On the Bitopological Nature of Stone Duality

On the bitopological nature of Stone duality

Achim Jung M. Andrew Moshier

December 4, 2006

Abstract Based on the theory of frames we introduce a Stone duality for bitopological spaces. The central concept is that of a d-frame, which axiomatises the two open set lattices. Exploring the resulting concept of d-sobriety we ﬁnd this to be a much more inclusive concept than usual sobriety. Spatial d-frames suggest additional axioms that lead us to deﬁne reasonable d-frames; these have an alternative presentation as partial frames. We explore natural notions of regularity and compactness for bitopological spaces, and their manifestation in d-frames. This yields the machinery to locate precisely within this general landscape a number of classical Stone- type dualities, namely, those of Stone for Boolean algebras and bounded distributive lattices, those of the present authors for strong proximity lattices (with negation), and the duality of classical frames. The general duality can be given a logical reading by viewing the open sets of one topology as positive extents of formulas, and those of the other topology as negative extents. This point of view emphasises the fact that formulas may be undecidable in certain states and may be self-contradictory in others. We also obtain two natural orders on the set of formulas, one related to Scott’s information order and the other being the usual logical implication. The interplay between the two can be said to be the main organising principle of this study.

1 Contents

1 Introduction and overview 3

2 Review of some Stone-type dualities 10 2.1 The dualities of Stone and Priestley ...... 10 2.2 Proximity lattices and (stably) compact spaces ...... 15 2.3 Frames ...... 20

3 Bitopological spaces and d-frames 25 3.1 Stone duality for bitopological spaces ...... 25 3.2 Logical order on a d-frame ...... 31

4 Sobriety of bitopological spaces 33 4.1 Bitopological analogues of topological concepts ...... 34 4.2 Hofmann-Mislove ...... 41

5 Reasonable d-frames and spatiality 44 5.1 Reasonable d-frames ...... 46 5.2 Biframes ...... 48

6 Regularity and compactness 54

7 Partial frames 64

8 From partial frames to distributive lattices 76 8.1 Removing the information order ...... 76 8.2 Reﬂexivity: Distributive Lattices ...... 85

9 Negation 89 9.1 Negation as additional structure ...... 89 9.2 Negation as a structural property ...... 94

10 Discussion 101

2 1 Introduction and overview

In his landmark paper — published in two parts as [Sto36] and [Sto37a] — Mar- shall Stone showed that every Boolean algebra is isomorphic to a subalgebra of a powerset. The powerset is that of the set of prime filters of the given algebra, which in this paper we will call the spectrum. Stone also showed that when the spectra are suitably topologised, homomorphisms between algebras correspond exactly to continuous functions between the corresponding spectra (but the direction is reversed). Furthermore, the spaces that arise as spectra are exactly the compact totally disconnected spaces, now called Stone spaces. In modern terminology, he thus established a dual equivalence between the category of Boolean algebras and Boolean algebra homomorphisms, and the category of Stone spaces and continuous functions. It was the first time that topology was used to resolve a problem in algebra and Stone’s work can rightly be called a milestone in the history of mathematics. Observing that the representation of Boolean algebras by their spectra does not require negation, Stone generalised his work to bounded distributive lattices in [Sto37b], but the resulting topological spaces are no longer Hausdorff and satisfy a list of axioms that must have seemed rather esoteric at the time. The morphisms, too, are not just the continuous ones but it must be further required that inverse images of compact open sets are compact. It is perhaps for these reasons that this work did not receive much attention until Hilary Priestley showed in [Pri70, Pri72] that the topology on the spectrum can be enriched in a natural way so that a compact Hausdorff space is obtained. Furthermore, the spectra carry an order relation and indeed the maps between spectra that are dual to lattice homomorphisms are the continuous order-preserving ones. To a topologist, this work offered the tantalising prospect of replacing the second-order structure of topological spaces by the first-order structure of lattices, except that the translation is only available for the very special spaces that appear in the theorems of Stone and Priestley. However, the spirit of the representation theorem can be maintained for a very large class of spaces if one axiomatises the set of all open sets (and not just the clopen or compact-open ones). According to

3 [Joh82, Notes on Chapter II] this step appears to have been first taken by Charles Ehresmann and his student Jean Benabou´ [Ben59],´ and it led to the development of frame and locale theory, sometimes referred to as pointfree topology. Although frame theory does succeed in replacing topology by algebra, it can do so only by considering the infinitary operation of “arbitrary join of open sets” and is therefore not first-order. Motivated by this fact and by applications in computer science, Michael Smyth in [Smy92a] tried to extend the original Stone duality without sacrificing the finitary nature of the algebras. He proposed to use proximity lattices, where the algebraic structure of bounded distributive lattices is extended with a “proximity relation” . In the paper [JS96], the first author in collaboration with Philipp Sunderhauf¨ showed that when the definition of proximity lattices is augmented with one further axiom, one obtains a perfectly self-dual algebraic structure for which the definition of the spectrum is furthermore much simplified and is seen to be a direct generalisation of Stone’s approach. Nonethe- less, the class of spaces that have a representation via strong proximity lattices is the same as that studied by Smyth, namely, the class of stably compact spaces.

These cover all compact Hausdorff spaces plus most of the ¡£¢ spaces that appear in

Dana Scott’s domain theory for denotational semantics, [Sco72, AJ94, GHK ¤ 03]. The representation of stably compact spaces via strong proximity lattices was successfully used in subsequent work, [JKM99, JKM01, MJ02] to provide a “continuous” version of Samson Abramsky’s Domain Theory in Logical Form, [Abr91], but at the fundamental level some important questions were left unan- swered. Foremost of these is the “correct” deﬁnition of a homomorphism between strong proximity lattices, which should be structure-preserving in a natural way and correspond to a topological morphism of some kind between spectra. At least three possibilities appear in the papers cited above, and each leads to a duality with certain topological morphisms, but in the absence of a clear understanding of the meaning of the proximity relation it is hard to choose between them. One might also hope to be able to express the duality by considering the maps into a dualising object (which is the set of truth values in the Boolean algebra case, and the two-element lattice in the case of frames), but none of the three existing

4 definitions works in this respect. Finally, one may wonder whether the duality in [JS96] can be modified so that locally stably compact spaces are covered, but despite several attempts no satisfying solution has so far emerged. The principal purpose of the present paper is demonstrate that the dualities of Stone, Ehresmann-Benabou,´ and Jung-Sunderhauf¨ are all special cases of a very general and very straightforward duality between bitopological spaces and what we will define below, d-frames. Priestley’s duality as such is not covered but her work is one of the sources of inspiration for bringing a second topology into the picture. Bitopology may at first sound rather abstract and perhaps technical but in fact there are at least three examples where two natural topologies coexist. The most important of these is the real line where we know that the usual topology

is the join of the upper and the lower topologies (with open sets all intervals

¥§¦©¨ ¥ ¨¦

and , respectively). The second example is given by the Vi-

etoris topologies for hyperspaces of a topological space ; one

! "$#&%')($#&* ,+ %-

is generated by the sets , , and the other by

. !45+ &!/"$#&%')($#102 -3 . Finally, in Scott’s domain theory one has the Scott topology and the weak lower topology, the join of which is known as the Lawson topology. An alternative and illuminating route to the constructions and results of this paper is given by the logical reading of Stone duality. For this recall that any Boolean algebra can be viewed as the set of formulas of a propositional theory factored by interprovability. The elements of the spectrum are then nothing other

than the models of the theory and Stone’s representation theorem is equivalent to ¥§78

the completeness of the proof calculus. The basic open sets 6 of the topology

¤ 7 consist of those models in which the formula holds, but the impact of this is less clear, unless there is an independent interpretation of this structure. The examples from computer science come in handy at this point, as domains bring a natural

notion of convergence with them, based on the approximation of computable ele- ¥97:

ments (e.g., functions) by partial ones. The fact that the sets 6 are open can

¤ 7 then be rephrased as saying that the property holds for a computable element if

5 and only if it holds for some finite approximating element already. Smyth emphasised this reading by pointing out that a Scott-open set in a domain corresponds to a finitely observable property of states of a computational system, [Smy83, Smy92b]. This is closely related to interpretations of open sets as semi-decidable properties in recursion theory, or as “states of knowledge” in in- tuitionistic logic. When motivating our definitions and constructions we will stick to “(finitely) observable” throughout; note that some qualification is necessary, as set-theoretically the extent of a predicate can be any subset of the state space. In the duality of propositional logic (Boolean algebras), the extents of formulas are clopen sets, in other words, both the extent and its complement are open. This is appropriate because the complement is the extent of the negated formula, but in general one would not expect the state space so neatly to separate into those states where a formula observably holds and those where it observably fails. To give an example from recursion theory, the set of halting Turing machines is finitely observable but its complement is not. There are Turing machines for which we can observe in finite time that they will run forever, namely those which return to a previous state of computation, but by Turing’s famous theorem no such finite observational criterion can exhaust all non-terminating machines. We hope that at this stage it is natural to consider separate topologies, one for

the subsets of a state space where a property observably holds, and one whose 7

open sets are those where the property observably fails. For every proposition

¥ ¥§78 ;¨ ¥978 7 6=<

we thus have the pair 6 . In general, there are states where can ¤ neither be established to hold, nor to fail. As a simple example, consider the

proposition “ >@?BA ” for real numbers; if numbers are presented as streams of

decimal digits, then we will be able to afﬁrm the property whenever > is indeed greater than zero, and refute it for negative numbers, but for any initial segment consisting of zeros we cannot afﬁrm or refute. It was phenomena such as these which led Stephen Kleene to consider three-

valued logic, where true and false are augmented with the value C , meaning that the proposition is (perhaps currently) undecided. The phenomenon is also well- known in knowledge representation, where an agent may not have enough infor-

6 mation to decide a given proposition. In this latter setting it is equally conceiv-

able that an agent is provided with conﬂicting information, that is, that its state

¥§78 ¥§78 6=<

of knowledge is contained in both 6 and . Thus we are led to Nuel ¤

Belnap’s four-valued logic, [Bel77], where there can be not only too little infor- D mation (denoted by C ), but also too much (denoted by ). As it will turn out, Belnap’s four-element lattice of truth values is the dualising object in our duality. So from the logical perspective, we are again led to consider bitopologies. In ad-

dition, the logic suggests that the following two special cases are of importance:

7 ¥978 ¥§78 6=<

some propositions do not self-contradict, that is, the extents 6 and ¤

on the state space are disjoint; we call such propositions consistent. Dually, some

¥ ¥

6 E 6=< E

propositions E are observable at every state, that is and cover the ¤

state space; these will be called total. The algebraic abstraction of the situation F <

consists of two frames F and plus two relations con and tot between them; ¤ we call this a d-frame. At this stage we hope we have given enough information to be able to outline the contents of the paper. In Section 2 we review the classical dualities mentioned above and present the mathematical machinery that underlies them. In Section 3 we begin our study of d-frames and the duality with bitopological spaces. The deﬁnitions and constructions will turn out to be straightforward adaptations of

the duality of frames. We show that d-frames can be deﬁned as algebraic struc- F <

tures on a single carrier set (to be thought of as F ). The most important ¤HG

outcome of this (mathematically simple) exercise is that d-frames carry an infor- J mation order I and a logical order . Semantically, the former can be thought of as increasing positive and negative extents of a formula, the latter as increasing the positive and decreasing the negative. The two orders are connected via simple algebraic laws. In Section 4 we look at those bitopological spaces that are faithfully represented by d-frames, and ﬁnd that this class is considerably wider than the class of sober spaces. The dual concept, spatial d-frames, is examined in Section 5. These satisfy a number of additional axioms (not necessary for the underlying duality) and we select a subset which appears to be both natural and powerful. We show

7 that for the resulting concept of a reasonable d-frame, free structures still exist. Some attention is given to the relationship between (reasonable) d-frames and the “biframes” of Banaschewski, Brummer¨ , and Hardie, [BBH83]. These are an alternative to d-frames but formalise the two topologies together with their join. Obviously, there is a forgetful functor from biframes to d-frames, and we show that it has a left adjoint. An interesting open problem is whether the d-frames that arise from biframes can be characterised independently. Section 6 examines two special cases of d-frames which are of particular importance to Stone duality, namely, the regular and compact regular ones. The latter can be shown to be spatial (using the Axiom of Choice) and are the duals of stably compact spaces. Many consequences of spatiality, however, can be shown directly and without reference to the Axiom of Choice.

In Section 7 we demonstrate that the structure of a (reasonable) d-frame K is

completely captured by the subset ML of consistent propositions. The axioma- tisation makes crucial use of both the information and the logical structure. In addition, the relation of strong proximity lattices is employed to capture the totality predicate. We call these structures partial frames. The equivalence with d-frames could now be summarised in the slogan that there is no difference be-

tween three-valued and four-valued logic. An important result from the transition

to partial frames is a logical interpretation of E : It means that every (infor-

7ON 7 N mation order) reﬁnement of implies (in the logical order) every reﬁnement E

of E . Sections 8 and 9 consider special cases of partial frames and their duality with bitopological spaces to locate precisely the classical dualities. To start with, the category of compact regular partial frames is equivalent to the category of strong proximity lattices, where morphisms of the latter category are adjoint pairs of consequence relations, ﬁrst considered in [JKM01]. The necessity for these emerges most naturally from the requirement to match partial frame homomorphisms. If

one assumes on top of compact regularity that the set of reﬂexive propositions

7 7 7 (that is, those for which ) is dense in the partial frame, then one obtains a category that is equivalent to bounded distributive lattices and lattice homomor-

8 phisms. If the language of (partial) propositions includes a negation operation (as, for example, considered by Belnap), then there is no distinction between positive and negative extents and one expects the two topologies on the spectrum to coincide. At the level of d-frames (or, equivalently, partial frames) the negation operation

must be speciﬁed explicitly, as the mere existence of an isomorphism between F ¤

and FH< is of little consequence. The theory then collapses to that of frames in the usual sense as we are really only dealing with one topology. Once regularity is assumed, there is at most one choice of negation on a d-frame, so here negation is a structural property rather than additional structure. Compact regular d-frames with negation then correspond to strong proximity lattices with negation, and these were shown by the second author to be the Stone duals of compact Hausdorff spaces, [Mos04]. If there is a sufficient supply of reflexive elements, finally, one obtains a Boolean algebra and Stone’s original duality with Stone spaces. This presentation of Stone dualities clarifies a number of issues, as promised at the start of this introduction, and we look at this more carefully in the concluding section.

Acknowledgements The research reported here was begun during a visit of the second author to Birmingham in the autumn of 2004. The Engineering and Phys- ical Sciences Research Council of England provided financial assistance which is gratefully acknowledged. The work was completed during the first author’s sab- batical at Chapman University in the academic year 2005/06. At Chapman, Peter Jipsen organised a weekly research gathering, of which Joanne Walters-Wayland and Melvin Henriksen were regular attendants. We are grateful to all three for the opportunity to report our on-going work to them, and for their perceptive feed- back. During a visit to Louisiana State University and Tulane University the first author had the pleasure to discuss the paper with Jimmie Lawson and Michael Mislove, which was of great help to get the central ideas into their final shape.

9 2 Review of some Stone-type dualities

For motivation and ease of reference we brieﬂy review the main components of some classical Stone-type dualities. This will allow us to point out how they ﬁt into the general picture that we are about to develop. We present them in a formulation that is most convenient for this purpose, not one that is most accurate historically.

2.1 The dualities of Stone and Priestley Stone’s original papers, [Sto36, Sto37a], were concerned with the representation of Boolean algebras as sub-algebras of powersets. Tarski and Lindenbaum had shown that a Boolean algebra is isomorphic to a powerset if and only if it is complete and atomic. In this case, the isomorphism is with the powerset of the set of atoms. Atoms occupy a special position in the Boolean algebra (just above the

element A ) but it is their property of being prime which is used in the proof: An

>PJRQTSVU >WJQ >WJXU element > is prime if implies or . Seeing an analogy with

Kummer’s ideals in the theory of rings, Stone realised that it was the properties

J Qd+ > of the set ZY\[]>_^`!a"$QV(b>c3 that were important, not those of itself, and, crucially, that there were potentially many more of the former than of the latter. The deﬁnitions don’t require the negation operation of Boolean algebras, so we ﬁrst state Stone’s representation theorem of 1937 for distributive lattices1 (pub-

lished as [Sto37b]). F Deﬁnition 2.1 A subset e of a lattice is called a ﬁlter if it is an upper set that

is closed under ﬁnite meets (including fgfh!jik4 ). A ﬁlter is called prime if it is

%pe >qSrQW%pe >p%pe inaccessible by ﬁnite joins: lk4m!Xn`n/o , and if then either

or Qs%'e . The notions ideal and prime ideal are deﬁned dually.

1 In this paper, “lattice” will mean bounded lattice: tvu£wyx8z|{8z~}}z] and “lattice morphisms”

will preserve this structure. Although Lemma 2.3 and Theorem 2.4 can be formulated without

boundedness, nothing in this paper is gained by the greater generality. We use }} and rather than

and because throughout the paper we are speciﬁcally interested in the connection with logic.

5` F

We denote the set of all prime ﬁlters of F with and the set of prime

¥ F ideals with .

Proposition 2.2 In any lattice, the complement of a prime ﬁlter is a prime ideal and vice versa.

The following is traditionally stated in terms of prime ideals and referred to as the Prime Ideal Theorem or “PIT”. However, because of the previous proposition

there is no real difference between using ideals or ﬁlters. J@Q

Lemma 2.3 In any distributive lattice, if >X3 then there is a prime ﬁlter con- Q taining > but not .

Theorem 2.4 (Stone 1937) Every distributive lattice is isomorphic to a sub- lattice of a powerset.

Proof. Consider the powerset of the set of prime ﬁlters, and map an element > of

¥ 2 >

the lattice to the set 6 of prime ﬁlters containing it. It is easily checked that

¥ ¥ ¥ ¥ ¥ ¥

>Q !X6 > 0q6 Q 6 >SQ !X6 > 6 Q

6 and . By the deﬁnition

¤ ¤ ¤ ¤ ¤ ¤

¥ ¥ ¥

nn !R4 6 fgf !5 F

of prime ﬁlters, it is clear that 6 and .

¤ ¤

This covers Boolean algebras also, as a prime ﬁlter on a Boolean algebra is the > same as an ultraﬁlter: it contains exactly one of > and . From this it follows

that the subset of prime ﬁlters associated with > is exactly the complement of those

that are associated with > , and so the negation operation is faithfully modelled by complement. Stone did not stop here but realised that it was just as important to represent the homomorphisms between the algebras. In order to single out the

right set-theoretic maps between collections of prime ﬁlters, he employed topol-

ogy. The basic opens of the Stone topology on 5 are exactly the sets

¥ ¥

> ^! "©e%5` F (©>V%'e+

6 which were used in the proof of the repre- ¤

sentation theorem. The topological space so obtained is called the spectrum of F (borrowing terminology from Ring Theory).

2 The choice of notation ¡ ¡t£¢ will become clearer later.

11 Theorem 2.5 (Stone 1936, 1937)

1. The spectra of Boolean algebras are precisely the totally disconnected compact Hausdorff spaces.

2. The spectra of distributive lattices are precisely those topological spaces that

(i) satisfy the ¡8¢ separation axiom; (ii) are compact; (iii) have a basis consisting of compact open sets;

(iv) are coherent, that is, intersections of compact open sets are compact;3 ¥¥¤§¦§ ¦©¨«ª

(v) are well-ﬁltered, that is, if the intersection of a ﬁlter base of ¤=¦ compact open sets is contained in an open set, then so is some already.

The spaces that appear in part (1) above are called Stone spaces and those in part (2) spectral spaces. The word “precisely” in the theorem refers to the translation from these spaces back to algebraic structures. In the case of Stone spaces one considers the collection of subsets that are both closed and open (“clopen”), for spectral spaces those that are both compact and open. Since a subset of a compact Hausdorff space is compact if and only if it is closed, (1) is a special case of (2). The characterisation of homomorphisms on the side of the spectrum works as follows. For Boolean algebras one considers continuous functions between the spectra in the opposite direction; for distributive lattices one uses perfect maps.4 These are continuous functions for which the inverse image of a compact open is compact. Again, any continuous function between compact Hausdorff spaces is perfect, so the Boolean algebra case is a special case. In the language of Category Theory we have the following:

3This is also called stably compact. 4There is some variability regarding terminology here with some authors insisting that the right adjective for these functions should be “proper”.

12 Bool, the category of Boolean algebras and Boolean algebra homo-

morphisms;

¥ ¨ ¨ ¨

S fgf n`n dLat, the category of bounded distributive lattices and - homomorphisms; Stone, the category of Stone spaces and continuous functions; Spec, the category of spectral spaces and perfect maps.

Theorem 2.6

1. The categories Bool and Stone are dually equivalent.

2. The categories dLat and Spec are dually equivalent.

As explained in the introduction, spectral spaces feature prominently in Scott’s theory of semantic domains, but from a mathematical point of view they carry rather a lot of conditions. Hilary Priestley recognised that there was an alternate description as certain ordered spaces.

