = ⊥ , a x ( t ) : = ( ⊤ , t x , ⊥ , t ≤ x , ( y ⊗ x )( t ) : = ⊤ , t x , y , ⊥ < t ≤ x , ⊥ , t = ⊥ , e y , x ( t ) : = ( y , t x , ⊥ , t ≤ x . Lemma 1.1.
For each f ∈ [ X , Y ] ∨ , x ∈ X , and y ∈ Y , f ( x ) ≤ y if and only if f ≤ y ⊗ x. Consequently, foreach f ∈ [ X , Y ] ∨ , f = ^ { y ⊗ x | f ( x ) ≤ y } = ^ x ∈ X f ( x ) ⊗ x , and the sup-preserving functions of the form y ⊗ x generate [ X , Y ] ∨ under arbitrary meets.
36 Dualizing sup-preserving endomaps of acomplete latticeThe tensor notation arises from the canonical isomorphism [ X , Y ] ∨ ≃ ( Y op ⊗ X ) op of ∗ -autonomouscategories. That is, [ X , Y ] ∨ is dual to the tensor product Y op ⊗ X and the functions y ⊗ x correspond toelementary tensors of Y op ⊗ X . For f ∈ [ X , Y ] ∨ , g ∈ [ Y , Z ] ∨ , and h ∈ [ X , Z ] ∨ , let us recall that there existsuniquely determined maps g \ h ∈ [ X , Y ] ∨ and h / f ∈ [ Y , Z ] ∨ satisfying g ◦ f ≤ h iff f ≤ g \ h iff g ≤ h / f . The binary operations \ and / are known under several names: they yield left and right Kan extensionsand (often when X = Y = Z ) they are named residuals or division operations [6] or right and left impli-cation [14, 5]. With the division operations at hand, let us list some elementary relations between thefunctions previously defined: Lemma 1.2.
The following relations hold: (i) y ⊗ x = c y ∨ a x = c y / c x = a y \ a x , (ii) e y , x = c y ∧ a x = c y ◦ a x ,(iii) c y = y ⊗⊤ = c y ◦ a ⊥ , (iv) a x = ⊥⊗ x = c ⊤ ◦ a x .Proof. The relations y ⊗ x = c y ∨ a x and e y , x = c y ∧ a x are well known, see e.g. [19], and ( iii ) and ( iv ) areimmediate consequences of these relations.Let us focus on y ⊗ x = c y / c x = a y \ a x and notice that, in order to make sense of these relations, weneed to assume c x : X ′ −→ X , c y : X ′ −→ Y , a x : X −→ Y ′ , and a y : Y −→ Y ′ . Observe that f ( x ) ≤ y if andonly if f ◦ c x ≤ c y and therefore, in view of Lemma 1.1 and of the definion of / , the relation y ⊗ x = c y / c x .Next, the condition f ≤ a y \ a x amounts to a y ◦ f ≤ a x , that is, for all t ∈ X , if t ≤ x then f ( t ) ≤ y . Clearlythis condition is equivalent to f ( x ) ≤ y , and therefore to f ≤ y ⊗ x . Finally, the relation e y , x = c y ◦ a x is directly verified. However, let us observe the abuse of notation, since for the maps a x and c y to becomposable we need to assume either X = Y or a x : X −→ and c y : −→ Y . Inf-preserving functions as tensor product.
Let [ X , Y ] ∧ denote the poset of inf-preserving functionsfrom X to Y , with the pointwise ordering. Observe that, as a set, [ X , Y ] ∧ equals [ X op , Y op ] ∨ . Yet, as aposet or a lattice, the equality [ X , Y ] ∧ = [ X op , Y op ] op ∨ is the correct one. As a matter of fact, we have f ≤ [ X , Y ] ∧ g iff f ( x ) ≤ Y g ( x ) , all x ∈ X , iff g ( x ) ≤ Y op f ( x ) , all x ∈ X , iff g ≤ [ X op , Y op ] ∨ f . Using standardisomorphisms of ∗ -autonomous categories, we have [ X , Y ] ∧ = [ X op , Y op ] op ∨ ≃ Y ⊗ X op . That is, the set of inf-preserving functions from X to Y can be taken as a concrete realization of thetensor product Y ⊗ X op . This should not come as a surprise, since it is well-known that the set of Galoisconnections from X to Y —that is, pairs of functions ( f : X −→ Y , g : Y −→ X ) such that y ≤ f ( x ) iff x ≤ g ( y ) —realizes the tensor product Y ⊗ X in SLatt , see e.g. [18, 12] or [5, §2.1.2]. Such a pair offunctions is uniquely determined by its first element, which is an inf-preserving functions from X op to Y .Notice now that [ X , Y ] op ∨ ≃ Y op ⊗ X ≃ X ⊗ Y op ≃ [ Y , X ] ∧ , (1)from which we derive the following principle: Fact 1.3.
There is a bijection beweeen sup-preserving functions from [ X , Y ] op ∨ to [ X , Y ] ∨ and sup-preservingfunctions from [ Y , X ] ∧ to [ X , Y ] ∨ . The map yielding the isomorphism in equation (1) is ρ , the operation of taking the right adjoint. Thebijection stated in Fact 1.3 is therefore obtained by precomposing with ρ ..Santocanale 337We exploit now the work done for [ X , Y ] ∨ to recap the structure of [ X , Y ] ∧ as a tensor product.Consider the maps γ y ( t ) : = ( ⊤ , t = ⊤ , y , otherwise , α x ( t ) : = ( ⊤ , x ≤ t , ⊥ , otherwise , y ⊗ x ( t ) = ⊤ , t = ⊤ , y , x ≤ t , ⊥ , otherwise . By dualizing Lemma 1.1, we observe that the relation y ⊗ x = γ y ∧ α x holds, the maps y ⊗ x realize theelementary tensors of the (abstract) tensor product Y ⊗ X op , [ X , Y ] ∧ is join-generated by these maps, andevery g ∈ [ X , Y ] ∧ can be canonically written as g = W x ∈ X g ( x ) ⊗ x .Recall that a bimorphism ψ : Y × X op −→ Z is a function that is sup-preserving in each variable,separately. This in particular means that infs in X are transformed into sups in Z . The universal propertyof [ X , Y ] ∧ as a tensor product can be therefore stated as follows: Fact 1.4.
Given a bimorphism ψ : Y × X op −→ Z, there exists a unique sup-preserving functions ˜ ψ : [ X , Y ] ∧ −→ Z such that ˜ ψ ( y ⊗ x ) = ψ ( y , x ) . For g ∈ [ X , Y ] ∧ , ˜ ψ ( g ) is defined by ˜ ψ ( g ) : = _ x ∈ L ψ ( g ( x ) , x ) . For g ∈ [ X , Y ] ∧ and f ∈ [ X , Y ] ∨ , define g ∨ ( x ) : = _ x t g ( t ) , f ∧ ( x ) : = ^ t x f ( t ) . It is easily seen that g ∨ has a right adjoint, so g ∨ ∈ [ X , Y ] ∨ , and that f ∧ has a left adjoint, so f ∧ belongsto [ X , Y ] ∧ . We call the operations ( · ) ∨ and ( · ) ∧ the Raney’s transforms, even if Raney defined thesetransforms on Galois connections. (In [16] we explicitly related these maps to Raney’s original wayof defining them). Notice that g ∨ ≤ f if and only if g ≤ f ∧ , so ( · ) ∧ is right adjoint to ( · ) ∨ . Raney’stransforms have been the key ingredient allowing us to prove in [17] that [ C , C ] ∨ is a Girard quantaleif C is a complete chain and, lately in [16], that the full-subcategory of SLatt whose objects are thecompletely distributive lattices is a Girard quantaloid.Consider the bimorphism e : Y × X op −→ [ X , Y ] ∨ sending y , x to e y , x = c y ◦ a x ∈ [ X , Y ] ∨ and its exten-sion ˜ e ( f ) = _ { c f ( t ) ◦ a t | t ∈ X } . By evaluating ˜ e ( f ) at x ∈ L , we obtain˜ e ( f )( x ) = _ { ( c f ( t ) ◦ a t )( x ) | t ∈ L } = _ x t f ( t ) = f ∨ ( x ) ,
38 Dualizing sup-preserving endomaps of acomplete latticethat is, ˜ e ( f ) = f ∨ . Remark now that e : Y × X op −→ [ X , Y ] ∨ is the (set-theoretic) transpose of the trimor-phism h y , x , t i = ( ⊥ , t ≤ xy , otherwise . Consequently, Raney’s transform ( · ) ∨ is the transpose of the map Y ⊗ X op ⊗ X ≃ −−→ Y ⊗ [ X , ] ∨ ⊗ X Y ⊗ eval −−−−−→ Y ⊗ ≃ −−→ Y . (2)In the category SLatt , is both the unit for the tensor product Y ⊗ X and its dual ( Y op ⊗ X op ) op ≃ [ X , Y op ] ∨ . ∗ -autonomous categories with this property are examples of isomix categories in sense of[3, 4, 2] where the transpose of the map in (2) is named mix . That Raney’s transforms are mix maps wasrecognized in [8] where also the nuclear objects—i.e. those objects whose mix maps are invertible—inthe category SLatt were characterized (using Raney’s Theorem) as the completely distributive lattices.The importance of this characterization stems from the fact that the nucleus of a symmetric monoidalclosed category—that is, the full subcategory of nuclear objects—yields a right adjoint to the forgetfulfunctor from the category of compact closed categories to that of symmetric monoidal closed ones,as mentioned in [15]. In particular, the full subcategory of
SLatt whose objects are the completelydistributive lattices is more than symmetric monoidal closed or ∗ -autonomous, it is compact closed [10].A key property of Raney’s transforms, importantly used in [17, 16], is the following. For g ∈ [ X , Y ] ∧ and f ∈ [ X , Y ] ∨ , the relations ρ ( g ∨ ) = ℓ ( g ) ∧ , ℓ ( f ∧ ) = ρ ( f ) ∨ , (3)hold. This property might be directly verified, as we did in [17, 16]. It might also be inferred from thecommutativity of each square in the diagram below: [ X , Y ] ∧ [ X , Y ] ∨ Y ⊗ X op X op ⊗ Y [ Y op , X op ] ∧ [ Y op , X op ] ∨ [ Y , X ] op ∨ [ Y , X ] op ∧ ( · ) ∨ X , Y ℓ ρσ mix Y , X mix Xop , Yop ( · ) ∨ Yop , Xop ( · ) ∧ Y , X Naturality of Raney’s transforms.