Deﬁnition 2.7 A Priestley space is a topological space with a (topologically

closed) partial order relation J¬ which is

(i) compact Hausdorff;

J¬MQ ®

(ii) totally order disconnected, that is, for >_3 there is a clopen upper set

Q¯3%P® such that >¯%P® and .

The category Pries has Priestley spaces as objects and continuous order- preserving maps as morphisms.

Theorem 2.8 (Priestley 1970) The categories dLat and Pries are dually equivalent. Proof. Stone’s topology on the spectrum of prime ﬁlters is enriched with open

sets

¥ ¥

6=< > ^!&"©e°%5 F (b>±3%'e+

13 which clearly results in a Hausdorff topology. The order is inclusion between prime ﬁlters.

In the reverse direction one considers the collection of clopen upper sets.

¥ ¥

> 6< >

The sets 6 constitute a topological basis, and so do the sets sep- ¤ arately. So Theorem 2.8 clearly suggests a bitopological view. Indeed, compact-

ness of the Priestley topology is established in two steps. Following Stone’s idea,

> ³ one ﬁrst shows that each set 6§² is compact with respect to the opposite ( ) topology. Then one invokes the Alexander Sub-base Lemma to see that the join of the two topologies is compact. So the “hard” part of the proof is located at the

transition from a bitopological setting to a topological one. F

Instead of “prime ﬁlter of F ” we could have said “homomorphism from fgf to ´ ^`!" nn¯µXfgf¶+ ” because prime ﬁlters are exactly the inverse images of under

such homomorphisms. Likewise, instead of “clopen subset of the Stone space ”

¥ ¨ ¨ ¹q

· ^! "$¸ Qd+

we could have said “continuous function from to , where ¹

is the discrete topology. For spectral spaces one uses Sierpinski space: º*^!

¥ ¨ ¨¶»5 »

Q+ "$Q+ "$¸ where is the only nontrivial open of , and for Priestley spaces one

uses the discrete topology together with the order ¸WµpQ .

· º The fact that ´ , and have the same number of elements is not a coincidence. It follows from the fact that all five categories introduced so far are concrete over Set, with the forgetful functor having a left adjoint. In such a situation, the underlying set of a dual object is always given by the set of morphisms into a dualising object, and the dualising objects of the two categories involved have “the same” underlying set. See [Joh82, SectionVI.4.1] for a concise discussion of this phenomenon. Let us also briefly review the logical reading of Stone duality. Every Boolean algebra is the Lindenbaum algebra of a propositional theory. A prime filter on the algebra corresponds to a model of the theory. The Prime Ideal Theorem (our Lemma 2.3) then states the completeness of propositional logic. The duality of bounded distributive lattices likewise corresponds to propositional logic without negation, or positive logic.

To every proposition > one can associate the binary partition of the spectrum

¥ >

into those models in which the proposition is true (its positive extent 6 ) and

¥ > those where it is false (the negative extent 6< ). Stone duality equips the spectrum with a topology in which both extents are compact and open. With the Haus- dorff assumption, this yields clopen extents for the Boolean algebra case. Priestley duality renders the positive extent a compact open (hence clopen) upper set, the negative extent a compact open (hence clopen) lower set.

2.2 Proximity lattices and (stably) compact spaces Priestley duality associates a compact ordered space with a bounded distributive

lattice, whereas Stone duality provides a certain ¡¡¢ -space. The two are closely linked as the underlying set in both cases is provided by the collection of prime ﬁlters. The connection between a Priestley space and the corresponding spectral space is, in fact, purely topological and holds more generally. We recall the main points of this.

Deﬁnition 2.9 A compact ordered space is a set with a compact topology and

an order relation J¬ that is closed in the product topology.

Compact ordered spaces were studied by Leopoldo Nachbin in a systematic

fashion in [Nac65]. He established the following central property for these spaces. J¬¼Q

Lemma 2.10 (Separation Lemma) For >3 in a compact ordered space there

>½%&® Qp%R is an open upper set ® and an open lower set such that , , and

®p02 &!4 .

One deﬁnes the upper topology as the collection of all open upper sets and ¤

likewise the lower topology ¾< .

Proposition 2.11 The topology of a compact ordered space is the join of the upper S2$<

and lower topologies: q!½ .

^`!½Y£ >¯%¯

Proof. For °%V , the complement is compact. Consider ; for

>13J¬ Q >13 ¬ Q any Q¿% , either or , so the Separation Lemma can be employed. The rest is standard.

15 From the Separation Lemma it also follows that J¬ is the specialisation order of the upper topology (and the reverse of the specialisation order of the lower topology). A set that is closed in the lower topology is an upper set and therefore saturated in the upper topology, that is, it is the intersection of upper open sets.

Furthermore, it is compact in the upper topology by the overall compactness of . The converse is again an easy consequence of separation. Thus:

Proposition 2.12 The sets that are upper and closed in a compact ordered space are precisely the compact saturated sets with respect to the upper topology.

In preparation for our general development later, we note that the bitopological

¥ ¨ ©<

space is compact and regular. Here is the deﬁnition:

¥ ¨ $<

Deﬁnition 2.13 A bitopological space is called compact if every

©<

cover of with elements from has a ﬁnite sub-cover.

¨ N N

® ® ®

Let ® be two elements of . We say that is well-inside (and write

NÁ N

® °%¿©< ® 02 &!4 ® &!½ ® ) if there is such that and .

A bitopological space is called regular if every open of is the union of those

¤ ]<

-opens well-inside it, and the analogous statement holds for the opens of . ¤

By Alexander’s Sub-base Lemma, a bitopological space is compact if and only if it is compact in the join of the two topologies. However, since most of our results do not depend on the Axiom of Choice, it makes sense to have a separate

notion that does not require the join topology.

¥ ¥ ¨

Proposition 2.14 If ~:ÂJ¬ is a compact ordered space then is ¤ compact and regular.

We will maintain that the bitopological point of view is the best one to understand Stone duality, but to complete our brief presentation of the link with Priestley duality we observe that Proposition 2.12 suggests that the second topology is actually redundant. Indeed, the topologies that arise as upper topologies of compact ordered spaces can be characterised independently.

16 Deﬁnition 2.15 A topological space is called stably compact5 if it is

(i) ¡O¢ ;

(ii) compact;

(iii) locally compact;

(iv) coherent, that is, intersections of compact saturated sets are compact; ¥¥¤§¦9 Ã¦©¨«ª

(v) well-ﬁltered, that is, if the intersection of a ﬁlter base of compact ¤=¦ saturated sets is contained in an open set, then so is some already.

Clearly, every spectral space, as characterised in Theorem 2.5, is stably compact, but the latter class is much bigger since zero-dimensionality is not required; for example, all compact Hausdorff spaces are included.

The following is an immediate consequence of the deﬁnitions.

Theorem 2.16 A topological space is stably compact if and only if the

¥ ¨

«Ä§Ä ÂÄ§Ä bitopological space c is compact and regular, where is the topology

whose closed sets are (generated by) the compact saturated sets with respect to .

Theorem 2.17 A topological space is stably compact if and only if it is the upper topology of a compact ordered space. Proof. We have done most of the work for the “if” direction explicitly. For the converse one considers the co-compact topology whose closed sets are deﬁned as the compact saturated sets of a given topology. It is easy to see that an ordered Hausdorff space is obtained this way, but to establish compactness one needs to call upon the Axiom of Choice in the form of Alexander’s Sub-base Lemma.

For more detail we refer to [AMJK04] and [GHK ¤ 03, Section VI-6]. For our present purposes we can say that the theorem establishes the topological link between the dualities of Stone and Priestley for bounded distributive lattices. The

5These spaces were called coherent in [JS96].

17 dualities themselves can be generalised to the setting of the theorem by considering a reﬁnement of bounded distributive lattices. Following [JS96], we outline the main results.

Deﬁnition 2.18 A strong proximity lattice is a structure Å consisting of a dis-

¥ ¨ ¨ ¨

S fgf nn tributive lattice together with a transitive binary relation such

that

> fgf

( – fgf )

n`n >

(nn – )

¨ N ÆÇ N

> Q > Q > Q¿Q

( – )

¨ N ÆÇ N

> Q > Q >S2> Q

( S – )

Ç É N N N

È2> Q ! È %VVÊÈ È È Ë> Q

( – ) and

Ç É N N N

> QS¿È ! È %VVÊÈ È > QSVÈ

( – S ) and

By letting >c!-fgf in ( – ) one sees that the relation is interpolative, that

Q U > U Q is, > implies that there is such that . Strong proximity lattices

generalise distributive lattices in that the lattice order J on a distributive lattice satisﬁes the deﬁnition. We will consider morphisms on strong proximity lattices in Section 8.1.

Deﬁnition 2.19 An upper subset # of a strong proximity lattice is called round if

N N

> %V# > > for every >¯%¯# there is with . A round lower set is deﬁned dually.

The carrier of the spectrum of a strong proximity lattice consists of all round

Å prime ﬁlters, the collection of which we denote by Ì¥5` . The following are

the analogues to Proposition 2.2 and Lemma 2.3: Å

Proposition 2.20 If e is a round prime ﬁlter on strong proximity lattice , then

¥ É N N

^`!Ï"b>¯%V ( > %VÐY=eÑÊ> > + BYÎe is a round prime ideal, and vice versa. Furthermore, the translations are inverses of each other.

6 Ò The qualiﬁer “strong” distinguishes the concept from its precursor in [Smy92a], where ( x – ) was not a requirement.

Lemma 2.21 In a strong proximity lattice, if e is a round ﬁlter not intersecting

e Ó

an ideal Ó then can be extended to a round prime ﬁlter still disjoint from .

> !

We topologise the spectrum as in Stone duality, that is, taking the sets 6

Å (b>¯%'e+ "©eÔ%PÌ|5` as the basic opens.

Theorem 2.22 The spectrum of a strong proximity lattice is a stably compact space, and every stably compact space arises in this way.

Proof. The details of the ﬁrst half are in [JS96]. For the second statement one

® Õ associates with a stably compact space all pairs ®&½Õ where is open and is

compact saturated. The lattice operations on these pairs are deﬁned component-

¥ ¨ ¥ N9¨ Ng N

Õ ® Õ ÕÖX® wise and the proximity is given by ® iff . While this result establishes an interesting link between a wide class of topological spaces and certain algebraic structures, it is not clear what the represen-

tation problem is that is solved by it. Sure enough, it is still true that the map

Ø 6 >

>Ë× preserves the bounded lattice structure, but how is modelled by it? ¤ And do we have a prior idea what it should be modelled by? The paper [JS96] did not address the second question, as the emphasis was on an algebraic (or logical) description of certain given spaces. The problem is also apparent when we look at morphisms. In [JS96] a duality is established between continuous functions on the side of stably compact spaces and certain approximable relations between strong proximity lattices. This choice was motivated by applications in Domain Theory. In later work, [JKM99], we introduced continuous consequence relations which correspond to certain continuous relations between the spaces; this was motivated by the logical reading of the duality. However, neither choice generalises Stone duality for bounded distributive lattices. Another problem with Theorem 2.22 is that the functors are not given by the set of morphisms into a dualising object. Indeed, the two types of morphism for strong proximity lattices mentioned above are not even functions so there is no forgetful functor into Set. Following the general methodology laid out in [Joh82, SectionVI.4.1], we should ﬁnd a dualising object by constructing the free stably

19 compact space over the one-point set (which is the one-point topological space)

and dualise this. The construction outlined in the proof of Theorem 2.22 yields

n`n > fgf >

a three-element proximity lattice n`nXµ-CÙµÚfgf with for any but

3 C C . If we were dealing with a concrete duality, then the dualising object among stably compact spaces would also have three elements but there is no such space that yields the right proximity lattice. Our answer to the riddles above is to suggest that the duality of stably compact spaces and strong proximity lattices is a special case of a general duality between certain algebraic structures, which we will introduce below, and bitopological spaces. Given the information above, the reader can perhaps guess the

general outline of how this might go. For example, in Stone duality one can look

¥ >

at the bitopological space where one topology has the collection of 6 as its

¥ > basis, and the other is generated by the 6< . Stone’s perfect maps between spectral spaces are then nothing else but bicontinuous maps.

The duality of Theorem 2.22 is also bitopological in nature; instead of thinking

¥ ¨ Õ of the tokens ® as consisting of an open and a compact saturated set, we

should view them as consisting of an open and a co-compact open ZYTÕ . The correct morphisms on the spatial side should again be bicontinuous maps and whatever we choose as morphism between strong proximity lattices should mirror these.

Finally, recall the logical reading of Stone duality as brieﬂy outlined at the end

¥ >

of Section 2.1. For strong proximity lattices we see that the positive extent 6

6< > of a lattice element > is disjoint from the negative extent but it is not necessarily the set-theoretic complement. Lattice elements can thus be seen as partial predicates which are true in some models, false in others, and whose status is unknown (or undecidable in ﬁnite time) in the remaining cases. A chief aim of the present paper is to convince the reader that this is indeed a fruitful point of view.

2.3 Frames The Stone dualities reviewed so far lead to rather special topological spaces. If one is interested in a duality that applies to all spaces then frame theory is the

20 answer. Although there is an excellent text available on the subject, [Joh82], we review its main components as we will make constant use of frame-theoretic ideas throughout the paper.

Deﬁnition 2.23 A frame is a complete lattice in which ﬁnite meets distribute over

Û Ü A Ý arbitrary joins. We denote with I , , , , and the order, ﬁnite meets, arbitrary joins, least and largest element, respectively.7 A frame homomorphism preserves ﬁnite meets and arbitrary joins; thus we

have the category Frm.

¥ ¥ ¥ ¥ N Nv

8Â Þ^ Ø ~

For a topological space, is a frame; for

<àß

^8 Ø

a continuous function, Þ is a frame homomorphism. These are the Ø

constituents of the contravariant functor áË^ Top Frm. It is represented by

¥T¨ º Top º where is Sierpinski space – the same space that arose in Theo-

rem 2.5(2). In this representation, an “open” is identiﬁed with a continuous map º

from to . The frame operations on such maps are deﬁned point-wise.

È È

The collection â of open neighbourhoods of a point in a topological

¥ áË

space forms a completely prime ﬁlter in the frame , that is, it is an

È äP0

upper set, closed under ﬁnite intersections, and whenever ãRä&%±â then

È !43

â . This leads one to consider the set of points (sometimes called “abstract F points” for emphasis) of a frame F to be the collection spec of completely prime

ﬁlters. Abstract points are exactly the pre-images of "Ý+ under homomorphisms å!Ï"©AæµRÝ+

from F to . So an alternative representation takes a point to be a å

homomorphism from F to .

F 6 > !

A frame F induces a topology on spec whose opens are of the form

FR(b>V%'e+ >V%'F F

"©eÔ% spec with . In the alternative representation, spec takes

çR×Ø ç >

the weakest topology such that for each >%ÔF , the evaluation map

º èÑ^:FØ F

is continuous as a map from spec F to . A frame homomorphism

F Ø F è e ^`!

induces a continuous function spec è§^ spec spec by letting spec

¥ N

<àß

e eÏ% F

è for spec . These are the components of the contravariant func-

¥T¨ tor spec from Frm to Top, represented by Frm å . 7This lattice notation is different from that chosen for distributive lattices. The reason for this will become clear as our theory unfolds.

21 Theorem 2.24 The functors á and spec constitute a dual adjunction between

Top and Frm. 6

The unit and co-unit of this adjunction are simply â and . That is, for any

¥ ¥

éêq^O Ø áË ÈW×Ø â È

space the map spec , given by , is continuous;

ë¶ìí^8FkØ á F

it is also open onto its image. Likewise, for any frame F the spec ,

Ø 6 > given by >Ë× is a frame homomorphism; it is also surjective. A brief comparison of the dual adjunction between frames and spaces and

Stone’s original theorems is instructive. In Stone’s duality for distributive lattices

Ø 6 >

and spectral spaces, the co-unit, > × , is clearly a surjective distributive ¤ lattice homomorphism onto the compact opens of the spectrum. The prime ideal

theorem is used to show that if >±î&Q holds in a distributive lattice, then there is

¥ >

a point in the spectrum (that is, a prime ﬁlter) showing that 6 is not a subset

¥ Q

of 6 . This establishes that the co-unit is injective. Armed with this, one then ¤ sees that the co-unit is in fact an isomorphism. The axioms for spectral spaces are engineered to ensure that (a) compact opens form a distributive lattice (a minimal requirement) (b) there are enough compact opens to distinguish points, so the unit is also injective and (c) there are enough points so that the unit is also surjective. For frames, the prime ideal theorem can not help us establish that the co-unit is an isomorphism. The abstract points of a frame are completely prime filters, not merely prime filters. But the complement of a completely prime filter is a

principle prime ideal, so the prime ideal theorem is powerless to ﬁnd an abstract

¥ ¥

> 6 Q >/3I@Q point that would separate 6 from for . So in general, the co-unit

of the áÙï spec adjunction may not be an isomorphism. Similarly, we have not assumed any separation on spaces that would ensure that the unit is injective

(although this is easily remedied as injectivity is precisely the ¡¡¢ axiom) nor that

spaces have enough points to ensure that the unit is surjective. áË

We can ask when a frame F is spatial in the sense that it is isomorphic to

F ð áË

for some space . The adjunction transfers isomorphisms: ! if and only

F F ë ì

if ! spec . So is spatial if and only if is a frame isomorphism. Because ë¶ì ëì is already a surjective frame homomorphism, this holds if and only if is injective. Examples of non-spatial frames are found in [Joh82].

Similarly, we can ask when a space is sober in the sense that it is homeo- F

morphic to spec F for some frame . By the same reasoning as in frames, this éê holds if and only if é¾ê is a homeomorphism. Because is already continuous

and open onto its image, it sufﬁces for éê to be a bijection. As mentioned above,

injectivity is precisely the ¡:¢ axiom. Surjectivity says that every completely prime ﬁlter of opens is the neighbourhood ﬁlter of a point.

Theorem 2.25 The functors á and spec restrict to a dual equivalence between sober spaces and spatial frames.

A key property of Frm is the fact that free frames exist; the construction for the free frame over a set of generators is described in [Joh82, Section II.1.2], and for a presentation with generators and relations in [Joh82, Section II.2.11]. Existence of free frames, but not the details of construction, will be the basis for various free constructions for d-frames in what follows. A frame can alternatively be regarded as a “hybrid” structure: a distributive lattice and directed complete partial order in which (a) the lattice operations are Scott continuous (preserve directed suprema) and (b) the lattice order and the directed complete order coincide. Then a frame homomorphism is a Scott continuous lattice homomorphism. So frames are special objects in the category of dcpo distributive lattices. On this view, a frame “thinks about its data” in two ways: ﬁrst, as propositions of a positive propositional logic where ﬁnitary meets and joins make sense; second, as data in an information order where accumulation of directed joins makes sense. The following two important concepts in frames

highlight this distinction.

>%/F Qk%/F

On any frame F , say that is well-inside provided there exists

>ËÛVñò!òA QæóVñ@!-Ý > Q some ñÐ%kF so that and . We write this as . Notice

that this relation is meaningful in any lattice; it says nothing about directed joins.

Q >kIôQ Q

On the other hand, with distributivity > implies . Also, for any the

Á Á NõÁ NõÁ

Q > Q > Q >mó¿> Q set of all > is directed because and implies . So far,

this depends only on the fact that F is a distributive lattice. On the other hand,

in a frame the join of elements well-inside Q exists due to directed completeness,

and the join is always below or equal to Q . A frame is called regular if the join Q of elements well-inside Q is always equal to . So one can take regularity to be a

condition on the interaction of logic and information within a frame.

> Q ö QcI ö

Say that is way below provided that for any directed set , if Üh÷

øs%ö >¿ù Q then >WIø for some . We write this as . This relation is meaningful

in any dcpo, as it says nothing about finite meets and finite joins. On the other >mùÏQ hand, because a frame has finite joins, the set of all > such that is directed.

A frame is called continuous if the join of elements way below Q is always equal

to Q . Again, continuity is a condition on the interaction of logic and information. Topologically, well-inside and way-below have very different intuitions. In

á , an open set is well-inside another provided that the closure of ﬁrst is con-

tained in the second. This splits the deﬁnition of clopen in the sense that ® is

Á ®

clopen if and only if ® . On the other hand, the way-below relation splits the

ú ú

deﬁnition of compactness: ®½ù provided every open covering of contains a

® ®ù ® ﬁnite sub-covering of ® . So is compact in the usual sense if and only if .

Because a frame is a certain kind of dcpo, the Scott topology on a frame plays ® an important role. A subset ® of a frame is Scott open if and only if is an upper

set and is inaccessible by directed joins: if ö is a directed subset of the frame,

%® öô0_® !Z43 then Üh÷;ö implies . For example, a completely prime ﬁlter is automatically Scott open. One important application of the Scott topology on frames is the Hofmann- Mislove Theorem. Because we will encounter bitopological versions of the theorem in Sections 5 and 6, we state it here for reference, along with its frame-

theoretic version.