Observe now that, for g : X ′ −→ X and f : Y −→ Y ′ , we have f ◦ c y = c f ( y ) , a x ◦ g = a ρ ( g )( x ) , and f ◦ e y , x ◦ g = e f ( y ) , ρ ( g )( x ) , implying that the following diagram commutes:.Santocanale 339 Y ⊗ X op [ X , Y ] ∨ Y ′ ⊗ X ′ op [ X ′ , Y ′ ] ∨ f ⊗ g op ( · ) ∨ Y , X [ g , f ] ∨ ( · ) ∨ Y ′ , X ′ That is, Raney’s transform ( · ) ∨ is natural in both its variables. Let us remark on the way the following: Proposition 2.1.
There are exactly two natural arrows from Y ⊗ X op to [ X , Y ] ∨ , the trivial one andRaney’s transform. In order to simplify reading, we use ψ both for a bimorphism ψ : Y × X op −→ Z and for its extensionto the tensor product ˜ ψ : Y ⊗ X op −→ Z . Proof. If ψ is natural, then ψ ( y , x ) = ψ ( c y ( ⊤ ) , γ x ( ⊥ )) = c y ◦ ψ ( ⊤ , ⊥ ) ◦ a x , since ρ ( a x ) = γ x . Let f = ψ ( ⊤ , ⊥ ) . If f = ⊥ , then ψ is the trivial map. Otherwise, f = c ⊥ and f ( ⊤ ) = ⊥ .Then, observing that f ◦ a x = c f ( ⊤ ) ◦ a x and that c y ◦ c z = c y for z = ⊥ , it follows that ψ ( y , x ) = c y ◦ f ◦ a x = c y ◦ c f ( ⊤ ) ◦ a x = c y ◦ a x . Remark . Similar considerations can be developed if naturality is required in just one variable. Forexample, if the bimorphism ψ : Y × X op −→ [ X , Y ] ∨ is such that ψ ( y , x ) ◦ g = ψ ( y , ρ ( g )( x )) , then ψ ( y , x ) = χ ( y ) ◦ a x for some χ : Y −→ [ X , Y ] ∨ . ♦ For f : X −→ Y in the category SLatt , let j = ρ ( f ) ◦ f and o = f ◦ ρ ( f ) . Denote by X j (resp. Y o ) theset of fixed points of j (resp., of o ). Then, we have a standard (epi,iso,mono)-factorization X YX j Y ofj ≃ Thus, f is mono if and only j = id X and f is epic if and only if o = id Y . Notice that Y o is the image of X under f , while X j is the image of Y under ρ ( f ) . We apply this factorization to Raney’s transforms. Definition 2.3.
A sup-preserving function f : X −→ Y is tight if f ∧∨ = f , or, equivalently, if it belongsto the image of [ X , Y ] ∧ via the Raney’s transform ( · ) ∨ . We let [ X , Y ] t ∨ be the set of tight functions from X to Y .By its definition, [ X , Y ] t ∨ is the sub-join-semilattice of [ X , Y ] ∨ generated by the c y ◦ a x . Moreover, itis easily seen that w ⊗ z ∈ [ X , Y ] t ∨ , for each w ∈ Y and z ∈ X , and that c y ◦ a x ≤ w ⊗ z if and only of y ≤ w or z ≤ x . From these relations, [ X , Y ] t ∨ yields a concrete representation of Wille’s tensor product Y b ⊗ X op ,see [19], which, for finite lattices, coincides with the Box tensor product of [7].Next, we list some immediate consequences of naturality of Raney’s transforms: Proposition 2.4.