¥ ¨ Theorem 2.26 [HM81, KP94] In a sober space , there is a bijection be-

tween the set of compact saturated subsets of and the set of Scott open ﬁlters

in .

Theorem 2.27 [GHK ¤ 03, Corollary V-5.4] A Scott-open ﬁlter in a frame is equal to the intersection of the collection of completely prime ﬁlters containing it. More- over, this collection is compact in the spectrum of the frame.

24 In later sections, we will consider structures in which a bounded distributive lattice and a dcpo interact but do not coincide in their orders. Indeed, we shall see that the relationship between frames and the duality theorems of Sections 2.1 and 2.2 hinges on the interaction of logical structure and information structure, and that bitopology allows us to make the needed distinction between the two.

3 Bitopological spaces and d-frames

We now begin to lay out the general framework within which the dualities reviewed above can all be seen as special cases. As mentioned several times, the correct setting for this is bitopological spaces. So we ﬁrst must establish a bitopological analogue of frames.

3.1 Stone duality for bitopological spaces ©<

Our notation for the two topologies of a bitopological space is and , which is ¤

meant to suggest that the opens of are the positive extents of predicates, that is, ¤ those models where a certain proposition is (perhaps observably) true. Likewise,

an open from ©< is the negative extent of a predicate, that is those models where

¥ ¨

®û< %

a certain proposition is false. With this understanding, pairs of opens ®

are the denotations of predicates; ® is the set of models in which the

¤HG ¤

predicate is true, ®ü< the set in which the predicate is false.

Classically, a predicate takes a deﬁnite truth value in every model. Here we

% ® 0c®û<

allow a predicate to be undeﬁned in a particular model ( >&o ), or to be

¤ 0r®û<

over-deﬁned ( >'%P® ).

¤ ®ü<

If ® and are disjoint, the predicate is consistent. That is, the predicate

®û<2!*

can not be both true and false in the same model. Similarly, if ® , then ¤ the predicate is total. It must be either true or false in any model, but of course it

may not be consistently so. ©<

Note that in general no relationship between and is assumed, nor is any ¤ separation axiom for either topology.

A bicontinuous map between bitopological spaces and is a function from

N to which is continuous with respect to both topologies separately. Thus we obtain the category biTop.

For the “Stone duals” of a bitopological spaces, we start with pairs of

¥ ¨ FH<

frames F , and pairs of frame maps. That is, we start in the category ¤

Frm Frm. This is not enough, however, as it makes no provision for linking G

the two frames; they are, after all, intended to describe the same set of points. For

¨ F <

this reason we add two relations con tot aF with the intended meaning

¤HG

Q5ÿH% > Q

that ýþ> con indicates that the “opens” and do not intersect. The intended

Q5ÿ% > Q meaning of ýþ> tot is that and cover the whole space. We will also refer

to these as the “disjointness” and “covering” relations.

¥ ¨ ¨

F¡ b As a preliminary (unofﬁcial) deﬁnition, call a structure F§ß con tot a d- frame. Certainly, “biframe” would have been more appropriate but unfortunately

8 N K

that terminology is already taken. Morphisms between d-frames K and are

N N ¨

è:ß ^F ßûØ F è¢ ^F¡ ÑØ F ýþ> Q5ÿ %

pairs of frame homomorphisms and such that

¥ ¥ ;¨ ¥ ~ N

> è£ Q % con implies è8ß con and similarly for the tot relation.

As an aside, the biframes of Banaschewski, Brummer¨ , and Hardie, [BBH83]

FHß F¤ F ß F¤ Fû¢ consist of three frames Fü¢ , and , where and are sub-frames of

and together form a generating set for it. Clearly, this is a frame-theoretic version

¥ ¨

$ß§S_¥ ¾ß ¦ of the common reﬁnement Â¢k^! for a bitopological space

together with the two given topologies. In biframes, the relation between the

F¡ elements of FHß and is made fully explicit by virtue of being included in the

common frame Fû¢ . D-frames, on the other hand, only encode when two “opens” are disjoint and/or covering (consistent and/or total). We will study the connection

between these two approaches in greater detail later.

F¡

Because con and tot are subsets of the product F=ß , we visualise a d-frame G concretely as the product (see the right column of Figure 1 for examples). This is much more than a heuristic aid, though, as the structure of the product frame itself will play an important role in the theory. This leads us to our “ofﬁcial” deﬁnition.

8In fact, “d-” is the next best thing, as you can take it to abbreviate the Proto-Indo-European word “dvo,ˆ ” meaning “two.”

¥ ¨ ¥ ¨ ¥ ¨

F¤ Ý A A Ý Ý A Û

If F ß and are frames, then and are complements –

¥ ¨ ¥ ¨ ¥ ¨ ¥ ¨ ¥ ¨

Ý ! A A Ý A ó A Ý ! Ý Ý FHß F¡

A and – in the product , and the pairs of

¥ ¨¨§5 N N

Þ FÑß F F¤ F

frame homomorphisms from to ß and from to are bijective with

N N ¥ ¨ ¥ ¨

FÑß F¡ F F Ý A A Ý

the frame homomorphisms from to ß that preserve and .

G G

¥ ¨ ¨ ¨

Ü Û C D fgf

Conversely, consider a frame FÑ with designated constants and

fgfTÛWn`nÚ! C fgfóWn`nÚ! D F

n`n that are complementary: and . Then is iso-

¨ ¨

C fgf ©C n`n

morphic to the product of the two intervals © and (both of which

¨ ¨

are frames themselves). The isomorphism from F to is given by

¥ ¨ ¨

×Ø Ûrfgf Ûhn`n Qÿ×Ø >VóPQ

. In the reverse direction, ýþ> . It is also eas-

N N

fgf n`n

ily shown that if F has designated complements and , then the frame ho-

N F

momorphisms è§^8F&Ø that preserve the constants are bijective with pairs of

¨ ¨ N § ¨ ¨

Frm Frm is equivalent to the category of frames equipped with complementary G

pairs.

¥ ¨ ¨

n`n£ F

Thus we take a d-frame ofﬁcially to be a structure FÑÂfgf con tot where

n`n *F is a frame, fgf and are complements, and con tot . The foregoing discussion shows that this is equivalent (categorically) to the “unofﬁcial” version, but has

the advantage of constituting a concrete category. A d-frame homomorphism is a n`n frame homomorphism that preserves fgf , , con and tot. We denote the category

of d-frames by dFrm.

¥ ¨ ¨ nn

Because a d-frame F§«fgf con tot is determined up to isomorphism by

¨ ¨

C fgf © C n`n the two frames © and together with con and tot, and because we of-

ten ﬁnd it easier to consider the two frames separately, we continue on occa-

¥ ¨ ¨

F¡ b

sion to use the “unofﬁcial” notation FÑß con tot to abbreviate the d-frame

¥ ¥ ¨ ;¨b¥ ¨ ¨

F¡ b Ý A A Ý

FHß con tot . This has the advantage of being more obviously

¥ ¨

A A

motivated by the two topologies on a bitopological space. Note that C-^`!

¥ ¨

DÔ^`! Ý and Ý are the least and greatest elements of the product d-frame.

Since the underlying frame of a d-frame K as isomorphic to the concrete prod-

¨ ¨

C fgf © C n`n

uct of frames, © , we will ﬁnd that explicit notation for the isomor-

¨ ¨

^`!© C fgf F <¯^!©C n`n

phism is needed. Speciﬁcally, let F and . The projections

F F < ×Ø ÛËfgf ×Ø Ûn`n

from F to and are given by and . The isomorphism ¤

F < F ýþ> Q5ÿ,×Ø >¿óWQ

from F to is given by . We denote these operations as

¤HG

%¯F Qs%'F <

follows, for , >V%'F and :

^`! Û'fgf

< ^`! ÛÎn`n

ýþ> Q5ÿ)^`! >óËQ

Ø K

Clearly, a d-frame homomorphism èÑ^¡K is determined by its operation

F < è èO<

on F and . These restrictions, denoted by and , are separately frame

¤ ¤

N N

F F

homomorphisms into and < . Together they preserve con and tot. That is, if

¨ ¥ ¶¨ ¥

Qÿ % ýyè > èO< Q ÿH%

ýþ> con, then con, and similarly for tot. ¤

The (contravariant) functor á from bitopological spaces to d-frames associates

¥ ¨ ¥ ¨ ¨ ¥ ¨

©< $<£ ® ú %

a space c with the d-frame con tot where con if and

¤ ¤

¥ ¨

! 4 ® ú % ® úÙ!Z only if ®0±ú and tot if and only if . The functor associates with a bicontinuous function the map determined by the two inverse image maps. A trivial bit of set theory will convince the reader that the disjointness

and covering predicates are preserved. Figure 1 shows some small examples. The º

bitopological space ºûÊ , which looks like a product of two copies of Sierpinski

9 ¥Ã ºíÊº

space , allows us to represent the functor á as biTop . Note how the four º

elements of ºüÊ correspond to the four ways in which an element of the space can ©<

be related to an open from and an open from : it can be in one of the two but ¤ not the other, it can be in both, or it can be in neither.

For a functor in the reverse direction, we continue to follow the theory of

¥ ¨ ¨

! F§«fgf n`n£ å Êå frames by considering d-frame morphisms from K con tot to ,

depicted in the upper right corner of Figure 1. Such morphisms are determined

^8F Ø å ç8<=^8F <_Ø å

by pairs of frame homomorphisms ç and that together

¤ ¤

preserve con and tot. So they correspond to pairs of completely prime ﬁlters

F eM< F <

e , such that

¤ ¤

! %'e3

(dpcon) % con or

¤ ¤

! %re <¿%'eM<õÊ

(dptot) % tot or

¤ ¤

9 We will make clear below in which sense can be seen as a product.

2.2:

: 3.3:

Figure 1: Some bitopological spaces and their concrete d-frames. (D-frame elements in the con-predicate are indicated by an additional circle, those in the tot-predicate are ﬁlled in.)

¥! ¨"!

e eû<

On F itself, a point manifests itself as a pair of completely prime ¤

ﬁlters on F that satisfy the analogue of (dpcon) and (dptot), plus

(dp ) fgfÑ%

¤ ¤

n`n¯% eM<£

(dp < )

¥! ¨#! eM<

Figure 2 illustrates the idea that e determines four “quadrants” so that ¤ con does not intersect with the “upper quadrant” and tot does not intersect with the ‘lower.”

e%$

e ¤

tot

fgf n`n

con

eû< ¤ 0

Figure 2: An abstract point in a d-frame.

In ordinary frame theory every point e is alternatively determined by the ele- Û

ment Ü1FæYe ; the elements that arise in this fashion are exactly the -prime ones.

2%cF &5¤±^!'&Tóæn`n

Translated to the setting of d-frames, we get elements & where

^!(&ó±fgf Û F ) &

and & are -prime in , no element below is in tot, and any el-

7 7 7

*& < I+&5<

ement in con satisﬁes either I or . However, this is not very

¤ ¤ helpful in actually ﬁnding points in a d-frame. A more “constructive” analysis of the situation is possible if we add further axioms to the deﬁnition of a d-frame; this will be presented in Section 5 below.

The set of d-points becomes a bitopological space by considering the collec-

¥ ¥ ¨

> ^`!" e eM< (b>¯%'e + >'%rF ,

tion of 6 , , as the ﬁrst topology , and the

¤ ¤ ¤ ¤ ¤

¥ ¥ ¨

Q ^!" e eM< (©Qs%'eM<¡+ Qs%'F < ,í<

collection of 6\< , , as the second topology .

¤ K Together, this is the spectrum of the d-frame K , which we denote as spec , following the usual notation for frames. The construction for objects is extended to a

30 Ø (contravariant) functor spec ^ dFrm biTop in the usual way, that is, by noting that the inverse image of a point under a d-frame morphism is again a point.

Theorem 3.1 The functors á and spec establish a dual adjunction between biTop and dFrm.

Our two categories are clearly concrete over Set. For bitopological spaces the left adjoint to the forgetful functor equips a set with two copies of the discrete topology. The free concrete d-frame over a given set is the product of the usual

free frame with itself. The two predicates are chosen minimally, that is, as 4 . It Ý follows that the dualising object in dFrm is given as á2e biTop and that of biTop

Ý å Êå ºíÊº

as spec e dFrm . Indeed, we obtain and this way.

º ¡¡¢ We note that the individual topologies on ºûÊ are not even , only their join distinguishes all four elements. This explains why there is no schizophrenic object

for the duality of strong proximity lattices and stably compact spaces, as ºûÊº is far from being stably compact. This also gives a clue as to why biframes are not

suitable as the Stone duals of bitopological spaces. If a schizophrenic object were Ý

to exist, on the biframe side it would be the biframe derived from e biTop . But

¥ ¨ ¨

å å this is å . In other words, the schizophrenic object would have to be a two element bitopological space. None of the ten non-bihomeomorphic candidates yields the required representable functor.

3.2 Logical order on a d-frame

¥ ¨ ¨

F¤ b

As mentioned above, in a d-frame FÑß con tot given by two frames sepa- F <

rately, the structure of the product F plays an important role in our devel- ¤HG opment. Recall that one can think of a frame as a logical structure (ﬁnite meets and joins) with an information structure (directed joins) where the two orders coincide. In a d-frame, a second distributive lattice that is “at 90 degrees” to the frame order also exists. This structure is a completely general phenomenon, known at least since [BK47]. Its proof is straightforward and can safely be left as an exercise.

10 - The free d-frame over the one-element set looks like -. in Figure 1, except that no elements should be marked as belonging to con and tot. The generator is the element in the middle.

¥ ¨ ¨ ¨ ¥0/¶¨

Û ó Ý A Þ

Proposition 3.2 Let FÑ be a bounded distributive lattice, and a

/ /

Û2Þ2!RA ó¿Þm!Ý

complemented pair in F , that is, and . Then by deﬁning

¥ ¥ ¥ ¥ ¥ ¥

>æ2Q ^! >,Û¿Þ ó QÛ¿Þ ó >ÛsQ ! >,ó¿Þ Û QTó¿Þ Û >,óËQ

¥ /~ ¥ /~ ¥ ¥ /~ ¥ /~ ¥

>æS2Q ^! >,ó Û Q ó Û >,óËQ ! >Û ó Q Û ó >,ÛËQ

¥ ¨ ¨¨/¶¨ ¥ ¨

S Þ Ý A one obtains another bounded distributive lattice FÑ¶ , in which is

a complemented pair. The original operations are recovered from it as

¥ ¥ ¥ ¥ ¥ ¥

>,ÛËQ ! >ÎVA S Q2A S >¿Q ! >S¿A QS¿A >ÎS¿Q

¥ ¥ ¥ ¥ ¥ ¥

>,óËQ ! >ÎSPÝ QSrÝ >S¿Q ! >WÝ S QWÝ S >Î¿Q

ó S

Furthermore, any two of the operations Û , , , and distribute over each other.

If F is a frame, then and are also Scott continuous. n`n

This justiﬁes our choice of symbols fgf and in a d-frame, and suggests that we

¥ ¨ ¨ ¨

S fgf n`n regard FÑ¶ as the logical structure of a d-frame. The logical structure makes a d-frame into a distributive “bilattice.” See [Gin92, Fit91, MPS00] for introductions to bilattices. Bilattices are motivated by Belnap’s four-valued logic [Bel77].

We can easily compute conjunction and disjunction in terms of elements of FH<

F and :

¨ N ¨ N N9¨ N

ýþ> Q5ÿ¡ýþ> Q ÿ)^`! ýþ>Ûq> QTóËQ ÿ

N N N N

¨ ¨ ¨

ýþ> Q5ÿ¡Sýþ> Q ÿ)^`! ýþ>óq> QTÛËQ ÿ Note the reversal of order in the second component. This makes sense, as we think

of the second frame as providing negative answers. ó The translation from the logical operations back to Û and means that we could have factored our definition of d-frames quite differently into a logical structure (a distributive lattice) and an information structure (a dcpo, not explicitly a frame) together with con and tot, and the needed axioms to make our definition re- coverable. This approach would clearly emphasise our point that the information order is actually separate from logic, and that frames conflate the two. Separate treatment of the two orders is the primary motivation of investigations into bilattices. That research program, however, has not taken directed completeness of information into account.

32 4 Sobriety of bitopological spaces

Following the terminology of frames, we say that a bitopological space is (d-) K sober if it is bihomeomorphic to spec K for some d-frame . As with frames and topological spaces, d-sobriety has an internal characterisation.

Theorem 4.1 For a bitopological space , the following are equivalent:

1. is d-sober; áË

2. is bihomeomorphic to spec .

¥ ¥ ;¨ ¥ ~

Ø ám > ×Ø â > âæ< >

3. The unit map éí^O spec given by is a bi- ¤ homeomorphism.12

4. The unit is a bijection.

Proof. Clearly, (3) implies (2) and (2) implies (1). Furthermore, it is clear that

¥ ¨

©< é

for any bitopological space the map is bicontinuous and bi-open ¤ onto the image. So it is a bihomeomorphism if and only if it is a bijection. Thus

(4) and (3) are equivalent. K

For (1) implies (4), assume is bihomeomorphic to spec . We prove that

é L

for spec K , the unit spec is a one-to-one correspondence between points on the

á K é

given d-frame K and points on the second dual spec . Then by naturality of éê

and the sobriety of , is also one-to-one.

¥ ¨

eM< K é

Let e be a point of ; we calculate its image under according to the ¤

deﬁnitions:

¥21 ¨31 ¥ ¨

< ! é e eM<

¤ ¤

¥ ¥ ¨ ¶¨ ¥ ¨

! â e eû< â,< e eû<

¤ ¤ ¤

¥ ¥ ¨ ¨ ¥ ¨

! "© %4, ( e eû< %¯ + "© <¿%4,û

¤ ¤ ¤ ¤ ¤

¥ ¥ ¨ ¥

! "¾6 > %4, (b>¯%'e + "¾6=< > %4,ûV%'eû<£+

¤ ¤ ¤

11We will usually leave out the qualiﬁer “d-” when it is clear that we are talking about a bitopological space. 12 “ 5 ” indicates the open neighbourhood ﬁlter.

33 K Injectivity of é is clear as different points on will give rise to different sets of

open sets in at least one of the two canonical topologies on spec K . For surjectiv-

¥61 ¨71

á K

ity, we assume that < is a point of spec . We claim that

¥ 1 ¥ 1

e ^! "b>¯%'F (¾6 > % + eû< ^`! "$Qq%'F <(¾6=< Q %

¤ ¤ ¤ ¤

¥ ¨ ¥21 ¨31

é e eû< ! <

deﬁnes a point of K such that . Let’s check the details:

¤ ¤

eû< 6 6=<

Both e and are completely prime ﬁlters because the maps are

¤ ¤

¨ QÿV%

frame homomorphisms. Next assume that ýþ> con; in this case, no point

¥28 ¨78 8 8 ¥68 ¨38

>X% Qk% < <

< can have and , and so can not be both in

¤ ¤ ¤

¥ ¥ ¥ ¥ ¥ 1

> 6\< Q 6 > 0'6=< Q !4 6 > % 3

6 and . This means that and hence

¤ ¤ ¤ ¤

¥ 1

Q % 3 < >c3%re Q'3%reM<

or 6=< . This, ﬁnally, means that either or . The argument ¤

for the tot-predicate is dual.

¥ ¨ eM<

By the calculation at the beginning of the proof it is clear that e has no

¥21 ¨31

neighbourhoods other than those in < . ¤

Example 4.2 All the bitopological spaces in Figure 1 are d-sober. For the one- point space this is clear, as the associated d-frame admits only one point. For the other four spaces one argues as follows: The underlying frame is the same in each

case and it admits four completely prime ﬁlters:

ß ß

e ^! [fgf e ^! [bn`n

¥ ¨ ¥ ¨

e ^! [ 4 e ^! [ 4 <

¤ ¤ The notation already indicates which of these can be used as the ﬁrst, respectively second, component of a point. From this we get four possible combinations, and

these are indeed all available in the last example. In the other three examples,

¥ ¨

the con/tot labelling of the element in the centre of the d-frame excludes

e e

certain combinations: if it belongs to con, then cannot be paired with < , and

ß ß

e e

if it belongs to tot then cannot be paired with < . ¤

4.1 Bitopological analogues of topological concepts

As the foregoing examples show, d-sobriety is a subtle constraint on the interac- ©<

tion between and . The remainder of this section explores this interaction, ¤

emphasising bitopological analogues of classical topological ideas.

©< ¡O¢

Lemma 4.3 Let and be two topologies on a set , and assume that

¥ ¨ ©<

the bitopological space is sober. Then the intersection of the two

0WJ<Ë! !

specialisation preorders equals identity: J ‘ ’.

¥ ¥ ¶¨ ¥

Q >PJ â,< Q

Proof. Assume >cJ , for two elements of . Then

¤ ¤

0¯

is a point of áË . Indeed, if then either or . In the

¤ ¤

%' >¯J Q

ﬁrst case, >c3 follows because .

¤ ¤

V%' >¯%¯ h<

Similarly, if then either or . In the second case,

¤ ¤

< >¯J

Qs%¯ follows because .

¥ ¥ ¶¨ ¥

> â,< Q

Sobriety implies that â must be a canonical point associated

¡¡¢ with a single element È of . However, the separation axiom says that different

points have different neighbourhood ﬁlters, so it must be the case that >m!RÈÎ!RQ .

¥ ¨

Corollary 4.4 Let be a topology on a set such that is d-sober. Then ¡õß

satisﬁes the separation axiom already.

¥ ¨ eû<

Proof. Any abstract point e of a d-sober space is of the form

¥ ¥ ;¨ ¥

> âæ< > >

â for some unique point . So when the two topologies are the

! eû< > Q

same, it must be the case that e . Then if and are two distinct

¥ ¥

â > !Ôâæ< >

points of it must be the case that their neighbourhood ﬁlters

¥ ¥

Q !½âæ< Q ¡¡¢

and â differ, in other words, the topology must be . By the pre- ¤

ceding lemma the specialisation order must be equality so the topology is even ¡Mß .

J 0pJ<

We take these results to indicate “bi- ¡ß ” ought to mean that is equal-

¥ ¨ ¨

ity. We refer to a bitopological space as symmetric. Obviously, symmetry makes the category of topological spaces equivalent to a full sub-category of

bitopological spaces. We will return to symmetric spaces in Section 9.

¥ ¥ ¨

c Counterexample 4.5 If is a sober space in the usual sense, then

is not necessarily d-sober. Indeed, by the previous statement this can only happen

¥ ¡õß if is a space, but sobriety does not imply this, the Sierpinski space being

35 the smallest example of a sober non- ¡ß space. Another way of expressing this is

to say that ﬁnite sets equipped with two ¡:¢ topologies are not necessarily d-sober.

¥ ¨ ¥

$< ~ Sm$<

Lemma 4.6 If ~ is a d-sober space then is sober.

¤ ¤

¥ ¨

S*$< e eû< ^!

Proof. For e a completely prime ﬁlter in , consider

¤ ¤

¥ ¨

e 0m ek0m©< 01 <ò! 4

. We show that this is a d-point. Indeed, if

¤ ¤

then this intersection can not be an element of e . Therefore we can not have

%/e e

and at the same time. If then

¤ ¤ ¤

%'e %'e <¯%'e %'e

because , we must have or by primality. Hence

¤ ¤ ¤

or .

¥ S©<

Of course, the sobriety of does not tell us anything about the

¥ ¨

©< ©<

d-sobriety of ; a counterexample is again provided by and both

¤ ¤ being equal to the Sierpinski topology on the two-element set.13 A more intricate

example is required to show that Lemma 4.3 can not be reversed:

^!:9 "¾C DÎ+ Counterexample 4.7 Consider the set that can usefully be

visualised as follows: ;

< 9

As the positive topology on we take all subsets of plus all co-ﬁnite subsets

¡õß ¡=

of . This is and sober, but not . For the negative topology we take the weak

?> >

upper topology, whose closed sets are all of the form Í with a ﬁnite subset

of . This, too, is a sober topology. The specialisation order with respect to is ¤

equality, and that with respect to ¾< is given in the diagram. Their intersection is,

of course, equality.

¥ ¨ $<

We claim that the bitopological space is not sober. Consider the

¥ ¥ ;¨ ¥ ~

C âæ< D

pair â . It satisﬁes the condition for total predicates because every

¤ D non-empty open set of ¾< contains . The condition for consistent predicates is

{A@3B This differs from the theory of biframes where sobriety is taken to mean sobriety of @ .

36 also satisﬁed because every positive neighbourhood of C intersects with every

negative neighbourhood of D in co-ﬁnitely many natural numbers.

On the positive side we have:

Proposition 4.8 Let be a sober space. The following are all d-sober

bitopological spaces:

¥ ¨7Cb ¥ C¾¨ C

1. and , where is the indiscriminate topology;

¥ ¨¶¹q ¥ ¹m¨ ¹

2. and , where is the discrete topology;

Proof. For the ﬁrst claim, remember that the set "bc+ is the only completely

prime ﬁlter in the indiscriminate topology, and when paired with a completely C

prime ﬁlter of always gives rise to a d-point: As has only two open sets,

0¯ <¯!4 !&4 h<¯!&4

can only happen if or . Likewise,

¤ ¤ ¤

can only happen if or . ¤ For the second claim, we note that the discrete topology is Hausdorff

and hence guaranteed to be sober. Any d-point, therefore, has the form

¥ ¥ ;¨ ¥ ~

> âæ< Q

â . The rest of the proof depends on playing the role of (the

¤ ¤

J Q

other case requiring a dual argument): If we had >X3 then there would exist

>R%X QR3%X

Ö%R with , . The complement of is an open set in and so

Q >¯J Q QqJ >

belongs to â,< , contradicting (dpcon). Thus and likewise .

¤ ¤

We interpret these results about d-sober spaces as telling us that it is more

0 <

appropriate to consider J as the specialisation (pre-) order of a bitopo-

0°J< ¡8¢ J 0 <

logical space, rather than J . So “bi- ” should mean that is

¤ ¤

a partial order. For this order the open sets of become upper sets, and those ¤

of ©< , lower sets. The bitopological spaces in Figure 1 have been drawn in this way. This view jibes with the situation in stably compact spaces where the specialisation order is opposite to that of the associated co-compact topology.

Let us pause to say a few more words about the bitopological space ºüÊº , the

dualising object in biTop. There is a forgetful functor from bitopological spaces

¥ ¨

c $<

to Top D , the category of pairs of topological spaces, which maps

¥~¥ ;¨b¥ ~ ¥¥2E ;¨$¥2EÎN N£ ~

to . It has a right adjoint which maps a pair to ¤

N N N

¥2E E ¨ E E

! ! ! !

$< ^! " ® ( ®%V + ©<½^!" ® ( ®&%¯©<:+

, where and .

G ¤ ¤ G ¤ G

E EÎN

We denote the resulting bitopological space with Ê . Notice that the usual Ty-

E EhN

! ! ©<

chonoff product topology on is precisely the join of and . The natural

G ¤

¥~¥¥ ;¨$¥ ¶¨$¥¥2E ;¨b¥6ETN Ng

$< ~

isomorphism between hom-sets Top D and

¥~¥ ¨ ¶¨3E ETN£

$< Ê ºûÊº

biTop ~ is obvious. The dualising bitopological space is ob- ¤

tained in this way from two copies of Sierpinski space.

¥ ¨ ©<

Deﬁnition 4.9 A bitopological space c is called order-separated if

0 < >3J°Q

J½!&J is a partial order and implies that there are disjoint open

%V h<¿%V$< >V%' Qs%' <

sets and such that and .

¤ ¤ ¤

Lemma 4.10 In an order-separated bitopological space the following are true:

! <

1. J ;

0PJ

2. J ‘ ’.

J Q >ò3J-Q

Proof. For the ﬁrst claim assume >ò3 . This implies and we get a

¥ ¨

< Q±%R h< >3% <

separating partial predicate . Since but we conclude

À À

>c3 . So and this is equivalent to the ﬁrst claim. ¤

The second claim follows immediately from (1) and anti-symmetry of J . D-sobriety is a surprisingly inclusive concept. We illustrate this with two examples.

Example 4.11 F with the usual upper and lower topology is d-sober. This is

¥§¦$¨¶ ¦

HG ^! " ( %IF§+ "JFÑ+Ô! Y "©45+

seen as follows: Although e is com- ¤

pletely prime and not the neighbourhood ﬁlter of any real number, there is no

¥ ¨

eHG eM< eü<

eM< such that the pair is a d-point. Indeed, all have the form

¥ ¨¦ ¦

" ( ?LK+ eû

e¬r^! or ; can not be paired with

¥ ¨ ¥ ¨¶ ¥ ¨

ôÝ 0 KOMôÝ ! 4 KNMôÝ %Ze¼¬3

because KNM but neither nor

¥ ¨¶

VÝ %'e3 G eü

K£M . The same argument shows that can not be paired with .

+ !P©A Ý¦ Y"

Example 4.12 Consider with the usual upper and lower topology. This is a d-sober space.

Indeed, the only pair of completely prime ﬁlters that could cause trouble is

¥þ¦$¨ ¦ ¨¦ ¦

ß ß ß ß

e ^`! " Ý¦ ]Y" +( µ + eû< ^! "?©A Y" +( ? +

¨ ¨ 5¿¥ ¨

ß ß ß

!Q©A ©A Ý¦ %'eM<

but this does not give rise to a d-point: but neither

¥ ¨

Ý¦ :%re

nor . ¤

Both of these examples are order-separated. The fact that they are sober generalises to all such spaces.

Theorem 4.13 Order-separated bitopological spaces are sober. Ø

Proof. Order separation clearly implies that the canonical map é ^¼

¥ ¨

e eû<

spec áË is injective; the real issue is surjectivity. So assume that is ¤

a point of áË . Consider the two sets

ú ^`!RãR"© %V (¾ %'e3 + ú:< ^`!ãR"©

¤ ¤ ¤ ¤ ¤

Ä Ä

ú ú ú ú:<

and their complements < . Because of condition (dptot), cannot be

Ä Ä

ú 0¿ú

the whole space, in other words, the intersection < is non-empty.

Ä Ä

ú ú

Next we show that every element of is below every element of < in the

Ä Ä

JB!ÏJ 0 < >Ô%Ôú Q½%Ôú >Ð3J Q

specialisation order . Indeed, if , < , and ,

¥ ¨

< >*%½

then by order separation there is a partial predicate with and

¤ ¤

< ú ú8< %ce

Q2%c . By deﬁnition of we have and , contradicting

¤ ¤ ¤

condition (dpcon) of d-points.

Ä Ä

0Wú È ú e

Finally, let be an element in the intersection < . We show that is

ÈV%± ú3

the neighbourhood ﬁlter of È in . Assume ; this implies and

¤ ¤ ¤ ¤

%re Rú3

the latter is equivalent to . For the converse we start at , which

¤ ¤ ¤ ¤

m%Rú 0P RmJÐÈ

gives us an element R about which we already know that . It

J È Èq%'

follows that R and hence .

¤ ¤

We see this result as being analogous to the well-known fact that ¡ spaces are sober in the traditional sense.

Corollary 4.14 Compact regular bitopological spaces are d-sober.

A perhaps less well-known (but easy to prove) fact is that sub-spaces of sober ¡¼ß ¡£ß spaces are sober. Recalling our understanding of “bi- ”, this, too, has an

analogue in the bitopological setting:

¥ ¨ E

©< J 0HJ<'! !

Theorem 4.15 Let be d-sober and ‘ ’. Let further be

¤ ¤

¥6E ¨

~ $<

any subset of . Then is d-sober, too.

¤TSVU SWU

¥ ¥

Í Í

È È ! È < È ! È

Proof. (i) Let be an element of . Since cl , cl < , and

¥ ¥ ¥

0ÑJ<*! ! È 0 < È ! "©Èà+ ú ^`!ZY È

J ‘ ’, cl cl . Now, cl is in but

¤ ¤ ¤ ¤ ¤

¥ ¥ ¥ ¥

È ú:< È ^`!ÏÙY < È %ô$

not in â . Likewise, cl . Furthermore,

¥ ¡ ¥

È ú:< È !*aY"©È+

ú .

E ¥ ¥ E ¥

ú È !Xú È 0 %V ú:< È !

(ii) Assume Èq%VÐY . Then and

¤ S ¤ ¤TS S

U U U

¥ E ¥ ¥ E

ú8< È 0 %¯$< È ú:< È !

. Furthermore, ú .

S ¤ S S

U U U

¥ E ¥ ¨

U U U

e ú È e %'e

(iii) Let < be a d-point on . Because of (ii), either or

¤ S

¤ ¤

ú8< È %'e

< .

<àß

¡^ Ø X

(iv) The embedding X is bicontinuous, hence restricts to frame

¥ E

<àß

Ø ©

homomorphisms and . Concretely, .

¤ ¤TS S

U U

It follows that the inverse image map to the frame homomorphisms maps a com-

pletely prime ﬁlter e in to a completely prime ﬁlter in . Concretely,

¤TS ¤ ¤

!/"© °%V (© *0 %'e +

e .

¤ ¤

¥ ¨ ¥ ¨

U U

e e e eM<

(v) For the d-point < consider on . This is again a d-point:

¥ Eh ¥ Eh E

0W <±!Ð4 0 0

implies , so either or

¤ ¤ ¤

<3%'e

. The covering condition is proved analogously.

¥ ¨ ¥ ¥ ¶¨ ¥

e eû< ! â È â,< È

(vi) By the assumption of being d-sober, for

¤ ¤

E ¥ E ¥

ÈP%1 È% e È ! "© 0 (¾ °%¯â È +

some . Case 1: . Then !ôâ

¤ SWU ¤

%°e 0 %°e

because can by deﬁnition only happen if . Conversely,

%X ® ! &0

®Ö%e implies that there exists such that by deﬁnition

e e

of the sub-space topology. This then belongs to . It follows that is the

È e

neighbourhood ﬁlter of in . Likewise for < .

E ¥ E ¥ E

% ú È 0 %¯e ú8< È 0 %

Case 2: È¯3 . We know from (ii) that either or

¥ ¥

ú È % e ú:< È % eM< e

< . This implies or . This is a contradiction, though,

¤ ¤

¥ ¥

È ú:< È È

because neither ú nor are neighbourhoods of in . ¤

40 E

(vii) The conclusion is that all d-points of á are pairs of neighbourhood E ﬁlters of points in .

It is tempting to conclude from this rather surprising result that every space

02J

which satisﬁes J ‘ ’ is d-sober, as it can be rendered as a sub-space of ¤

its d-sobrification. However, the condition does not survive d-sobrification: 9 Example 4.16 Consider the co-finite topology on . Its specialisation order is

trivial, yet the sobriﬁcation (in the usual sense) adds a new point Y (correspond-

ing to the ﬁlter Z of all non-empty co-ﬁnite subsets) which sits above all other

elements: [

¥ ¨ ¥ ¨

=~ Y Z Z

The d-sobriﬁcation of 9 adds (represented as the d-point ) and

¥ ¥ ;¨ ¥ ~

â ¸ â ¸

triplicates each natural number: apart from ¸@! there are also

¥ ¥ ¶¨ ¥ ¨ ¥ ~

^! â ¸ Z ¸±^! Z â ¸ J 0RJ<

¸ and . The order between these is quite ¤

rich:

¢ ß

¢ ß 0 1 2 3

0 1

¨ ¨ ¨

\Y ¸ ¸ ¸£+ ºûÊ º Each sub-structure " carries the same bitopology as the space on the bottom of Figure 1.

4.2 Hofmann-Mislove We conclude this section with a discussion of Hofmann-Mislove type theorems

for bitopological spaces. For motivation and comparison we look at an ordinary

¥ ¥

â #

topological space ﬁrst. The collection of open neighbourhoods of

Ð any subset # is always a filter in the frame , but in general there are far more filters than neighbourhood filters. In three cases we can say more:

â # ![# 1. If # is an open set then , that is, the neighbourhood ﬁlter is

principal. All principal ﬁlters in arise in this way.

> â # 2. If #½!½[]> for an element of then is completely prime. The space

is sober if and only if all completely prime ﬁlters arise uniquely in this way.

â # 3. If # is compact then is Scott-open. The Hofmann-Mislove Theorem

states that in a sober space all Scott-open ﬁlters arise in this way.

¥ ¨ »

$< #&^!

Consider #&k for a bitopological space . The collection

¤ ¤

]_^ ¥

¨¦`

È #

â is equal to the set of all -neighbourhoods of , which from the

¤ ¤ #

point of logic corresponds to all those predicates a for which every elements of

^ ¥

¨¦`

satisﬁes a . On the negative side, one should consider which

# a

is all those predicates a for which some element of fails . A moment’s consid-

¥|» ¨

b<#

eration will convince the reader that the pair # still satisﬁes the axioms

¤ »

(dpcon) and (dptot) for d-points. However, the collection # is merely a ﬁlter

b<#

in , while is a completely prime upper set (but not necessarily a ﬁlter). As ¤ with ordinary topological spaces we can ask which special cases of such pairs can

be characterised by properties of the subset # . D-sobriety, obviously, is case (2)

generalised to bitopological spaces; case (1) takes the following form:

Proposition 4.17 For any topological space ~ , there is a bijection between

closed sets and completely prime upper subsets of .

b#cX^!Ð"]ä&%±1(ed&0äÐ3!°45+

Proof. For a closed set # , let . For a completely

b #Af¯^!½ÐY "© °%Vr(© Úo%Ib+

prime upper set , let ã .

¥ ¨

©< bT<

Proposition 4.18 Let ~ be a bitopological space. Let be a com-

# ¾<

pletely prime upper set in ©< , and the corresponding -closed set according to

» #

the previous proposition. Then the -neighbourhood ﬁlter of satisﬁes:

¤ ¤

%V <¿%V$

(hmtot) for all

¤ ¤

Ç »

¤ ¤ ¤

N »

% ^ ú:<Ë!*

(hmcon) for all

¤ ¤ ¤

N »

Moreover, is uniquely determined by (hmtot) and (hmcon). ¤

Proof. The ﬁrst part is immediate.

N N »

For uniqueness suppose is a ﬁlter in satisfying (hmtot) and (hmcon). From

N N N

» » »

(hmcon), #T

¤ ¤ ¤

inclusion, suppose # . Therefore , and since we

¤ ¤

»N %

must have by (hmtot). ¤

From the preceding discussion, the reader may have expected to ﬁnd the fol- N

lowing instead of (hmcon):

%¯ <¿%V$<¯^

(hmcon) for all

¤ ¤

Ç »

0¿

¤ ¤ ¤

N »

However, this easily follows from (hmtot) and (hmcon): if then ú:<Ë!

¤ ¤ ¤

0±

and if also then must follow. Without additional ¤

assumptions on , however, the stronger condition is needed for uniqueness.

Proposition 4.19 With the terminology of the previous proposition:

¥ » #<

1. is Scott-open if and only if sat is -compact.

¤ ¤ ¤

¥ ¨ ¥

$< J < #

2. If satisﬁes then sat .

¤ ¤ ¤

¥ #<

Proof. Part (1) is trivial. For the second claim assume >/% sat . Since

¥ ¥

#T< !*[ #T< Q > >VJT

sat there is Q%¯#T< with , so by assumption and

¤ ¤

< Q >¯%¯#T< >V% cl . This forces . We doubt that Proposition 4.18 together with part (1) of the preceding result qualiﬁes as a bitopological version of the Hofmann-Mislove Theorem, since its proof is so easy and d-sobriety is not required. We obtain a more satisfactory result when we assume regularity (Deﬁnition 2.13). First of all, it is not hard to show that every regular bitopological space is order-separated and therefore sober.

The interesting bit for us, however, is that condition (hmcon i ) can be replaced

by (hmcon):

¥ ¨ $<

Theorem 4.20 Let be a regular bitopological space. There is a one- ¤ to-one correspondence between

©<

(i) subsets # of which are -closed and -compact; and

¥¥» ¨ »

< b<

(ii) pairs b where is a Scott-open ﬁlter in and is a completely

¤ ¤ ¤

prime upper set in ©< , satisfying (hmcon) and (hmtot).

Proof. We show that (hmcon i ) follows from regularity and (hmcon). To this end,

» %

let . It is the directed union of open sets well-inside it, and so by Scott-

¤ ¤

Á »

h<°%ò$<

openness some belongs to , too. The witness satisﬁes

¤ ¤

, so by (hmcon) cannot belong to , in other words, it is a subset

jb<

of ú:

ú8<Ë!*

as required. ¤

5 Reasonable d-frames and spatiality

K áË We say that a d-frame K is spatial if is isomorphic to for some bitopological

space . As with d-sobriety, spatiality has internal characterisations.

¥ ¨ ¨

FÑÂfgf n`nõ ëL

For any d-frame K ! con tot , the co-unit is determined by the

¥ ¥

^OF Ø á K ë;<\^F <¿Ø á K <

two frame homomorphisms ë spec and spec

¤ ¤ ¤

¥ ¥

Ø 6 > Qq×Ø 6\< Q

deﬁned by >'× and . Clearly, both of these are surjective and ¤

so ë~L itself is surjective.

Theorem 5.1 For a d-frame K , the following are equivalent:

1. K is spatial.

á K 2. K is isomorphic to spec .