The following statements hold:(i) [ X , Y ] t ∨ is a bi-ideal of [ X , Y ] ∨ .(ii) For a complete lattice L, the transform ( · ) ∨ : [ L , L ] ∧ −→ [ L , L ] ∨ is surjective if and only if id L ∈ [ L , L ] t ∨ , that is, if id = id ∧∨ .
40 Dualizing sup-preserving endomaps of acomplete lattice (iii) For each complete lattice L, the pair ([ L , L ] t ∨ , ◦ ) is a quantale. Let us recall that Raney’s Theorem [13] characterizes completely distributive lattices as those com-plete lattices satisfying the identity z = _ z x ^ y x y . This identity is exactly the identity id = id ∧∨ or, as we have seen in Proposition 2.4, the identity f = f ∧∨ holding for each f ∈ [ L , L ] ∨ . Since complete distributivity is autodual (at least in a classical context), wederive that Raney’s transform ( · ) ∨ : [ L , L ] ∧ −→ [ L , L ] ∨ is surjective if and only if it is injective.We conclude this section with a glance at the quantale ([ L , L ] t ∨ , ◦ ) of tight maps, where L is anarbitrary complete lattice L . We pause before for a technical lemma needed end the section and later onas well. Recalling the equations in (3), let us define f ∗ : = ℓ ( f ∧ ) ( = ρ ( f ) ∨ ) , and observe the following: Lemma 2.5.
For each x ∈ X , y ∈ Y , and f ∈ [ X , Y ] ∨ , the following conditions are equivalent: (i) for allt ∈ X , x ≤ t or y ≤ f ( t ) , (ii) c y ◦ a x ≤ f (iii) y ⊗ x ≤ f ∧ (iv) y ≤ f ∧ ( x ) (v) f ∗ ( y ) ≤ x (vi) f ∗ ≤ x ⊗ y.Proof. ( i ) ⇔ ( ii ) : direct verification. ( ii ) ⇔ ( iii ) : since c y ◦ a x = ( y ⊗ x ) ∨ , c y ◦ a x ≤ f ∧ and by theadjunction ( · ) ∨ ⊣ ( · ) ∧ . ( iii ) ⇔ ( iv ) : by the dual of Lemma 1.1. ( iv ) ⇔ ( v ) : since f ∗ ⊣ f ∧ . ( iv ) ⇔ ( v ) : by Lemma 1.1. Proposition 2.6.
Unless L is a completely distributive lattice (in which case [ L , L ] t ∨ = [ L , L ] ), the quantale ([ L , L ] t ∨ , ◦ ) is not unital.Proof. Let u be unit for ([ L , L ] t ∨ , ◦ ) and write u = W i ∈ I c y i ◦ a x i . For an arbitrary x ∈ L , evaluate at x theidentity a x = a x ◦ u = _ a x ◦ c y i ◦ a x i and deduce that, for each i ∈ I , ⊥ = a x ◦ c y i ◦ a x i ( x ) . This happens exactly when x ≤ x i or y i ≤ x , thatis, when c y i ◦ a x i ≤ x ⊗ x . Since x ∈ L and i ∈ I are arbitrary, we have, within [ L , L ] , u = W i ∈ I c y i ◦ a x i ≤ V x ∈ L x ⊗ x = id L . Again, for y ∈ L arbitrary, evaluate at ⊤ the identity c y = u ◦ c y = _ c y i ◦ a x i ◦ c y . and deduce that y = W y x i y i . Considering that c y i ◦ a x i ≤ id L , then we have y i ≤ id ∧ ( x i ) and therefore y = _ y x i y i ≤ _ y x i id ∧ ( x i ) ≤ _ y t id ∧ ( t ) = id ∧∨ ( y ) . Since this holds for any y ∈ L , id ≤ id ∧∨ and since the opposite inclusion always holds, then id = id ∧∨ .By Raney’s Theorem, L is a completely distributive lattice.Recall that a dualizing element in a quantale ( Q , ◦ ) is an element 0 ∈ Q such that 0 / ( x \ ) = ( / x ) \ = x , for each x ∈ Q . As consequences of Proposition 2.6, we obtain:.Santocanale 341 Corollary 2.7.
Unless L is a completely distributive lattice,(i) the quantale ([ L , L ] t ∨ , ◦ ) has no dualizing element,(ii) the interior operator ( · ) ∧∨ obtained by composing the two Raney’s transform is not a conucleuson [ L , L ] ∨ .Proof. (i) If 0 is dualizing, then 0 \ ( g ◦ f ) ∧∨ ≤ g ∧∨ ◦ f ∧∨ , required for ( · ) ∧∨ to be a conucleus on [ L , L ] ∨ , does not hold, unless L is completelydistributive. The opposite inclusion g ∧∨ ◦ f ∧∨ ≤ ( g ◦ f ) ∧∨ holds since g ∧∨ ◦ f ∧∨ belongs to [ L , L ] t ∨ , g ∧∨ ◦ f ∧∨ ≤ g ◦ f , and ( g ◦ f ) ∧∨ is the greatest element of [ L , L ] t ∨ below g ◦ f . If ( g ◦ f ) ∧∨ = g ∧∨ ◦ f ∧∨ for each f , g ∈ [ L , L ] ∨ , then 1 ∧∨ is a unit for [ L , L ] t ∨ and L is completely distributive. [ L , L ] ∨ We investigate in this section dualizing elements of [ L , L ] ∨ . Proposition 2.6.18 in [5] states that if id L ∨ is dualizing, then L is completely distributive. Recall that a cyclic element in a quantale ( Q , ◦ ) is anelement 0 ∈ Q such that 0 / x = x \
0, for each x ∈ Q . Trivially, the top element of a quantale is cyclic. Ourwork [16] proves that if [ L , L ] ∨ has a non-trivial cyclic element, then this element is id L ∨ and, once more,cyclicity of id L ∨ implies that L is completely distributive. It was still open the possibility that [ L , L ] ∨ might have dualizing elements and no non-trivial cyclic elements. This is possible in principle, since thetool Mace4 [11] provided us with an example of a quantale where the unique dualizing element is notcyclic. The quantale is built on the modular lattice M (with atoms u , d , a , b , c ) and has the followingmultiplication table: ⊥ u d a b c ⊤⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ u ⊥ u d a b c ⊤ d ⊥ d ⊤ ⊤ ⊤ ⊤ ⊤ a ⊥ a ⊤ ⊤ ⊤ d ⊤ b ⊥ b ⊤ d ⊤ ⊤ ⊤ c ⊥ c ⊤ ⊤ d ⊤ ⊤⊤ ⊥ ⊤ ⊤ ⊤ ⊤ ⊤ ⊤ It is verified that d is the only non-cyclic element and that, at the same time, it is the only dualizing ele-ment. For the quantale [ L , L ] ∨ we shall see that existence of a dualizing element again implies completedistributivity of L (and therefore existence of a cyclic and dualizing element).As this might be of more general interest, we are going to investigate how divisions · \ f and f / · byan arbitrary f ∈ [ L , L ] ∨ act on the e y , x and y ⊗ x . To this end, we start remarking that Raney’s transformsintervene in the formulas for computing left adjoints of the maps c and a . Lemma 3.1.
The functions c : Y −→ [ X , Y ] ∨ and a : X op −→ [ X , Y ] ∨ have both a left and a right adjoint.Namely, for each x ∈ X , y ∈ Y , and f ∈ [ X , Y ] ∨ , the following relations hold:c y ≤ f iff y ≤ f ∧ ( ⊥ ) , f ≤ c y iff f ( ⊤ ) ≤ y , a x ≤ f iff f ∗ ( ⊤ ) ≤ x , f ≤ a x iff x ≤ ρ ( f )( ⊥ ) . Using the relations stated in Lemma 3.1 computing divisions becomes an easy task.42 Dualizing sup-preserving endomaps of acomplete lattice
Lemma 3.2.
For each x ∈ X , y ∈ Y , and f , g ∈ [ X , Y ] ∨ , we havef / a x = f ∧ ( x ) ⊗⊤ ( = c f ∧ ( x ) ) , c y \ f = ⊥⊗ f ∗ ( y ) ( = a f ∗ ( y ) ) , f / c y = f ∧ ( ⊥ ) ⊗ y , a x \ f = x ⊗ f ∗ ( ⊤ ) . Proof.
We compute as follows: g ≤ c y \ f iff c y ◦ g ≤ f iff c y ◦ a ⊥ ◦ g ≤ f iff c y ◦ a ρ ( g )( ⊥ ) ≤ f iff f ∗ ( y ) ≤ ρ ( g )( ⊥ ) iff g ≤ a f ∗ ( y ) . where we have used Lemma 2.5. Verification that f / a x = c f ∧ ( x ) is similar. The other two identities areverified as follows: h ≤ f / c y iff h ◦ c y ≤ f iff c h ( y ) ≤ f iff h ( y ) ≤ f ∧ ( ⊥ ) iff h ≤ f ∧ ( ⊥ ) ⊗ y , and, dually, h ≤ a x \ f iff a x ◦ h ≤ f iff a ρ ( h )( x ) ≤ f iff f ∗ ( ⊤ ) ≤ ρ ( h )( x ) iff h ( f ∗ ( ⊤ )) ≤ x iff h ≤ x ⊗ f ∗ ( ⊤ ) . The relations in the following proposition are then easily derived.
Proposition 3.3.