3. The co-unit ëL is an isomorphism.

4. The co-unit is injective and reﬂects con and tot.

5. K satisﬁes the following four conditions:

N É¡¥ ¨ ¨ N

d>3Ik> %'F e eû< % KÊ >'%'e > %'e3

(s ) k spec

¤ ¤ ¤ ¤ ¤

N N

É¡¥ ¨ ¨

kOQ'3I Q %'F < e eM< % KÊ]Qs%'eû< Q %'eM<£3

(s < ) spec

É¡¥ ¨ ¨

% 3 e eM< % KÊ %¯e

(scon) k con spec

¤ ¤ ¤

É¡¥ ¨ ¨

% 3 e eM< % KÊ %¯e3 <3%'eû<õ

(stot) k tot spec

¤ ¤ ¤

Proof. Clearly, the d-frame ám associated with a bitopological space satisﬁes

Ç Ç Ç the four other conditions. So (1) (5). Also (3) (2) (1) are trivial. As ë L is a surjective frame homomorphism that preserves con and tot, if it is also injective

and reﬂects con and tot, then it is an isomorphism. Ç

For (5) (4), observe that conditions (s ) and (s < ) imply that the assign-

¥ ¥

Ø 6 > Q@×Ø 6\< Q ëL

ments >-× and are injective. Thus is injective, and ¤

we only need to check that the two predicates are reﬂected. If we assume that

¥ ¥ ¥ ¨

0p6=< < ! 4 e eû<

6 then we know that for every abstract point ,

¤ ¤ ¤

¥ ¥

%òe3 % 3 6 eM

either e or , which by deﬁnition means

¤ ¤ ¤ ¤ ¤

%'eM< or

We collect some properties of the con- and tot-predicate on spatial d-frames.

¥ ¨ ¨

n`n£ IL)

Lemma 5.2 Let FÑÂfgf con tot be a spatial d-frame, and . Then

! )%

% tot tot

% ! %

) con con

Ú3%

Proof. We show the contrapositive: ) tot implies by (stot) that there is an

¥ ¨

eû< ) %1e3 ):

abstract point e with and , so the same is true for

¤ ¤ ¤ ¤

and < , which shows that can not belong to tot.

The proof for con is analogous.

¥ ¨ ¨

n`nõ fgf% fgf%

Lemma 5.3 Let FÑÂfgf con tot be a spatial d-frame. Then con, tot, n`nV%

n`nV% con, tot and

¨ Ç ¥ ¥

W% ! g) % Sg) %

) tot tot and tot

¨ Ç ¥ ¥

% ! g) % Sl) %

) con con and con

¥ ¨

% e eM< fgf;

Proof. Suppose fgfpo con. Then there is a d-point such that

¤ eû< Ac%Reû< . This is impossible as is a completely prime ﬁlter. The other three

memberships are proved similarly.

For closure of tot under S (and by a symmetric argument, ), we show the

¨ N ¨ N

Qÿ )k!Úýþ> Q ÿ Sm)k! ýþ>Ëó contrapositive: Suppose !Úý§> and and assume

N N N

¨ ¥ ¨

QíÛQ ÿ3% e eû< >\ó> %re3

> tot, then by (stot) there is an abstract point with ,

¤ ¤

N N

%reM<3 > > e

QÛmQ from which we get that both and do not belong to and either

N ¨ N ¨ N

Q eû< ýþ> Qÿæ3% ýþ> Q ÿæ3%

Q or does not belong to . It follows that either tot or tot. eû<

The proofs for con are analogous, using the primality of e and . ¤

Lemma 5.4 Let K be a spatial d-frame. Then con is Scott closed with respect

to I .

ö % o ö ÷

Proof. If were a directed subset of con with Ü con, there would be an

¥ ¨ ¥ ¥

e eû< Ü ö %_e Ü ö <±%_eû< ÷

abstract point with ÷ and . Both ﬁlters are

¤ ¤ ¤

%We )c%Wö ):<'%Weû<

completely prime. So for some %rö , and for some , .

¤ ¤

!n) But ö is directed, so we can choose . This contradicts the deﬁnition of

abstract points. We already checked that con is closed downward.

¥ ¨ ¨

n`n£ % )°% Lemma 5.5 Let FÑÂfgf con tot be a spatial d-frame, con and tot.

Then

!o) ! IL)

¤ ¤

Ip)8<

Proof. Toward a contradiction for the ﬁrst implication suppose <3 . By

¥ ¨

eM< <%_eM< )8<o%_eû<

spatialilty, there is an abstract point e for which and .

%°eo )ô% ) %@e ) I 3

But % con implies and tot implies ; so ,

¤ ¤ ¤ ¤ ¤ ¤ contradicting the assumption. We believe that the properties expressed in the lemmas in this section are independent of each other (except for the trivial fact that con being Scott closed implies con is closed downward), but we do not have a proof for it.

5.1 Reasonable d-frames The properties stated in Lemmas 5.2 through 5.5 constitute a good restriction on general d-frames without necessarily requiring spatiality. All of them, except for Lemma 5.4 are ﬁrst-order properties. The one non-ﬁrst-order property is never- theless constructive, as it simply involves closure of con under directed suprema.

L)4qr)% ! %

(con– ) I con con

IL)4q % ! )P%

(tot– [ ) tot tot fgfÑ%

(con– fgf ) con n`n¯%

(con– n`n ) con

Ç ¥

% qr)c% ! s) %

(con– ) con con con

Ç ¥

% qr)c% ! Ss) %

(con– S ) con con con fgfÑ%

(tot– fgf ) tot n`n¯%

(tot– n`n ) tot

Ç ¥

% qt)P% ! l) %

(tot– ) tot tot tot

Ç ¥

% qt)P% ! Sl) %

(tot– S ) tot tot tot

Ü #& I ! Ü #/% ÷

(con– ÷ ) con directed w.r.t. con

Ç ¨ ¨ ¥

(con–tot) con ) tot or

¤ ¤ Table 1: The deﬁning properties of reasonable d-frames.

From now on, we will generally concentrate on d-frames that satisfy these conditions.

Deﬁnition 5.6 A d-frame which satisﬁes the properties stated in Lemmas 5.2 through 5.5, is called reasonable. For ease of reference they are collected and named in Table 1. The category of reasonable d-frames is denoted by rdFrm.

Taking Lemmas 5.2 through 5.5 together, we see that the adjunction between bitopological spaces and d-frames co-restricts to reasonable d-frames. So there is no real loss in restricting our attention to rdFrm.

A reasonable d-frame need not be spatial: take a frame F without any points

¥ ¨ ¨ ¨ ¨

FÑ ýþ> Qÿ% >sÛ¿Qm!°A ý§> Qÿ% and consider F con tot where con if , and tot if

>æómQæ!Ý . It is a trivial exercise to prove that the resulting d-frame is reasonable, but it obviously can’t have any points. Below we will combine reasonableness with compactness and regularity, and then spatiality will follow.

Proposition 5.7 The forgetful functor from rdFrm to Set has a left adjoint.

¥ ¨ ¨

eT# eT# e#

Proof. The free reasonable d-frame over a set # is con tot where

¥ ¨

È È Èq%¯#

is the free frame over # . Generators are the pairs , . The two relations

¨ ¨

Qÿ% >¯!ÔA QË!A ýþ> Q5ÿ%

are chosen minimally: ýþ> con if and only if or ; tot if QV!aÝ and only if >P!ÐÝ or . The conditions for a reasonable d-frame are proved

by case analysis.

5Êvu As an example, the structure labelled u in Figure 1 is the free reasonable d-frame generated by a one-element set.

5.2 Biframes

We return to the question of how d-frames are related to the biframes of Ba-

¥ ¨ ¨

F ß F¤

naschewski, Brummer¨ , and Hardie. If Fí¢ is a biframe then we can deﬁne

¥ ¨ ¨ ¥ ¨ ¨ ¨

Fû¢ F ß F¡ F ß F¡ b ýþ> Qÿ % ^ x

a d-frame by setting w to be con tot where con

ýþ> Q5ÿ% ^ x >2óVQr! Ý FÑß F¡

>ËÛ'Q'!ôA , and tot , exploiting the fact that and w are subsets of the frame Fü¢ . Clearly, this extends to a functor from biframes to d-frames. The following is an easy exercise:

Proposition 5.8 Every d-frame derived from a biframe is reasonable.

One can think of w as a forgetful functor from biframes to d-frames. Specif-

FÑß F¡

ically, it forgets everything about Fí¢ except for the pairs of elements in

G Ý that meet at A or join at .

Theorem 5.9 The functor w has a left adjoint.

¥ ¨ ¨

FÑÂfgf n`nõ Fí¢

Proof. Given K*! con tot , we generate a frame using the set

^`!&"zy¾>{m(b>¯%¯F F <£+ ¤

subject to the relations

N N N

¾>{ I y¾> { >¯Ik> %rF >¯Ik> %'F <

y for all or

|}^ |

> > >

¨~

È{ I y { 3F 3\FH<

y for all or

¨`

#*F #*F < ]Üp#T{ I Ü y È{

y for all or

¾>{ÛsyQ¢{ I A ýþ> Q5ÿ %

y for all con

I y¾>{ósyQ¢{ ýþ> Q5ÿ % Ý for all tot

48 Note the similarity of this construction with the coproduct of frames; the only dif-

ference is in the last two rules which ensure that the two predicates are respected.

y F {

As with the coproduct, here too we ﬁnd that Fí¢ contains subframes and

F <{ F FH<

y given by the equivalence classes of the generators from and . Thus

¥ ¨ ¨

y F { y F <{ >@×Ø y¾>{

the triple FM¢ is a biframe, and restricts to frame ho-

y F { F < y F <{

momorphisms from F to and from to . By the last two rules,

¤ ¤

these maps preserve con and tot, so we have a d-frame homomorphism from K to

¥ ¨ ¨

FM¢ y F { y FH< {

w .

¥ ¨ ¨

ÕËß Õ} è è K

If Õ¢ is a biframe and a d-frame homomorphism from to

¥ ¨ ¨

Õ,¢ ÕËß Õ Fü¢ Õ¢

w we get a map from the set of generators of to by setting

¾>{&×Ø è > y . Since this map clearly respects the above relations, it extends

uniquely to a frame homomorphism; the result sends equivalence classes of gen-

Õmß F¤ erators from FHß to and similarly for . An alternative construction of the free biframe highlights, once again, the in-

terplay between logic and information. For this we use a known folk theorem that

F¡ FÑß F¡

a cartesian product FÑß of frames is also a coproduct of and in the cat-

G F <

egory of meet semilattices. In terms of a d-frame, the injections from F and

Ø)>óhn`n Qæ×ØÏQTó'fgf

take the form >m× and , respectively.

¥ ¨ ¨

FÑÂfgf n`n£ ¢ For a given d-frame K ! con tot , we consider the collection of

subsets #/*F satisfying

(bif–con) con #

¨ ¨ ¨ Ç ¨

Û) )8

(bif–tot) % tot

¤ ¤ ¤

# I

(bif– I ) is Scott closed with respect to with respect to

# J

(bif– J ) is a sub-lattice with respect to

¤¥

Û F Considering F as a -semi-lattice, we deﬁne families of subsets of in-

dexed by members of F as follows:

¤q¥

)I 4,% )

1. % con and implies ;

¨ ¨ ¨ ¤¥

" ý Û) )8

2. % tot implies ;

¤ ¤ ¤

¤q¥

IRÜ ")ËÛ ()P% +Î%

3. ÷ implies ;

¤¥ fgf

4. 4,% ;

¤¥ nn

5. 4,% ;

¨ ¨ Nv¨ ¤q¥ N9¨

ýþ> Q5ÿ ýþ> Q5ÿ;+T% ýþ>,óq> Q5ÿ

6. " ;

¨ ¨ ¨ N ¤q¥ ¨ N

ýþ> Q5ÿ ýþ> Q ÿ;+ % ýþ> QTósQ ÿ 7. " ;

These families form a covering as per [Joh82, Section II.2.11]. That is, they are

¤¥ ¤q¥

_% "sÛs)p(.r%m +Î% ÛT)

stable under meets: implies that . Deﬁne a

F _*Ó

-ideal to be a subset Ó of which is closed downward and for which and

¤q¥

±% %¯Ó #%h ¢

imply . Consider . By deﬁnition it is downward closed and ¤

satisﬁes the closure conditions for -ideals – the last two conditions being special

cases of the closure under S and . So is contained in the collection of - ¤

ideals. Conversely, the ﬁrst three conditions on -ideals are essentially (bif–con),

¤ J

(bif–tot), and (bif– I ). To see that a -ideal is also closed under (bif– ), recall

¨ N ¨ N N9¨ N ¤

Qÿ]S,ýþ> Q ÿü!Ôýþ>Ñó> QüÛQ ÿ

that ýþ> and that a -ideal is closed downward. Thus

¤ ]¢

is the collection of -ideals.

¤ ¢

Following Johnstone’s general construction, the set of -ideals is guar-

anteed to form a frame with three useful properties: First, letting # denote ¤

the principle -ideal containing , the resulting frame is generated by the sets

¥ ¥

# F ¢ Ø # . Second, the map × preserves the ﬁnite meets of . Third,

is free with respect to transferring covers to joins. That is, if Õ is a frame and

¤¥

F Õ Ú%

è is a meet semi-lattice morphism from to for which implies

¥ ¥

!Xè è ^ ¢HØ Õ

Ükè , then extends uniquely to a frame homomorphism

¥ ¥ ¥

! # %rF

so that è for all .

F F <

Because F , as a meet semi-lattice, is a coproduct of and , it is generated

F Fû¤

by the sub-semilattices Fí¤ and , where consists of elements of the form

óæn`n F 5ß Fü¤

¤c^`! and similarly for . Deﬁne and to be the images of and

¥ ¤

< #

F with respect to . The last ﬁve clauses in the deﬁnition of ensure that

5ß ¢ >Ë×Ø # >O¤ F 5ß

and are subframes of , and the map from to is a frame

¥ ¨ ¨

]¢ ß

homomorphism, and similarly for . Thus is a biframe. The ﬁrst two

;¨ ¥ ¥ ¥

# Ø #

clauses ensure that the map × is a d-frame homomorphism

¥ ¨ ¨

w ]¢ ß from K to .

¥ ¨ ¨

Õmß Õ} è K

Given a biframe Õq¢ and a d-frame homomorphism from to

¥ ¨ ¨

Õ,¢ ÕËß Õ è

w , the map cuts down to two semilattice homomorphisms from

F Õ¢ è

Fû¤ and into . So these extend uniquely to a semilattice homomorphism

N9¥ ¨ ¥ ¥

Õ¢ è ýþ> Q5ÿ !Rè > Û¿èO< Q

from F to . Speciﬁcally, .

¤ N

Now it is fairly easy to check that è transfers covers to joins. For example,

¨ ¥ ¥ ¨ N§¥

Qÿ % è > Û§èO< Q !RA Õ¢ )Iýþ> Q5ÿ è ) !A

ýþ> con implies that (in ), so for , . ¤

The other conditions are just as routine, except perhaps the condition involving

¨ ¥ ¥

!Ïýþ> Q5ÿs% > óèO< Q !ÙÝ Õ¢

tot. Suppose tot. So, è in . For any other

N N

¯!°ýþ> Q ÿ

) , we have

N N N

¥ ¨ ¥ ¥ ¥

è ý Û) )8 Ûmè > Û2è8< Q

¤ ¤ ¤ ¤

N N N

¥ ¨ ¥ ¥ ¥

è ý2) <,Û)8 Û2èO< Q Ûmè8< Q Ê

¤ ¤

¥ Ng ¥ N£ N9¥

> ó¿èO< Q !Xè )

So the join of these is simply è .

F Õs¢

Finally, because è is a semilattice homomorphism from to that transfers

¢ Õ,¢

covers to joins, it extends uniquely to a frame homomorphism from to .

¥ ¥ ¨ ¥ ¨ ¥

# ýþ> Ý$ÿ !è > Ý !è > %Õmß

This sends >&%F to . Likewise,

¤ ¤

sends elements of FÑ< to , so it is a biframe homomorphism.

¥ ¨ ¨

¢ ß

One can wonder whether the d-frame that one obtains from (equiv-

¥ ¨ ¨ ¥ ¨ ¨

y F { y F <{ K ! FÑÂfgf n`nõ

alently, Fû¢ ) is isomorphic to the original con tot . ¤

This need not be the case, even if K is reasonable:

¥2T

w K Lemma 5.10 Let K be (isomorphic to) for a biframe . Then satisﬁes

the cut rules14 listed in Table 2.

¥2

Proof. The cut rules in w are simply instances of laws that hold in any frame.

Counterexample 5.11 The ﬁnitary cut rule (cuttot) does not follow from reason- F <

ableness. Consider the d-frame in which both F and are the powersets

¥ ¨ ¨ ¨

"©A Ý+ ý§> Qÿ2% ( >M(\Ma( Qõ(íJBå ýþ> Q5ÿ¿% . We take con if and only if and tot

} x { The “logical” terminology for } and and for the operations and was justiﬁed by Propo- sition 3.2. Similar terminology here will be justiﬁed in Section 7.

¨ ¨ ¨

Qÿ % q@ýþ> QTósR;ÿÑ% q°ý|È R;ÿÑ%

(cuttot) ýþ>,ómÈ tot tot con

Ç ¨

ýþ> Q5ÿÑ%

! tot

¨ ¨ ¨

Qÿ % q@ýþ> QTÛlR;ÿÑ% qÔý|È R;ÿ§%

(cutcon) ýþ>,ÛmÈ con con tot

Ç ¨

ýþ> Q5ÿÑ%

! con

¨ ¦ ¦¨ ¦|¨ ¦

¦©¨«ª

ýþ> QTó R ÿÑ% qkû%¯ÓÊdýþ>æómÈ Q5ÿ % q@ý|È R ÿH%

(CUT ) Ü tot tot con

Ç ¨

ýþ> Q5ÿÑ%

! tot

¦y¨ ¨ ¦ ¦|¨ ¦

¦©¨«ª

ýþ>,ó2Ü È Qÿ % qkí%¯ÓÊý§> Q ósR ÿH% q@ý|È R ÿH%

(CUT ) tot tot con

Ç ¨

ýþ> Q5ÿÑ% ! tot

Table 2: The ﬁnitary and inﬁnitary cut rules.

¨ ¨

Ý+ Q!&"©A Ý+ S

if and only if >m!/"©A or . The axioms (con– ) and (con– ) can be

>M(¥M( Q£(Jå

checked by a case analysis. To see that (con–tot) holds, note that if (

"©A Ý¾+ Q*!Ù4 and >! , then and similarly for their roles reversed. The other

axioms are trivially satisﬁed. So this d-frame is reasonable.

¥ ¨ ¥ ¨ ¥ ¨

+hó±"Ý+ "Ý+ "©A+ "©A +hó±"Ý¾+ "Ý+ "©A +

Now consider the pairs "©A , and .

¥ ¨

+ "Ý+ % These meet the pre-conditions of the cut rule, which would require "©A tot. On the other hand, it is easy to check that the other ﬁnitary cut rule, (cutcon), holds in this d-frame.

The order dual of this example shows that (cutcon) also does not follow from reasonableness. Together, these show that the two ﬁnitary cut rules are independent of each other.

Counterexample 5.12 To show that the inﬁnitary cut rules are independent and

strictly stronger than (cuttot) requires a more subtle counter-example. Consider

¥ ¨

ú ®

the d-frame consisting of pairs ® where is an open in the relative Sor-

A Ý¦ ú

genfrey topology on the interval © and is an open in the standard topology

¨ ¨ ¥ ¨ ¥ ¨

A Ý¦ ® ©È R È R È Ý¦

on © . So is a disjoint union of sets of the forms , and

¥ ¨

R¯JBÝ ® ú % ®0ú !Ú4

where A±JZÈpµ Take con if and only if and take

¥ ¨ ¥ ¨ ¨

® ú % # % ® #X!©A Ý¥

tot if and only if there exists a pair con so that

¨ ¨

ú !©A Ý¥ ©A Ý¦ and . Clearly, con is exactly the same as in the spectrum of

with the two given topologies, so it satisﬁes the conditions for reasonableness.

¥ ¨ ¨

ú % ® úa!(©A Ý¦ Also, tot is closed upward and ® tot implies . So con and

tot satisfy the axiom (con–tot). Finally, tot is closed under the logical operations:

¥ ¨ ¶¨$¥ N©¨ N£ ¥ ¨ ¥ N9¨ Ng

® ú ® ú % # #

if tot, then there exist pairs and in con as wit-

¥ N9¨ N£ ¥ N ¨ Nv

#R0P# ® ® ú&0cú nesses; thus witnesses that also belongs to

tot.

¥ ¨3EÎ ¶¨$¥ ¨3E ¥þ¤T¨ ;¨$¥þ¤ N¨ Ng

# % ö ö %

Furthermore, suppose tot. Let

¥ V¤ ¤TN ¥ ¯¤Ng N

# 0 0¯ö

con be the needed witnesses. Then and are disjoint

¨7E ÿ\% and witness that ýþ tot. So this d-frame satisfies the first of the finitary cut

rules.

¨ E ¨ ¨ ¥ ¨

ß ß ß ©<àß

!(© Ý¦ !(©A #AË!©A s! Ý¦

Next, consider , , and for

¨ Ek ¨

¸ ã # !©A Ý¦ ¸ !Q©A Ý¦

positive integers . Clearly, and for each , .

¨3E

ý§ ÿho%

So these fulﬁl the conditions for the rule (CUT ). But we claim that tot.