The following relations hold:f / ( c y ◦ a x ) = f ∧ ( x ) ⊗ y , ( c y ◦ a x ) \ f = x ⊗ f ∗ ( y ) , f / ( y ⊗ x ) = f ∧ ( x ) ⊗⊤ ∧ f ∧ ( ⊥ ) ⊗ y , ( y ⊗ x ) \ f = ⊥⊗ f ∗ ( y ) ∧ x ⊗ f ∗ ( ⊤ ) . From Proposition 3.3, it follows that c f ∧ ( x ) ◦ a y ≤ f / ( y ⊗ x ) , c x ◦ a f ∗ ( y ) ≤ ( y ⊗ x ) \ f . If f ∗ is invertible, then its inverse is also its right adjoint. From the uniqueness of the right adjointit follows that f ∗ is inverted by its right adjoint f ∧ . Thus, f ∗ is invertible if and only if f ∧ is invertible,in which case we have f ∗ ( ⊤ ) = ⊤ and f ∧ ( ⊥ ) = ⊥ , since both f ∗ and f ∧ are bicontinuous. In someimportant case, the expressions exhibited in Proposition 3.3 simplify: Corollary 3.4.
If f ∧ ( ⊥ ) = ⊥ (resp., f ∗ ( ⊤ ) = ⊤ ), thenf / ( y ⊗ x ) = c f ∧ ( x ) ◦ a y ( resp., ( y ⊗ x ) \ f = c x ◦ a f ∗ ( y ) ) . These relations hold as soon as either f ∧ or f ∗ is invertible. Example 3.5. If L is a completely distributive lattice, then the relation id L ∧∨ = id L holds, by Raney’sTheorem [13]. Let o = id L ∨ , then o ∧ is invertible, since it is the identity. Necessarily, we also have o ∗ = id L and therefore: o / ( c y ◦ a x ) = ( c y ◦ a x ) \ o = x ⊗ y , o / ( y ⊗ x ) = ( y ⊗ x ) \ o = c x ◦ a y . ♦ Theorem 3.6.
If f ∈ [ L , L ] ∨ is dualizing, then f ∧ and f ∗ are inverse to each other and L is completelydistributive. .Santocanale 343 Proof. If f is dualizing, then, for all x , y ∈ L , c y = f / ( c y \ f ) = f / a f ∗ ( y ) = c f ∧ ( f ∗ ( y )) , a x = ( f / a x ) \ f = c f ∧ ( x ) \ f = a f ∗ ( f ∧ ( x )) , and since both c and a are injective, then y = f ∧ ( f ∗ ( y )) and x = f ∗ ( f ∧ ( x )) . Thus, f ∧ and f ∗ are inverseto each other. Remark now that f ∗ ∈ [ L , L ] t ∨ , since f ∗ = ℓ ( f ∧ ) = ρ ( f ) ∨ . Then id L = f ∧ ◦ f ∗ ∈ [ L , L ] t ∨ ,since [ L , L ] t ∨ is an ideal of [ L , L ] ∨ . Then L is completely distributive by Raney’s Theorem.It is not difficult to give a direct proof of the converse, namely that if f ∗ is invertible, then f isdualizing. We prove this as a general statement about involutive residuated lattices. Notice that the mapsending f to f ∗ is definable in the language of involutive residuated lattices, since f ∗ = f \ o , where o = id L ∨ is the canonical cyclic dualizing element of [ L , L ] ∨ (if L is completely distributive). The statementin the following Proposition 3.7 is implicit in the definition of a (symmetric) compact closed category in[10] (see [20, §5] for the non symmetric version of this notion). Proposition 3.7.
In every involutive residuated lattice Q, f is dualizing if and only if f ∗ is invertible.Proof. If f ∈ Q is dualizing, then f / ( x ∗ \ f ) = ( f / x ∗ ) \ f = x ∗ , for each x ∈ Q . Using well known identitiesof involutive residuated lattices, compute as follows: x = x ∗∗ = ( f / ( x ∗ \ f )) ∗ = ( x ∗ \ f ) ◦ f ∗ = ( x / f ∗ ) ◦ f ∗ . Letting x =
1, then 1 = ( / f ∗ ) ◦ f ∗ . We derive 1 = f ∗ ◦ ( f ∗ \ ) similarly, from which it follows that f ∗ is inverted by 1 / f ∗ = f ∗ \ f ∗ is invertible, say f ∗ ◦ g = g ◦ f ∗ =
1. It immediately follows that g = / f ∗ = f ∗ \
1. Again, for each x ∈ Q , x ∗ = x ∗ ◦ ( / f ∗ ) ◦ f ∗ ≤ ( x ∗ / f ∗ ) ◦ f ∗ , and then, dualizing this relation, we obtain x = x ∗∗ ≥ (( x ∗ / f ∗ ) ◦ f ∗ ) ∗ = f / ( x \ f ) . Since x ≤ f / ( x \ f ) always hold, we have x = f / ( x \ f ) . The identity x = ( f / x ) \ f is derived similarly. Example 3.8. If L = [ , ] , then f is dualizing if and only if it is invertible. Indeed, f is dualizing iff f ∗ is invertible iff f ∧ is invertible. Now, if f ∧ is invertible, then it is continuous and f ∧ = f ; therefore f isinvertible. Similarly, if f is invertible, then it is continuous and f ∧ = f ; therefore f ∧ is invertible. ♦ Example 3.9.
Consider a poset P , the complete lattice D ( P ) of downsets of P , and recall that D ( P ) is completely distributive. The quantale [ D ( P ) , D ( P )] ∨ is isomorphic to the quantale of weakeningrelations (profuctors/bimodules) on the poset P . These are the relations R ⊆ P × P such that yRx , y ′ ≤ y ,and x ≤ x ′ imply x ′ Ry ′ (for all x , x ′ , y , y ′ ∈ P ). Thus, weakening relations are downsets of P × P op and thebijection between [ D ( P ) , D ( P )] ∨ and D ( P × P op ) goes along the lines described in previous sections,since [ D ( P ) , D ( P )] ∨ ≃ D ( P ) ⊗ D ( P op ) ≃ D ( P × P op ) . Explicitly, this bijection, sending f to R f , is such that ( y , x ) ∈ R f iff y ∈ f ( ↓ x ) iff c ↓ y ◦ a ↓ x ≤ f , (4)44 Dualizing sup-preserving endomaps of acomplete latticewhere, for x ∈ P , ↓ x : = { y ∈ P | y ≤ x } . We have seen that dualizing elements of [ D ( P ) , D ( P )] ∨ are inbijection with automorphisms of D ( P ) which in turn are in bijection with automorphisms of P . Givensuch an automorphism, we aim at computing the dualizing weakening relation corresponding to thisautomorphism. To this end, let us recall that, when L is completely distributive, o = id ∨ = ℓ ( id ∧ ) is theunique non-trivial cyclic element. Observe that since o = id ∗ , o is also dualizing and that ( y , x ) ∈ R o iff x y . For f ∈ [ D ( P ) , D ( P )] ∨ , recall that f ∗ = f \ o . We use the relations in (4) to compute the dualizing elementof D ( P × P op ) corresponding to an invertible order preserving map g : P −→ P , as follows: ( y , x ) ∈ R D ( g ) ∗ iff c ↓ y ◦ a ↓ x ≤ D ( g ) ∗ = D ( g ) \ o iff D ( g ) ◦ c ↓ y ◦ a ↓ x = c D ( g )( ↓ y ) ◦ a ↓ x = c ↓ g ( y ) ◦ a ↓ x ≤ x g ( y ) . ♦ Even when Raney’s transforms are not inverse to each other, it might still be asked whether there are otherisomorphisms between [ L , L ] ∧ and [ L , L ] ∨ . By Fact 1.3, this question amounts to understand whether [ L , L ] ∨ is autodual.Let us discuss the case when L is a finite lattice. We use J ( L ) for the set of join-irreducible elements of L and M ( L ) for the set of meet-irreducible elements of L . The reader will have no difficulties convincinghimself of the following statement: Lemma 4.1.
A map f ∈ [ L , L ] ∨ is meet-irreducible if and only if it is an elementary tensor of the formm ⊗ j with m ∈ M ( L ) and j ∈ J ( L ) . The following statement might instead be less immediate:
Lemma 4.2.
For each j ∈ J ( L ) and m ∈ M ( L ) , the map e j , m is join-irreducible.Proof. Let m ∈ M ( L ) and j ∈ J ( L ) , and let us use m ∗ to denote the unique upper cover of m . Suppose that e j , m = W i ∈ I f i . By evaluating the two sides of this equality at m ∗ , we obtain j = W i ∈ I f i ( m ∗ ) and therefore j = f i ( m ∗ ) for some i ∈ I . If t ≤ m , then f i ( t ) ≤ e m , j ( t ) = ⊥ . Suppose now that t m , so m < t ∨ m and m ∗ ≤ t ∨ m . Observe also that f i ( t ) ≤ e j , m ( t ) = j , since e j , m = W i ∈ I f i . Then j = f i ( m ∗ ) ≤ f i ( m ∨ t ) = f i ( m ) ∨ f i ( t ) = ⊥ ∨ f i ( t ) = f i ( t ) , so j = f i ( t ) . We have argued that f i ( t ) = e j , m , for all t ∈ L , and thereforethat f i = e j , m . Theorem 4.3.