E& ¨

ö !n©A Ý¥

Consider any open ö in the standard topology for which . Then

¥ ¨ ¤

ö ë Ý¥

must cover some open set of the form . No open in the Sorgenfrey topology that is disjoint from ö can cover the complement of . This example does double duty by showing that the two inﬁnitary cut rules

are independent: because the standard topology is compact, (CUT ) reduces to

(cuttot).

Spatiality is respected by the translation to biframes and back. For this dis-

cussion, we need to consider the dual adjoint functors between biTop and biFrm. R

We use subscripts ø and to distinguish these from the dual adjunction between

¥ ¨ ¥ ¨ ¨

$< ! SW$< $<

biTop and dFrm. That is, á yields a biframe,

¤ ¤ ¤

¥ ¨

~ ©<

whereas á yields a d-frame, similarly for spec.

2!w+há# Checking deﬁnitions, one sees immediately that á . An equally

easy exercise in type checking for the adjunctions involved shows that spec is naturally isomorphic to spec . This leads to the following corollary.

Corollary 5.13 If a d-frame K is spatial, then it is derived from a biframe.

This also provides an alternative proof of Lemma 4.6.

We have found no application for (cutcon) in anything that follows, but note

that in the alternative construction of the free biframe over a d-frame, the condition ¢ (bif–tot) in the deﬁnition of the elements of is formally similar to (cutcon). In

fact, if a reasonable d-frame satisﬁes (cutcon), then the set con itself is the least

¢ K element of . So for a reasonable d-frame , (cutcon) is a necessary and sufﬁcient

condition for con to be reﬂected in the translation from K to a biframe and back.

It is tempting to claim “by symmetry” that (cuttot) is likewise sufﬁcient for the reﬂection of tot, but a closer look shows that the Scott closure of con is needed. Since tot is an upper set and can not be Scott closed (except in a trivial d-frame), the symmetry fails.

6 Regularity and compactness

Deﬁnition 2.13 can easily be adapted for d-frames:

¥ ¨ ¨ nn£

Deﬁnition 6.1 Let F§«fgf con tot be a reasonable d-frame. For two elements

¨ N N NõÁ

> % F > > > > Q% F <

> we say that is well-inside (and write ) if there is

N9¨ ¨

Q5ÿ% ý§> Qÿ%

such that ý§> con and tot. To avoid lengthy verbiage, we will

¦3¡ ¢ usually write for the “witnessing” element Q (although it is not uniquely

determined). On FH< the well-inside relation is deﬁned analogously.

A d-frame is called regular if every >'%¯F is the supremum of elements well- ¤

inside it, and similarly for elements of F§< .

We note that the elements well-inside a ﬁxed element > of a reasonable d- S

frame form a directed set; this follows from (con– S ) and (tot– ). That they are

Ý Ý all below > is a consequence of (con–tot). Finally, is always true.

As an exercise in reasoning with the logical structure of a d-frame K , consider

)

the following deﬁnition: for elements and ) , say that is well-inside (and

Á Á Á

) < )8<

write ) ) if and only if and . Then one can easily show

¤ ¤ )

that the elements well-inside ) form a directed set with as an upper bound, and )

that L is regular if and only if every ) is the supremum of elements well-inside .

Á 7 7 7

% S %

Moreover, ) holds if and only if there is a so that con, con,

7 7

s % )qS % ) tot and tot. Regularity allows us to derive a lot more information about d-points. This will come in handy later, so it is useful to formulate a couple of lemmas.

54 >¯%¯F

Lemma 6.2 Let K be a reasonable d-frame and . Deﬁne

¥ É ¨ ¥ ¨

^`!/".R%rF <( È¯3Ik>£ÊOý|È R;ÿH% + > ^`!&".R%'FH R;ÿT3% +

P > con and C tot

¥ ¥

1. P > C ;

¥ ¥

Ü > !Ü >

2. If K is regular then P C .

R;ÿ/%

Proof. (1) is a direct consequence of (con–tot): if we have ý|È con

R;ÿH% ÈqIk>

and ýþ> tot then follows.

NÁ ¥ ¦

R% > >

For (2) let R C . The witness cannot be below as otherwise we

¨ ¦ v ¨ ¦ v ¨ N

R;ÿ% ý R;ÿ% ý R ÿ%

could conclude ýþ> tot from tot. We also have con

N ¥ ¥

% > Ü > R

and so ﬁnd that R P . By regularity, P is above itself. It follows that

¥ ¤£ ¥

> > Ü

P Ü C , and by (1) the two suprema are in fact the same.

¥ ¨ ¨

FÑÂfgf n`n£ ¥ %RF

Lemma 6.3 Let K*! con tot be a reasonable d-frame and ,

¤ ¤ ]<¯%¯FH<

¥ . Consider the following statements:

¥ ¥

]<Ë!o¦§.¨ ¥ ¥ !r¦§.¨ ¥]<

(i) ¥ C and C ;

¤ ¤

¥ ¨

Í Í

Y ¥ F <ÎY ¥]<

(ii) F is a d-point;

¤ ¤

¨ £ ¥ £ ¥

ý2¥ ¥

(iii) tot, Üh÷ P , and P ;

¤ ¤ ¤

¨ £ ¥

ý6¥ ¥

(iv) tot and ÷ P ;

¤ ¤

¨ ¥

6¥ ¥

(v) ý is a maximal element of tot.

¤ ¤ÑG

The following are true:

Ç Ç Ç Ç

1. (i) (ii) (iii) (iv), and (i) (v). Ç

2. If K is regular then (iv) (i). Ç

3. If K satisﬁes the (cuttot) rule then (v) (ii).

Ç ¨

Qÿm% >Ð3I©¥ Q3Iª¥<

Proof. Part (1), (i) (ii): If ýþ> tot then either or as

¨ ¨

6¥ ¥ Q5ÿ% >*3I«¥

otherwise we would have ý tot by (tot– ). If con and

¤ ¤

¥ ¥

¥ ¥ QI¥<

then QW% P C by the previous lemma; hence . Thus we have

¤ ¤

¥ ¨

Í Í

Y ¥ F < Y ¥]<

shown that the pair F satisﬁes conditions (dptot) and (dpcon) for

¤ ¤

d-points and it remains to show that we have two completely prime ﬁlters. This

¥< Û ¥<1!-QæÛrQ S

will hold if ¥ and are -irreducible. So let ; by (tot– ) either

¨ ¨ N N

6¥ Q5ÿTo% ý6¥ Q ÿho% Qæ!r¥< Q !o¥]<

ý tot or tot, which means that either or .

¤ ¤

Ç ¨

I'¥ ý§> Qÿ% QVI¬¥<

(ii) (iii): If >k3 and con then by (dpcon). So we have

£ ¥ ¨ ¥

]< Ü ¥ ý6¥ ¥]

¥ P . tot follows from (dptot). The set P is directed

¤ ¤ ¤

Y ¥

because F is a ﬁlter and (con– ) is assumed for reasonable d-frames.

¤ ¤

Ç Ç

(iii) (iv) and (i) (v) are trivial.

Ç £ ¥

¥< Ü ¥

Part (2), (iv) (i): On the side of FÑ< we already have C by

It¥

the previous lemma. For the other side side, assume >k3 . By regularity there

NíÁ N N ¨¦¦3¡ . ¦3¡ .

> > I©¥3 ýþ> ÿ2% I©¥<

is > with . Because of con we have by

¨¦J3¡ . ¨

ÿ,% ýþ> ¥<8ÿæ% [

assumption, and then from ý§> tot we infer tot by (tot– ). It

¥ ¨

< ¥ ý6¥ ¥]

follows that C ¥ . Together with tot this is exactly (i).

¤ ¤

Ç Ç

¥]< Û

Part (3), (v) (ii): As in (i) (ii) we get that ¥ and are -prime, and

¥ ¨

Í Í

Y ¥ F <ÑY ¥]<

that condition (dptot) is satisﬁed for F . In order to show (dpcon)

¤ ¤

Qÿ % >R3IQ¥ Qp3IQ¥<

assume ýþ> con. If we had and then by (the contrapositive

¨ ¨

2¥ ¥]<æómQÿ3% ý6¥ óË> ¥]

of) the (cuttot) rule we would have either ý tot or tot,

¤ ¤

6¥ ¥

contradicting the maximality of ý .

¤ K Proposition 6.4 For K a regular d-frame, the bitopological space spec is

order-separated (cf. Deﬁnition 4.9). J<

Proof. The specialisation orders J and on the spectrum are given by inclu-

¥ ¨ ¥28 ¨38

eû< <

sion of ﬁrst, respectively, second component of d-points. Let e ,

¤ ¤

8 8

3 >%Ôe Y

be two points with e . There is some and by regularity

¤ ¤ ¤ ¤

NÁ 8 ¦J z

> e Y

some > also belongs to . The corresponding witness must

¤ ¤

< J

lie in

¥ ¨ ¥ Nv ¥68 ¨38 ¥þ¦¥3v .©

eM< %&6 > < %&6=<

the same witnesses show that e , , and

¤ ¤ ¤

¥ Nv ¥§¦¦3¡ z©

> 0¯6=< !½4

6 . ¤

56 The frame-theoretic version of the Hofmann-Mislove Theorem, cf. [GHK ¤ 03, Corollary V-5.4], states that a Scott-open ﬁlter in a frame is equal to the intersection of a compact collection of completely prime ﬁlters. Assuming regularity and

one of the inﬁnitary cut rules, we have an analogue for d-frames:

Lemma 6.5 Let K be a regular d-frame that satisﬁes the inﬁnitary rule (CUT ).

Íe

Assume that is a Scott-open ﬁlter in F and is a completely

¤ ¤

prime upper set in FH< such that:

Ç »

(hmcon) con ! or

¤ ¤

Ç »

! %

(hmtot) % tot or

¤ ¤

Then the following are true:

É » ¨

+ b< Ü ".R ( Èq% Ê$ý|È R;ÿH%

1. <! con , that is, is uniquely determined

¤ »

by .

» ¨ »

ý|È

2. !&"©È¿( tot , that is, is uniquely determined by .

¤ ¤

» ¥ ¨

x > < %

3. >¯I con.

¥ ¨ »

eM< % K *e x eM<¯b<

4. For any point e spec , .

¤ ¤ ¤

» ¥ ¨ »

e eû< *e

5. is the intersection of all e where is a point and .

¤ ¤ ¤ ¤ ¤

¥ ¨

< eü< e eû< eü

6. b is the union of all where is a point and .

¥ ¨ » ¥ ¨

^! " e eû< ( ½e + ! " e eM< (¾eû<¿Lb<:+

7. The set #

¤ ¤ ¤ ¤

,í<

is , -compact saturated and -closed in the bitopological space

¥ ¨

¨, ,ü<

spec KT .

Proof. (1) The element < can not be any smaller because of (hmcon). For

Á ¦®¯ .° »

the converse assume Q . The corresponding witness belongs to

É » ¨

.R ( Èq% Êbý|È R;ÿÑ% + <òI

by (hmtot) and so Q%j" con . By regularity, then,

É » ¨

+ .R ( Ès% Ê$ý|È R;ÿH%

Üh÷Â" con .

» ¨

ý|È

(2) Because of (hmtot) it is clear that must contain all È%'F with

¤ ¤

» »

tot. For the converse let >½% . By regularity and Scott-openness of there

¤ ¤

Á » ¦J3¡ z

> <

is > still in . The corresponding witness must be below because

of (hmcon), but then ýþ> tot by (tot– ).

» ¥ ¨

ÈP% > R %

(3) Assume >kIòÈ for all . By (con– ) we have con for all

É » ¥ ¨ ¥ ¨

Rh%p".R ( Èq% Ê È R % + > < %

con , so con by (con– ÜÎ÷ ) and part (1). For

¥ ¨ » ¥ ¨

< % ÈË% > < %

the converse, remember that È tot for all by (2), so con ¤

implies >¯I*È by (con–tot).

¥ » ¥ I±

! Ü F YÎe Be ¥

(4) Let ¥ . From and (hmcon) we get P

¤ ¤ ¤ ¤ ¤ ¤

¥ ¥ ¤£

FH<ÎY#b<

, so ¥ P and hence .

²b< ¥<¯!ÔÜ F <,YeM<

Starting with the right hand side, eí

¥ » ¥ » »

]< 0 !4 ¥ !RÜÎ÷ ¥< % 3 *e

(hmcon) we get P ¥ . So P and hence .

¤ ¤ ¤ ¤ ¤

» » %

(5) Assume that >@3 . Because is assumed to be Scott-open, we can

¤ ¤ >

apply Zorn’s Lemma to obtain a maximal element ¥ above that does not be-

^! F Y ¥

long to . The set e is a completely prime ﬁlter that sepa-

¤ ¤ ¤ ¤ »

rates > from , and it remains to show that it is the ﬁrst component of a d-

¥<

point. According to Lemma 6.3 the right candidate is eí<!)F

¥ ¥ ¥

¥]<2!&Ü ¥ !/Ü ¥ ¥ <¯Ir¥]< <'%

÷ P C . Note that as C by (hmtot). Using

¤ ¤ ¤

6¥ ¥]

Lemma 6.3(iv) we only need to show that ý tot. For this we employ

¨ ¨

ý|È R;ÿ% ÈP% e ý6¥ ó'È ¥

(CUT ): for all con with we have tot by (2); if

¤ ¤

¨ ¨ ¥ ¨

6¥ ¥

it was the case that ý P tot, then tot

¤ ¤ ¤ ¤

would follow, contradicting (hmtot).

²b< bT<

For part (6) let Qk% . By regularity and the assumption that is com-

Á ¦® .® »

Q b<

pletely prime, some Q also belongs to . The witness is not in

¦® ³ .®©¨ N

Q ÿ*%

because of ý con and assumption (hmcon). By part (5) there is a

¥ ¨ ¦® z® »

eû< eü

point e that separates from . By (4) we have that and

¤ ¤

¦® v .®¨

Qÿ % Qs%'eü< because of ý tot it must also be the case that .

Finally, consider the last claim; the two descriptions of # are equal because

¥ »

> # 6 >¯%

of (4). Any , -open neighbourhood of has the form with by (5).

¤ ¤ ¤

, < FH<üYb<

It follows that # is -compact. Only the maximality of in is required

6< <

to see that # is the complement of .

¥ ¨ ¨

FÑÂfgf n`n£ Theorem 6.6 For a regular d-frame K! con tot that satisﬁes the in-

ﬁnitary cut rule (CUT ) there is a one-to-one correspondence between

F å Ê`å fgf ÜÎ÷ 1. maps ´ from to the four-element d-frame which preserve , , con,

tot, and the logical operation , and K 2. subsets # of spec which are compact saturated in the positive and closed

in the negative topology.

» ¥

<àß

!«´ fgf 0¯F

Proof. Given a map ´ as described in part (1), consider and

¤ ¤

¥ ¥|» ¨

<àß

<¯!«´ nn 0¯F < b<

b . It is straightforward to show that the pair satisﬁes ¤ the assumptions of Lemma 6.5. The translation in the opposite direction is equally easy.

A few comments on this result are in order: Given a consistent predicate a ,

±% ´ a fgf n`n C

that is, a con, then the value of at can only be , , or . The ﬁrst outcome

a # indicates that all elements of # satisfy , the second that some element of

fails a , and the last that neither holds (which is a possibility because a consistent

predicate does not need to be Boolean). This means that ´ acts like a universal quantiﬁer for partial predicates.

Generally, one would expect a universal quantiﬁer to preserve fgf but not neces- #

sarily n`n , because could be the empty set. Also, one would expect it to preserve S conjunction ( ) but not disjunction ( ), and certainly one would not want it to be

inconsistent (returning D ) for a consistent predicate, or to be undecided (return-

ing C ) for a total predicate, that is, one expects it to preserve con and tot. Ü The preservation of ÷ can be seen as a computability condition on the uni-

versal quantiﬁer: If a (partial) predicate a is the directed supremum of (partial)

¦ a

predicates a , and if the universal quantiﬁer applied to returns a deﬁnite answer, nn

that is, either fgf or , then computability requires the same answer to be obtained ¦ from an approximant a already. All in all, then, Theorem 6.6 is a generalisation of Mart´ın Escardo’´ s theory

of computable quantiﬁcation on topological spaces, [Esc04], to a logic in which fgf predicates are allowed to have value n`n as well as .

Let us now turn to a notion of compactness for d-frames.

¨ E

Ü ÿh%

Deﬁnition 6.7 A d-frame is called compact if whenever ýÜp tot, then

N ¨ EN N E N E

Ü ÿÑ% k ýyÜ± tot for some ﬁnite subsets and .

59 Lemma 6.8 A reasonable d-frame K is compact if and only if the set tot is Scott-

open in F .

Deﬁnition 6.9 A distributive continuous lattice is called stably continuous if its

¨ N Ç

Ý > ù Q Q > ù way-below relation is multiplicative, that is, Ý¯ù and N 15

QÛmQ . F

Lemma 6.10 For K a compact regular d-frame, the well-inside relation on

F F <

(respectively, FH< ) is the same as the way-below relation. Furthermore, and ¤

are stably continuous lattices.

NdÁ N

> > > ù > >IÔÜ #

Proof. Let ; we show that also holds. Indeed, assume ÷ ;

¨¦¦3¡ z ¨¦¥3v .

ý ÷ # ÿ§% [ ý|È ¢ ÿH%

then Ü tot by (tot– ). Compactness implies that tot for

N N

¨¦¦3¡ .

ýþ> ÿH% > I*È]¢

some È¢=%¯# , and since con, follows from (con–tot).

N N Á

ù)> > I*È È >

On the other hand, > implies for some because of regularity,

Á > so > as well.

Closure of the well-inside relation against inﬁma on the right follows from

Ý Ý (con– ) and (tot– ); holds in any d-frame.

Theorem 6.11 Compact regular d-frames are spatial.

Proof. We check the conditions (s ), (s < ), (scon), and (stot) of Theorem 5.1.

Ia>R%/F3 F

Let > . Since is a continuous lattice, there is a Scott-open ﬁlter

¤ ¤

» N »

> ¥ >

that contains > but not . Let be maximal above outside , and set

¤ ¤ ¤

¥ ¥

¥]

÷ P . As the complement of tot is Scott-closed, and P is a

¤ ¤ G ¤

6¥ ¥]

directed subset of it, we have ý tot. By Lemma 6.3(iv) we have a d-point

N >

that separates > from .

Q5ÿs3% ¥

For the condition (stot) assume that ýþ> tot. Pick a maximal element

ý6¥ Q5ÿ'3% ¥5<*!

above > such that tot. As in the paragraph above, the element

¥ ¥ >¯IL¥ QsI

Üh÷ P partners up with to deﬁne a d-point. By construction, and

¤ ¤ ¤ ]<

¥ .

Q5ÿ3%

For (scon) let ýþ> con. By regularity and the fact that con is Scott-closed,

N N ¨ ¦3¡ z

> > ý§> Q5ÿs3% (con– Üh÷ ), we must have well-inside with con. The witness

15These structures were called arithmetic lattices in [JS96].

ýþ> Qÿ,% Q3I

can’t be above Q as otherwise con would follow by (con– ). So

¦¦ . ¥ ¨

eû<

and we can apply the ﬁrst part of the proof to obtain a d-point e with

¦¦ z

%eû<3 > e

Q_%XeM< , . The latter fact implies that must belong to because

¨¦¦3v . ÿH% ýþ> tot.

Corollary 6.12 The spectra of compact regular d-frames are exactly the compact regular bitopological spaces.

The next three results are an immediate consequence of spatiality but it is of some interest that they can in fact be derived without using the Axiom of Choice.

Proposition 6.13 Compact regular d-frames satisfy the cut rules (cuttot), (CUT ),

and (CUT ).

Proof. First of all, the two inﬁnitary cut rules reduce to (cuttot) because of com-

¨ Q5ÿ%

pactness, so this is all that we need to show. Assume, then, that ýþ>só¿È tot,

¨ ¨ N5Á

QTólR;ÿ=% ý|È R;ÿ=% > >

ýþ> tot, and con. By regularity and compactness there are

N\Á N ¨ ¨ N

Q ýþ> ócÈ Q5ÿr% ýþ> Q óµR;ÿP% and Q such that tot and tot are still valid. A semi-formal derivation is best suited for the somewhat involved argument that

follows:

N ¨

ýþ> ómÈ Q5ÿÑ%

Ý tot assumption

¨ N

ýþ> Q ósR;ÿ§%

å tot assumption

ý|È R;ÿ§%

u con assumption

¦ ¦¦3¡ z ¨¦

<Ë!

ýþ> tot regularity,

¨¦

· <ÿ§%

ýþ> con regularity

¦ ¨ ¦ ¦¦® v .®

Q5ÿ % !