If L is a finite lattice and [ L , L ] ∨ is autodual, then L is distributive.Proof. If ψ : [ L , L ] op ∨ −→ [ L , L ] ∨ is invertible, then ψ restricts to a bijection M ([ L , L ] ∨ ) −→ J ([ L , L ] ∨ ) , sothese two sets have same cardinality. For m ∈ M ( L ) and j ∈ J ( L ) , the e j , m as well as the elementarytensors m ⊗ j are pairwise distinct. Therefore, we have | M ( L ) | × | J ( L ) | ≤ | J ([ L , L ] ∨ ) | = | M ([ L , L ] ∨ ) | = | M ( L ) | × | J ( L ) | and | M ( L ) | × | J ( L ) | = | J ([ L , L ] ∨ ) | . That is, the elements e j , m are all the join-irreducible elements of [ L , L ] ∨ and therefore the set { e j , m | j ∈ J ( L ) , m ∈ M ( L ) } generates [ L , L ] ∨ under joins. It follows that [ L , L ] ∨ = [ L , L ] t ∨ and that L is distributive..Santocanale 345We do not know yet if the theorem above can be generalized to infinite complete lattices or whetherthere is some fancy infinite complete lattice L that is not completely distributive and such that [ L , L ] ∨ isautodual. It is clear, however, that in order to construct such a fancy lattice, properties of bimorphisms ψ : L × L op −→ [ L , L ] ∨ need to be investigated. What are the properties of a bimorphism ψ forcing L tobe completely distributive when ˜ ψ is surjective? Taking the bimorphism e as example, let us abstractpart of Raney’s Theorem: Proposition 4.4.
Let ψ : L × L op −→ [ L , L ] ∨ be a bimorphism such that for each x , y ∈ L, the image of Lunder ψ ( y , x ) is a finite chain. If id L belongs to the image of ˜ ψ : [ L , L ] ∧ −→ [ L , L ] ∨ , then L is a completelydistributive lattice.Proof. Let z ∈ Z such that z = V i ∈ I W j ∈ J i z j . We aim at showing that z ≤ W s V i ∈ I z s ( i ) , with the index s ranging on choice functions s : I −→ S i ∈ I J i ( s is a choice function if s ( i ) ∈ J i , for each i ∈ I ). Since id L = W { ψ ( y , x ) | ψ ( y , x ) ≤ id L } , we also have z = W { ψ ( y , x )( z ) | ψ ( y , x ) ≤ id L } and therefore, in orderto achieve our goal, it will be enough to show that for each y , x ∈ L , if ψ ( y , x ) ≤ id L , then ψ ( y , x )( z ) ≤ W s V i ∈ I z s ( i ) . Let y , x be such that ψ ( y , x ) ≤ id L , fix i ∈ I , and observe then that ψ ( y , x )( z ) ≤ ψ ( y , x )( _ j ∈ J i z j ) = _ j ∈ J i ψ ( y , x )( z j ) , since z ≤ W j ∈ J i z j . Since the set { ψ ( y , x )( z j ) | j ∈ J i } is finite and directed (it is a finite chain), it has amaximum: there exists j ( i ) ∈ J j such that W j ∈ J i ψ ( y , x )( z j ) = ψ ( y , x )( z j ( i ) ) . It follows that ψ ( y , x )( z ) ≤ ψ ( y , x )( z j ( i ) ) ≤ z j ( i ) , since ψ ( y , x ) ≤ id L . By letting i vary, we have constructed a choice function j : I −→ S i ∈ I J i such that ψ ( y , x )( z ) ≤ V i ∈ I z j ( i ) , and consequently ψ ( y , x )( z ) ≤ W s V i ∈ I z s ( i ) .Bimorphisms satisfying the conditions of Proposition 4.4 might be easily constructed by taking f ∈ [ L , [ L , L ] ∨ ] ∨ , g ∈ [ L op , L op ] ∨ (resp., f ∈ [ L , L ] ∨ and g ∈ [ L op , [ L , L ] ∨ ] ∨ ), and defining then ψ ( y , x ) : = f ( y ) ◦ a g ( x ) ( resp., ψ ( y , x ) : = c f ( y ) ◦ g ( x ) ) . These bimorphisms satisfy the conditions of Proposition 4.4, since they only take two values. As aconsequence of the proposition, they cannot be used to construct a fancy dual isomorphism of [ L , L ] ∨ . References [1] Michael Barr (1979): ∗ -autonomous categories . Lecture Notes in Mathematics 752, Springer, Berlin,doi:10.1007/BFb0064582.[2] R. F. Blute, J. R. B. Cockett & R. A. G. Seely (2000): Feedback for linearly distributive categories: tracesand fixpoints . J.PureAppl.Algebra154(1-3), pp. 27–69, doi:10.1016/S0022-4049(99)00180-2.[3] R. F. Blute, J. R. B. Cockett, R. A. G. Seely & T. H. Trimble (1996):
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