ý tot regularity,

¤ ¤

¦ ¨ N

Q ÿ§%

ý con regularity

¨¦ N

<æómQ ÿÑ% [

ýþ> tot from (4) by (tot– )

¨ N ¥ ¦

Q ó RûÛ < ÿ§% S

ýþ> tot from (8) and (2) by (tot– )

¥ ¦ ¨

ý > ómÈ Û Q5ÿ %

ÝbA tot from (1) and (6) by (tot– )

¦ ¨ N

Ý ý|ÈÛ Q ólR;ÿ§%

Ý con from (3) and (7) by (con– )

N N

¦ ¨ ¦

ýþ> Û Q ó

Ý$å con from (5) and (7) by (con– )

¥ N ¦ ¨ N ¥ ¦ ~

Ju ý ÈóË> ÿõÛ Q ó RMÛ < % S

Ý con from (11) and (12) by (con– )

ý|ÈóË> ÿõÛ Ik>

Ý from (13) and (9) by (con–tot)

ýþ> Q5ÿÑ% [ Ý tot from (14) and (10) by (tot– )

Lemma 6.14 Let K be a compact regular d-frame. »

1. To every Scott-open ﬁlter in F there exists a unique completely

¤ ¤

< FH< prime upper set b in such that the conditions (hmcon) and (hmtot) of

Lemma 6.5 are satisﬁed.

T< F <

2. To every completely prime upper set b in there exists a unique Scott- »

open ﬁlter in F such that (hmcon) and (hmtot) are satisﬁed.

¤ ¤

3. The translations in (1) and (2) are inverses of each other. h<

Proof. Lemma 6.5(1) states that there is a unique candidate for b , namely

É » ¨

Íe

F <Y < + .R%'F

, where

Q5ÿ% QVI <

is satisﬁed by construction, so let us look at (hmtot). If ýþ> tot with ,

NhÁ ¨ N N

Q ýþ> Q ÿp% Q then let Q with tot, too. The existence of is guaranteed by

N N

ù Q Q I²R

regularity and compactness. By Lemma 6.10, we have Q , and so for

É » ¨ ¨

T%p".R%'F

some R con . Now we know that con for

» »

ÈqIk> >V%

some Èq% , and since by (con–tot), we obtain as required.

¤ ¤

¥ »

< ! F

For the second statement let Ü and deﬁne

"©È,%rF ( ý|È

tot + as prescribed by 6.5(2). This is a ﬁlter by (tot– ); it

¨ »

Q5ÿ2% È*%

is Scott-open because of compactness. If ý|È con for some , then

< QsI by (con–tot). Part (3) is an immediate consequence of uniqueness. The preceding lemma allows us to conclude a frame-theoretic analogue of

Theorem 2.16 characterising stably compact spaces in bitopological terms:

¥ ¨ ¨

FÑÂfgf n`nõ Theorem 6.15 A d-frame K*! con tot is compact regular if and only if the following conditions are satisﬁed:

(i) F is a stably continuous lattice;

¤ F

(ii) F < is isomorphic to the Lawson dual of , that is, the set of Scott-open ¤

ﬁlters of F ordered by inclusion;

¨ ® ®

QÿÑ% >rIÈ Ès%Pe e (iii) ý§> con if and only if for all , where is the Scott-open

ﬁlter associated with Q according to (ii);

¨ ®

Q5ÿ % >V%'e (iv) ýþ> tot if and only if .

Furthermore, d-frame homomorphisms from K to another compact regular d-

N N

F F

frame K are in one-to-one correspondence to frame maps from to which

¤ ¤ preserve the way-below relation.

Proof. “Only if:” We showed in Lemma 6.10 that F is a stably continu- ¤

ous lattice. From the preceding lemma we get the order-reversing bijections

¥ ¥ » ¥

½¼ b< Q ^`!F Y Q b< Q ¼ Q

Q and which together establish an order-

¤ ¤

ý§> QÿË%

preserving bijection between F and its Lawson dual. If con then by

» ¥ Q

studying the construction in the proof of 6.14(2) one sees that for all ÈV% ,

¨ » ¥

Q5ÿ=% >WIXÈ >WIÈ Èm% Q

ý|È tot, so by (con–tot). Conversely, for all implies

¨ É » ¨

R ÿÑ% R ".R%'FH<( Èq% Êdý|È R;ÿH% +

ýþ> con for all belonging to con , by (con– ). ¤

QÿÑ% ÜÎ÷

This in turn implies ýþ> con by (con– ). The equivalence in (iv) is true by

» ¥

construction of Q .

IÐ> F

“If:” In order to show regularity, assume that >Ô3 in . By continuity,

> È/3I-> e >/%&e

there is È±ù with , so there is a Scott-open ﬁlter such that

¨ ¨

ýþ> Q5ÿ\% ý¥È Qÿ\% Q¿%PF§<

and È2IXe , in other words, tot and con for the element

e Q¦¾

associated with . Compactness is trivial as the ﬁlter associated to Üh÷ is the ¾ directed union of the ﬁlters associated with each Q .

Regarding morphisms, we have that the two components of d-frame homo-

Á è

morphisms preserve , so by 6.10 they preserve ù . For the converse, let be a

F F

ù -preserving frame homomorphism from to , the ﬁrst components of two

¥ ¥ ®Â

K è8< Q ^!*[ è e

compact regular d-frames K and . We set . The result is again

¤ F=<

a Scott-open ﬁlter by the preservation of ù . The frame operations of viewed Û

as the collection of Scott-open ﬁlters of F are: intersection for , directed union

N ¨ N N

Ü "©ÈÛ2È (©Èq%'e È %'e + ó ù for ÷ , and for (multiplicativity of is used in this

formula). With this knowledge, their preservation by è¡< is easily checked.

7 Partial frames

We have alluded several times to the fact that con corresponds to the consistent predicates on the spectrum of the d-frame. From a logical point of view, these are preferable to general elements of the d-frame as they do not give conﬂicting answers. In this section we will demonstrate that it is possible to replace a reasonable d-frame by its set of consistent predicates without any loss of expressivity.

To emphasise that we are dealing with a new structure in its own right, we will

L ¼L denote the set con by ¿ (and the resulting structure by ).

Clearly, the totality relation tot has been just as important as con so far, and L

we need a way to represent it within ¿ . This is the rationale behind the following

¨¨À

%Á¿L

deﬁnition. For a we set

À ¥ À

^ x aû<æó %

a tot

À ¨ ¨

ý6¥ ñÿ ýþ> Q5ÿ and say that a strongly implies . In the context of pairs, holds if

logical order

n`n

fgf

F <

7 7 7 7

! Ûrfgf Ûn`n

¤ C

Figure 3: The structure of the set of partial predicates associated with a d-frame.

ñÿH% ¿¼L

and only if ýþ> tot. Altogether, there is quite a bit of structure on :

K ¿L

Binary inﬁma ( Û ) are inherited from ; they stay in because of (con– ).

Ü ¿õL Ü ÷

By (con– ÷ ), is closed under directed sups .

nn S ¿¼L

Because of (con– fgf ), (con– ), (con– ) and (con– ), contains the two S

constants and is closed under and .

Á¿L

We also have the constant C°% .

Û Ü I S

and ÷ induce the information order ; and induce the logical or-

der J . The strong implication was introduced above.

Figure 3 presents a graphical representation of the set of consistent predicates

as a sub-structure of a d-frame.

¥ ¨ ¨

FÑÂfgf n`n£

Proposition 7.1 Let K*! con tot be a reasonable d-frame.

¥ õL£ÂI

1. ¿ has binary meets, directed suprema, and a least element (denoted

Û C

by , Üh÷ , and , respectively). Meets distribute over directed suprema.

¥ ¨ ¨ ¨

õL£¶ S fgf n`n 2. ¿ is a distributive lattice.

3. The relation is contained in J ; it is transitive and interacts with the

logical operations as follows:16

n`n

(n`n – )

fgf

( – fgf )

N N

7 ¨û7 ÆqÇ 7 7

E E S E

( S – )

N N

7 ¨û7 ÆqÇ 7

E E EÑ2E ( – )

4. The following mixed laws hold:

7 ¥§7 ¥ ¥§7

! ÛmE! VC S EÑ¯C S VE

( Û – )

7 7dN ¨ N ¨M7 Ç 7N N

I EÎI*E E ! E

( – I )

÷ C ¿L

Proof. (1) We have a dcpo by (con– Ü ). The element is a member of F because CÔI/fgfÑ% con. Meet-continuity is inherited from the frame .

(2) This is immediate from the “logical” axioms.

7 ¨7 7

ý|E

Regarding (3), let E ; so tot by deﬁnition. Since

¤ ¤

7 7 IÏE

% con, follows from (con–tot). In the same way, one concludes

¤ ¤

7 £ 7

E$< E2% E ë

< from con. Transitivity uses a similar argument: if then

7 7

I*ë <,óËE % <æó2ë % [

E and tot. So tot by (tot– ).

¤ ¤ ¤ ¤

7 ¨

ý Ý$ÿ% [

Since n`np! tot in any reasonable d-frame, we have tot by (tot– ).

7 78N

S E E

This shows (n`n – ). Regarding ( – ), assume and . This means

7 7

<æó2E % ó2E % S ó

tot and < tot by deﬁnition. By (tot– ) and distributivity of

¤ ¤

¥§7 7

S S

and Û over , we have tot as well. The reverse direction is an ¤

application of (tot– [ ). The proofs of the other rules are similar. (4) The ﬁrst law is precisely the second part of Proposition 3.2. The other law

follows from (tot– [ ).

N a

For two partial predicates a and to be related in the information order, that

N N

Ipa a a

is, a , means that will always give the same answer as , whenever the

I latter gives an answer at all, and may answer where a doesn’t. Rule ( – ) can

16We re-use the labels of Deﬁnition 2.18 because the axioms are formally the same, but note

{ x Ò

that in the present situation we neither have ( Ò – ) nor ( – ). On the other hand, here we have hÃÅÄ Ò which is not a requirement for strong proximity lattices. The exact relationship between the two is explored in Section 8.1.

therefore be read as saying that with a , every (information order) reﬁnement À

of a implies every reﬁnement of . Hence the terminology “strong implication.”

¥ ¨ ¨ ¨ ¨ ¨

-! ¿, Û Ü Cæ S fgf nn Deﬁnition 7.2 A structure ÷ which satisﬁes the properties of Proposition 7.1 is called a partial frame. Morphisms between partial frames preserve all operations and the strong implication relation. The resulting category is denoted by pFrm. Before we embark on the proof that every partial frame arises as the struc-

ture L of a reasonable d-frame, we note some consequences of the axioms.

Lemma 7.3 Let be a partial frame.

N N N N

7 7 7 7 ¨

EÑSVE E

1. Whenever E , then also for arbitrary .

2. Any one of the operations Û , , and distributes over any other.

S I Û J 3. The operations and are -monotone in each argument, and is -

monotone.

¨7À 7 ¥ 7: ¥0À 78 ¥ À

I a¯S S aVS

4. If a then is the least upper bound of

¨7À ¨7À

za + Æ¨ÇÉÈ Ê"za +

" in the information order, denoted by . Furthermore, for

¨7À ¨ À

E E=ÛsÆ¯Ç¢È "za +!rÆ¯Ç¢È "©E=Ûga E§Û + Ê

any , Ê .

¥ æÂI

5. ¿ is bounded-complete, that is, every bounded subset has a supremum.

7 7 7

m¿ !&"©Eq(©EhI +

Furthermore, for every % , the set is a frame.

Í fgf

6. In the frame Ê , logical order and information order coincide; in particu-

Û S Æ¨Ç¢ÈÊ Ê]n`n lar, the coincides with , and with . In the frame , the logical

order is the opposite of the information order; in particular, coincides

S Û

with Æ¯Ç¢È=Ê , and with . n`n

7. The logical constants fgf and satisfy the laws:

7 7

ÛrfgfÖ! S¯C

7 7

ÛÎn`n ! ¯C

C ! fgfÛn`n

7 7 ¨M7

! Æ¨Ç¢È " Ûrfgf Ûhn`n£+

8. Information order and strong implication are related by:

N N

7 ¨û7 Ç 7 7

E E ! Û E

7 ¨û7 N Ç 7 N

E E ! E§Û2E

7 ÆÇ 7

E Ûhn`n E=Û'fgf

7 7 7 £

Û¯fgfÑI*E\Û'fgf Ûhn`n E=ÛÎn`n

9. J½E if and only if and . S

10. The operations and are Scott-continuous in each argument.

Proof. The ﬁrst statement is a consequence of ( S – ) and ( – ), because

7 ¥§7 7N£ ¥ Nv

E!REH EÑS2E S and .

The second statement is part of Proposition 3.2 but in any case only requires

¥ ¨

, S

some straightforward computations in the distributive lattice ¿ .

N N

7 7 7 7 7

Û !

The third statement holds because of (2), e.g., if I then and

N N N

7 ¥97 7 ¥97 ¥§7 7 7

Û 2E! ¿E Û VE ¿EÎI VE ¿E! , hence .

(4) To see that the given expression is an upper bound, compute

¥¥ 7: ¥2À 7: ¥ À

aVÛ amS S amS !

¥¥ ¥ 78 ~ ¥¥ ÀH ¥ 78 ¥¥ ¥ ÀH ~

a¿Ûga S a¿Û a¿Û S a¿Û aVÛla S aVÛ !

¥ ¥¥ À ¥ ¥ ÀH ~ ¥¥ À

amS4a a¿Û SIa amS a¿Û !²am a2Û S4a !²a

À ¥ 78 ¥0À 78 ¥ À

a aWS S a'S I

Next, if E is an upper bound for and , then

¥ 7: ¥ 7: ¥ ¥ 78

E¼S E¼SæE ! E¼ S,E!½E E E¼S . Finally, let be an arbitrary element

of ¿ ; we get

¨7À ¥¥ 7: ¥2À 7: ¥ À

E§ÛsÆ¨Ç¢È "za +!E=Û a2S S amS !

¥~¥ ¥ 78 ¥¥ ÀH ¥ 7: ~ ¥~¥ ¥ ÀH ~

E=Ûsa S E=Û E§Û S E=Û E=Ûga S E\Û !

¨ À

Æ¯Ç¢È "©E=Ûga E§Û +

7 ¨ À

"©E§Ûga E=Û + where the last step holds because E\Û is an upper bound for .

(5) The supremum of the empty set is C . A nonempty set is the directed

union of its ﬁnite subsets, so bounded-completeness follows from the previous

¥ æÂI item and the fact that ¿ is a dcpo. It is then automatic that the elements below some ﬁxed bound form a complete lattice; the frame distributivity law is

68 satisﬁed because inﬁma distribute over directed suprema by assumption, and over

ﬁnite suprema by the previous item.

¨ N ¨ N ¥ ¥ N

a IÖfgf Æ¨Ç¢ÈÊT"za a +P! aPSkfgf a S1fgf

(6) Let a . By (4) we have

¥ Nv ¥ N£ N

WSËa !fgfüpfgfü aPSÅa !ªaWSËa

a and we can conclude that information

Í ]n`n

order and logical order coincide. In the frame Ê the analogous calculation reads

¨ N ¥ ¥ N ¥ N£ N ¥ N£ N

Æ¯Ç¢È "za a +! aæSn`n a Sn`n aæSa !²aa aæSa !taæa Ê from

which it follows that the logical order is the opposite of the information order.

7 ¥97

! Û±fgfæ! ¯C S

For (7) we only need to compute according to ( Û – ):

¥ ¥97 ¥§7 7 7

S Wfgf ! VC SpC°S ! S1C

fgf£¯C , etc. The last equation re- 7

quires the formula for the supremum given in (4), with as the upper bound:

7 ¨7 ¥¥97 78 ¥¥97 78 ¥¥97 ¥97

Æ¯Ç¢È " Ûrfgf ÛÎn`n£+ ! S¯C S 'C S S¯C ¯C !

7 ¥§7 7

VC !

S .

7 7dN 7 7N

E æC E E

(8) One ﬁrst uses (1) to obtain ,C , , and from the

assumptions. The left hand sides are put together by ( S – ), and one obtains the

7 7dN

expression for Û there. The second implication is completely analogous if one

¥§7 ¥ ¥§7 ¥97 ¥ ¥97

S EC S ,E ! S,C ES,C SæE remembers the equality C quoted in Proposition 3.2.

In the last law in (8), the direction from left to right is an application of what

fgf I we just showed plus (n`n – ) and ( – ). The other direction follows from ( – ).

(9) The direction from left to right is trivial by (3) and (6). For the converse

¥§7 ¥§7 ¥ ¥97 ¥ 7

Û'fgf ! Û'fgf E=Ûrfgf ! Ûrfgf Û E\Û'fgf ! Ûrfgf

we use (6) to get VE

¥§7 ¥97 ¥ 7 ¨ 7

Ûn`n_! Ûn`n EÛsn`n !*Æ¯Ç¢È=ÊT" ÛÎnn E=ÛÎn`n£+Ë! Ûqn`n and WE . Two

applications of the last law in (7) complete the proof.

I ¿

(10) Let Ì be a -directed set of elements of . We have

¥ 78

¨¥Î

fgfÛ2Ü Û Ü a2 !

vÍ ÷

÷ ( distributes over and )

¥¥ ¥ 78 ~

¨¥Î

Ü fgfÛla fgfÛ ! ¡Í

÷ (6)

¥¥ ¥ 78 ~

¨¥Î

ÜÎ÷ fgfÛla Û fgfÛ !

Í (distributivity)

¥ ¥ 78

¨¥Î

fgfÛsa Û fgfõÛ2Ü ! vÍ

÷ (6)

¥ ¥ 78

¨¥Î

fgfÛsa fgfõÛ2Üh÷ !

Í (distributivity)

¥ 78

¨¦Î

fgfÛ a2VÜ

÷¡Í

¥ 78 ¥ 78

¨¦Î ¨¥Î

n`n ÛsÜ aæ !±n`n Û aqÜ

ÏÍ ÷ÏÍ A similar computation shows ÷ and the last

law in (7) completes the argument.

KjÐ ^!

Proposition 7.4 For a partial frame consider the structure

¥ ¨ ¨ F <£

F con tot where

F ^! © C fgf d!"zac%Á¿ô(CÔIoaI/fgf;+

¨ À À

F < ^! © C n`n O!" %Á¿ô(CÔI Icn`n£+

¥ ¨7ÀH É 7 ¨7À 7

% ^ x %4¿§Êa I

a con

¥ ¨7ÀH À

% ^ x a a tot

Then

KÐ ML¢Ñ

1. is a reasonable d-frame and ! .

K KAÐ\Òð K

2. If is a reasonable d-frame then ! .

¨ ¨

F < ýfgf CTÿ ýyC n`n8ÿ

Proof. (1) F and are frames by Lemma 7.3(5). The pairs and

¨ ¨

CÎ+ "¾C nnõ+

are in con because "fgf and are (trivially) bounded. They are in tot by

nn [

( fgf – ) and ( – ). Condition (con– ) is trivially satisﬁed, and (tot– ) reduces

¨7À 7 N ¨¨ÀüN 7dN

"za + "za +

to ( – I ). For (con– ) assume that is bounded by and by .

N N N N

7 7

PÛha !©arÅa I a a fgf

Then a as and are elements below . Likewise,

N N N N N N

À\¨¨À À À 7 7 7 7 ¨ À\¨¨À

" +! I "zaVÛsa Æ¯Ç¢È=Ê" +]+

Æ¯Ç¢È=Ê . So is a bound for .

Condition (con– S ) is shown in the same way. For (tot– ) assume and

ÀûN N À=¨7ÀûN À\¨¨ÀüN N

Æ¯Ç¢È=Ê " + a Æ¨Ç¢È=ÊT" + a I

a . We get and by ( – ), and

À=¨7ÀüN N

Æ¨Ç¢È " + a¯ÛIa ÜÎ÷

then Ê by 7.3(8). Next consider (con– ); by assumption,

¥97¨ # each pair E in is bounded, and since we are in a bounded-complete dcpo,

the suprema exist and form a directed set. The supremum of the latter is an up-

¥ 7 É 7¨ ¨ É 7 7õ¨

Ü #! Ü " ( EbÊ8ý EÿH%¯#T+ Ü "©E( Êdý EÿH%¯#T+

÷ ÷

per bound for ÷ . Finally,

¨¨À N©¨7À À

a ÿË% ýa ÿË% a

consider (con–tot), so let ý tot and con. This means

À N À\¨ N

Æ¨Ç¢È " a + a I

and that the supremum of and a exists. We get by ( – )

N N N

À=¨ À\¨ À=¨

T" a +ÎJoa Æ¨Ç¢È=ÊT" a +}aP!Æ¨Ç¢È=ÊT" a +

which implies Æ¯Ç¢ÈÊ or . Taking the

¥ À\¨

Æ¨Ç¢È " a +H4a Ûrfgfí!

meet with fgf on both sides yields for the left hand side

¥ À=¨ N ¥ ¨ N N

Æ¨Ç¢È " a +¼Ûqfgf aæÛfgf !rÆ¯Ç¢È "¾C a +õ_aW!²a _a Ê

Ê and for the right hand

N N

a Ioa side just a , so indeed as desired.

¢Ñ

To see that the structure ¼L is isomorphic to , consider the translations

¨ 7 7 ¨7

eP^dØ L¢Ñ ×Ø ý Ûrfgf Ûn`n8ÿ

8 ¨ ¨7À ¨7À

^OL¢ÑhØ ýa ÿû×Ø Æ¯Ç¢È=ÊT"za +

8 ¥ ¨7À %

is well-deﬁned because the pairs a con are bounded and Lemma 7.3(4)

8 8

\e! Ó e ! ÔÕ³Ó Ö³×

applies. We have Id by 7.3(7), and Id by 7.3(4) and (7).

8 8

e C Û e ÷

and preserve the information order, so , , and Ü are preserved by

8 7

Îe JE

and . They also preserve the logical order: in is equivalent to

7 7 ¥§78 ¥97 ¨7 ¥¥97 ¨$¥§7

e ! ÛRfgf Ûnn ! ½E ÛRfgf E ÛWn`n !

EÔ! , so

¥~¥§7 ¥ ;¨$¥97 ¥ ¥~¥§7 ¥ ;¨ 7 ¨

Ûfgf EÛfgf Ûn`n EàÛn`n ! Ûfgf Û EÛfgf Æ¯Ç¢ÈÊT" Ûhn`n E§ÛÎn`n£+ !

¥§78 ¥ ¥978 ¥ 8 ¥ ¨7À ¥ N ¨7ÀûN£

e e E JZe E a J a

, in other words, e . For , assume

N Ào£«ÀûN ¨¨À N ¨¨ÀüN

1I²a Æ¨ÇÉÈÊ"za +,JtÆ¯Ç¢È=ÊT"za + which is equivalent to a and . We get

by the characterisation of J in 7.3(9). Finally, consider strong implication:

N N N

¥ ¨7À ¥ ¨7À ¥ ¨7ÀH

a L¢Ñ a % KÐ

a in is by deﬁnition equivalent to tot in

which is, again by deﬁnition, equivalent to a in . The latter implies

N N

¨7À ¨¨À

Ê Ê

"za + Æ¯Ç¢È "za + I

Æ¯Ç¢È by ( – ), and is also implied by it because of

À ¨¨À N ¨¨ÀüN N

!_n`nÛÆ¨Ç¢È "za + fgfÛsÆ¯Ç¢È "za +!ta Ê

Lemma 7.3(8): Ê .

Ð\Ò F F < F

(2) It is clear that K returns which is isomorphic to , so we only

¤ÑG

¨ ¨

ýþ> Qÿ%o©C fgf

need to check the two relations. Now, in ML we have if and only

¨ ¨ ¥ ¨ ¨ ¨

ý§> QÿH%Ø©C n`n >Ë!RA ýþ> A]ÿ ý|A Q5ÿ

if Q!RA , and if and only if . The pair belongs

Ð ML

to con in K if and only if it is bounded in , which happens if and only if

¨ ¥ ¨ ¨ ¨

Qÿ§% K ý§> A]ÿ ý¥A Q5ÿ KTÐ

ýþ> con in . For the covering relation, belongs to tot in

¨ ¨ ¨

Q5ÿ ýþ> A]ÿ ¼L ýþ> Q5ÿH% K if and only if ý¥A in , which happens if and only if tot in .

Theorem 7.5 The categories rdFrm and pFrm are equivalent.

Proof. We extend the construction of the previous proposition to morphisms, so

¥ ¨ N ¥ «¥

è è8< K K % è !

let èm! be a d-frame map from to . For con we set

¥ ;¨ ¥

èO< < ÿ ML ¼L è

ýyè . This is a well-deﬁned function from to because

¤ ¤

preserves the con-relation. It is easy to check that it preserves all partial frame

è operations, and the preservation of follows because è preserves tot. Thus is a morphism in pFrm.

fgf

I J

n`n

Êå ´¤Ù ^! H ¨Ú

Figure 4: The dualising object å as the partial frame . Note

that strong implication contains only those pairs that are required by (n`n – )

C C

and ( – fgf ), in particular, does not hold.

è èO<

Vice versa, if èÑ^BØ in pFrm, we let and be the restrictions

¨ ¨

©C fgf ©C n`n C fgf nn Û Ü

to and , respectively. Apart from , , , , and ÷ they also pre- è serve Æ¨Ç¢È=Ê because of Lemma 7.3(4), so they are frame homomorphisms. Since

is monotone with respect to I , bounded pairs are mapped to bounded pairs, hence

¥ ¨

èO< KAÐ

è preserves the con-relation on . The preservation of the tot-relation is

consequence of being preserved by è . 7

The translations are inverses of each other because every element of a partial

7 7 Ûn`n frame is the supremum of ÛWfgf and by 7.3(7), and suprema are preserved because they are computed from the logical operations, 7.3(4). Because of this equivalence we can from now on pretend that any partial frame is given concretely as the set con of a reasonable d-frame with the operations

deﬁned as at the beginning of this section. Ê`å

The effect of the equivalence on the dualising object å in dFrm is to chop

¡Ù1^!/H ¨Ú off the top element. Figure 4 depicts the resulting partial frame ´ in the

information order and the logical order.

¥ ¨

e eM<

Points on a d-frame K are given by pairs or by dFrm-maps into the

¥28 ¨38 <

dualising object; the equivalent for the partial frame ûL is a pair of

¤ õL

subsets of ¿ that satisﬁes:

8 8

Â I

and < are disjoint, non-empty, and Scott-open with respect to .

8 8

Â <

With respect to J , is a prime ﬁlter and a prime ideal.

À 8 À 8

ac% < %

Whenever a then either or . ¤

8 8

These properties follow immediately from the characterisation of and < as

¤ nn

the inverse image of fgf and , respectively, under the pFrm-equivalent to the Êå

morphism into å . It is also clear that every pair of subsets with these properties

¡ ¨Ú deﬁnes a pFrm-morphism into . The forgetful functor from pFrm to Set does not factor through the equivalence with d-frames, as the underlying set in general is smaller. Nonethe-

less, the construction of a left adjoint is similar to that in the proof of Propo- e#

sition 5.7. For a set # we again consider the free frame and the d-frame

¥ ¥ ¨ ¨

# ^! e# e#h fgf

K con tot where tot is chosen minimally to satisfy (tot– ),

[ ýþ> Q5ÿæ% >p!ÚÝ QW! Ý

(tot– n`n ) and (tot– ), that is tot iff or , but con is now the

¥ ¨

e# È È È%¯#

smallest Scott-closed subset of eT# that contains all pairs , , and G is closed under the logical operations. It is obvious that the axioms for a reason-

able d-frame are satisﬁed, except possibly (con–tot). For this we need to analyse

K # e# and more carefully.

First of all, the set eT#ÔY2"Ý+ contains the generators and is closed under all frame operations except empty meet. One sees this by studying the concrete

construction of the free frame or by considering the extension of the assignment

Ø A eT# åÎ^!AµXÝ eT#

#R× from to the two-element frame ; only the top element of

eT# p^!" ýþ> Q5ÿ(b>3!Ý,3!RQ+

is mapped to 1 by it. In the frame e# the subset G

is likewise closed under binary meets and arbitrary sups, and it contains the diag-

¥ ¨

È ÈP%k#

onal elements È , . In particular, it is Scott-closed. is further closed

¥ ¨ ¥ ¨

Ý A n`nk! A Ý under the logical operations but lacks the constants fgf! and .

Adding them does not change Scott-closedness (think of taking the union with

Í Í n`n fgf ) but binary suprema can no longer be taken. The logical operations are

ﬁne, though, as one can check by a case distinction. By this consideration we see n`n that the only elements of con that lie above an element of tot are fgf and . So

(con–tot) is valid. Þ

For the extension property let be any partial frame and a function from

¥ ¥

¿ Þ ^OÈ2×Ø Þ È ÛPfgf Þ]<=^8È2×Ø Þ È Û,n`n

# to in Set. We also have and . These

¨ ¨

^deT#Ø ©C fgf¯ èO<=^8e#&Ø © C n`n

lift to frame homomorphisms è and , respec-

¥ ¨ ¥

èO< K #

tively. The pair è is a d-frame homomorphism from to the d-frame ¤

Û ÜÝÜ

Þ Þ

Figure 5: The free partial frame over one generator and its Stone dual.

¥ ¥

<àß

è èO< ç

associated with , because is Scott-closed, contains the diagonal ¤HG elements, and is closed under the logical operations, and so preserves con; it pre-

serves tot because this was chosen minimally.

§ + Figure 5 shows the free partial frame over the one-element set #R!" on the left. Its Stone dual is the three-element bitopological space shown on the right. As we should expect, it has the same number of elements as the dualising partial frame shown in Figure 4. We conclude this section with a characterisation of three special cases of d- frames.

Proposition 7.6 The equivalence of Theorem 7.5 cuts down to one between reasonable d-frames that satisfy axioms (cuttot), (Cut ) or (Cut ) and partial frames

that satisfy the following corresponding Gentzen-style cut rules:

7 7dN N

Ia a2 E

(G-cuttot) EHS and

N N

Ç 7 7

! EÑS2E

N N

7 ¥ 7

EHS Ü 6 ka%r6Ê³a¿ E

(G-Cut ) and

Ç 7 7N N

! EÑS2E

7 ¥ 7N N

ka%r6Ê EÑS4a Ü 6 E

(G-Cut ) and

Ç 7 7N N

! EÑS2E

Proof. Start with a d-frame K that satisﬁes (cuttot), and assume the two hypothe-

7õ¨7ON ¨ ¨ N9¨

E a/%¿£L

ses in Gentzen’s rule for elements E . Unwinding the deﬁnition

7 ¨7 Ç

EÎSàa x ý|E óØa

of and using (tot– [ ), we get tot

¤ ¤

N N

7 ¨ 7 Ç ¨7 7

a' E x ý|E aû

< tot and tot

¤ ¤ ¤

N N

¨ 7 7

ý|E ócE aj% aû

< tot. Since con, we conclude by (cuttot) that

N ¨~7 7dN 7 7dN N

ý|E ómE <ó ÿ§% EÑS2E

< tot or .

¤ ¤

For the converse assume that the partial frame satisﬁes Gentzen’s (ﬁnitary)

À=¨ ¨ À ¨ À\¨

a )PIcn`n ý ó a¼ÿH% ý aó#)ÿ %

cut rule, and let I&fgf and , and assume tot,

)ÿ\% KjÐ

tot and ý con. By deﬁnition of the two predicates on , Proposition 7.4,

À À ¨ 7 7

S a2g) )cI %h¿ I

this means a , and for some . By ( – ) we

À 7 7 À

S aË

infer a , from which Gentzen’s cut rule allows us to conclude

À ¥2À ¨

aû< %

a , or, equivalently, tot.

7 ¥

Ü16

The proofs for the inﬁnitary rules are similar, noting that E¼S if and

¥ ;¨~7

ó Ü 6

only if ý|E tot and similarly for on the left.

¤ ¤ Next we characterise partial frames that arise from regular d-frames (Deﬁni- tion 6.1). We begin with the following observation which is an immediate consequence of the deﬁnitions.

Lemma 7.7 Let be a partial frame. The following are equivalent:

1. a

ÁáÀ

aû<

2. a with witness

¤ ¤

À Á À

aû<

3. < with witness ¤

The following technical lemma relates logical order to information order, and is used in the characterisation of regularity in partial frames and later in Section 9

when we consider symmetric d-frames.

e J

Lemma 7.8 In any d-frame, if Ó is directed and is ﬁltered with respect to ,

)1%pe " Û)p( %'Ó )W%'eæ+ J«) and for each %Ó and , , then is directed

with respect to I .

¨ ¨ N ¨ N

Proof. First note that J implies . Next, if ,

N N N N N N

¥ ¥ £ ¨

Û )Tâ) ! ó ó%)8<ó%) Û%) S Û%)

then < by Proposition 3.2.

7 À 7 À

zaVÛ (¢a +

Lemma 7.9 For any in a partial frame, the set " is di-

7£ã 7Éã 7 rected. Hence the supremum exists (denoted by ). Moreover, I .

N N N N

¨ 7 À\¨¨À 7 À À

a aWSÅa S

Proof. Because a implies by ( – )

7 À 7 À

a a I I

and ( – ), Lemma 7.8 applies. From we obtain and

¤ ¤ ¤

À 7 À ¨7À 7

<¿Iraû< aVÛ !°ýa

<¯I , so .

7 7¢ã 7 ! Proposition 7.10 A d-frame K is regular if and only if for every element

of the associated partial frame ML .

7 Á7

za (za + "za (za +

Proof. By Lemma 7.7, we have that the sets " and

¤ ¤ ¤ ¤

À 7 À À À Á7

are the same, and likewise " . The equivalence

7 7¢ã between regularity and ! is now obvious. Now, compactness (Deﬁnition 6.7):

Proposition 7.11 A reasonable d-frame K is compact if and only if for any two

¥ ¦þ ¦©¨Âª ¥0À ¨Jä

a ¾ ¾

I -directed collections and of elements in the associated partial

¦ À ¦ À

¦©¨Âª ¨ä

¼L Üh÷ a Üh÷ ¾ a ¾ q%&Ó

frame it is true that ¾ implies for some ,

å Å

% .

¦ À

¦©¨«ª ¨ä

Üh÷ a Üh÷ ¾

Proof. The forward direction follows because ¾ is equiva-

¥ ¦§ ¥2À ¨

¦¨Âª ¨Jä

÷ ÷

lent to ¾ tot. For the reverse direction note that in- ¤

stead of arbitrary suprema in the deﬁnition of compactness we can always use

ý # ÿÑ%

÷ Üh÷æ

directed suprema, and that the assumption Ü tot can be rephrased as

¨ ^ ¨

¨ç ¨`

Ü ý|A R ÿ Ü ý|È A]ÿ

÷ ÷

Another way of expressing the characterisation in this proposition is to say

õL ¿õL

that as a subset of ¿ is Scott-open. G

8 From partial frames to distributive lattices

8.1 Removing the information order As Proposition 7.1 and Lemma 7.3 demonstrate, partial frames have a very rich algebraic-relational structure. It is desirable to explore under which conditions a simpler structure can take their place without any loss of expressivity. Speciﬁ- cally, we will aim to get rid of all aspects of the information order, that is, binary

n`n

fgf

7 7

F I <

aM<

C spec

Figure 6: Representing the elements of a partial frame by ideal-ﬁlter pairs.

Û Ü C meets , directed joins ÷ , and the least element , leaving only the logic-related

structure (including strong implication). 7 We take our cue from Proposition 7.10 and deﬁne for an element of a partial

frame ,

7 7 7 À 7 À

za±%4¿ô(za + ^`!" %m¿ô( +

I ^`!/" F 7

It is clear that I is an ideal (in the logical sense), that is, a J -lower set that is

closed under S , and that F is a ﬁlter, that is, a -upper set that is closed under .

7 7dN 7 7dN 7 7dN

I By ( – I ) it is also clear that implies I I and F F , and by 7.10

the converse holds if and only if the partial frame is regular. Figure 6 illustrates 7

the situation. It is worthwhile to note that the positive part of a partial predicate

¤ 7

is captured by I , which is located on the negative side of . The sketch of the

spectrum in Figure 6 indicates why this is the right approach, in that a is the

7 I

condition that guarantees that a .

¤ ¤

So under the assumption of regularity the problem that we set ourselves ap-

S fgf n`n pears to be solved; the operations , , , and , and the relation encode all

information about the partial frame. However, it would be very cumbersome to

¥ ¨ ¨ ¨

æ¶ S fgf n`nõ write down the conditions under which a given structure ¿ can

be guaranteed to have been derived from a partial frame, and would amount to

7 7 a re-introduction of the information order via the sets I and F . Specifying the morphisms would also require that the conditions that pertain to the information order be encoded. Instead, our strategy is different; we will use the fact that there is a construction that, when given a strong proximity lattice (Deﬁnition 2.18), always returns a compact regular partial frame, and deﬁne morphisms between strong proximity

77 lattices that always yield homomorphisms between the associated partial frames. Furthermore, every compact regular partial frame can be shown to arise in this way

(up to isomorphism), and overall this will lead to an equivalence of categories.

ÅAÐ ^!

Lemma 8.1 If is a compact regular partial frame, then

¥ ¨ ¨ ¨

¿,¶ S fgf nn is a strong proximity lattice that additionally satisﬁes /J .

Proof. Comparing Deﬁnition 2.18 with Proposition 7.1(3), we see that there are

only two additional conditions to check, ( – ) and ( – ). We show the ﬁrst:

7 À À

E Ü "za¿Û (\a +

Assume ; by regularity we can write as ÷ ,

¥ ÀH 7 À

cÛ E a I

so by compactness a for some , which by ( – ) À implies that is an interpolant.

Recall that a round ideal of a strong proximity lattice is a nonempty sub-

set ÓX which is closed under , downward closed with respect to , and

Néè

a aÐ%Ó which contains an element a for every . A round filter is defined dually. It was shown in [JS96] that the set of round ideals (respectively, round filters) forms a stably continuous lattice. For our purposes we will need the concrete

deﬁnition of inﬁmum and supremum in these lattices:

N N N N N N

7 É ¨ 7

ÓTÛmÓ !½Ó0¿Ó ÓTósÓ !" ( a%'Ó a %¯Ó Ê amSIa +

N N N N N N

7 É ¨ 7Áè

e½Û2e !½e 0¿e e½óme !&" ( a%'e a %'e Ê am4a +

¥ ¨ À À

e aÔ%½Ó %Re a

We say that Ó is a round ideal-ﬁlter pair if for all , , .

¥ 7õ¨ 78

Because of interpolation and transitivity of , I F is a round ideal-ﬁlter pair 7

for any %V .

¥ ¨ ¨ ¨

S fgf n`n£ ¤ê Proposition 8.2 Let ¶ be a strong proximity lattice. The set

of round ideal-ﬁlter pairs carries the structure of a compact regular partial frame:

¥ ¨

^! n`n fgf

C I F

¥ ¨ ¥ N9¨ N` ¥ N ¨ N£

Ó e Û Ó e ^! Ó0¿Ó ek0Ve

¥ ¨ ¥ ¨

¨ì ¨ì ¨ì

ë ë ë ë

Ü Ó e ^! ã Ó ã e

÷³ë ÷³ë ÷³ë

¥ ¨ ¥ N9¨ N` ¥ N ¨ N£

Ó e Ó e ^! Ó0¿Ó e*ó2e

¥ ¨ ¥ N9¨ N` ¥ N ¨ N£

Ó e S Ó e ^! ÓTómÓ ek0Ve

¥ ¨

fgfÏ^!

F fgf

¥ ¨

^! n`n

n`n I

N N N

¥ ¨ ¥ ¨

Ó e Ó e ^! e 02Ó !R43

Proof. This is very easy to show, except perhaps regularity. For this observe

¥ ¨ À

e a %°Ó %òe

that for a given round ideal-ﬁlter pair Ó and elements , we

¥ ¨ ¥ ¨ ¥ À=¨ À ¥ ¨

Ó e Û a a a

have I a F I F because of roundness, and hence I F

¥ À\¨ ÀH ¥ ¨ À ¥ ¨

a Ó e I F ! I F . The (directed) supremum of these pairs yields , again by roundness.

We remark that the characterisation of can also be given as

¥ ¨ ¥ N ¨ Ng É ¨7À N À

Ó e Ó e ^! a%'e %¯Ó Êæa

Proposition 8.3 Every compact regular partial frame is isomorphic to its par-

tial frame of round ideal-ﬁlter pairs.

¥ ¨ e

Proof. We begin by showing that every round ideal-ﬁlter pair Ó is of the

¥ 7õ¨ 78 7 7 À ¨7À

%¿ ^! "za¿Û (za±%¯Ó %'eæ+

form I F for some . For this let ÜÎ÷ . If

N 7 N À N

a aÛ I(a aÔ%RÓ a %RÓ

a then by compactness for some , hence

aj%°Ó

already. This shows I Ó . To see that the other inclusion holds, let

N¤è À À N À

a %½e a a a Û

and a by roundness. For any we have , so by

À 7 7

Û I a% I

Lemma 7.3(8). Since a , we get I by ( – ).

7 ¥ 7õ¨ 78 Ø The proof is complete if we can show that the function × I F preserves

the partial frame structure. We check each line of the deﬁnition in Proposition 8.2:

C a n`n CÔIcn`n a n`n a n`n\Ûæfgfí!C

a implies because , and implies

by ( – fgf ) and 7.3(8).

7 7

ÛËE a a E I a if and only if and by 7.3(8) and ( – ).

7 7

¨ì

ë ë

Üh÷vë a í% Õ

a if and only if for some by compactness

and ( – I ).

7 7

2E a a E

a if and only if and by ( – ).

7 7ON 7 N 7dN N

a ShE E a ShE

if and only if there exist , E with . The “if”

direction is ( S – ), ( – ), and transitivity of , “only if” is Lemma 8.1

and axiom ( – S ) of strong proximity lattices.

a a fgf

nn and are always true.

7 7

a a E E if and only if there is with .

Strong proximity lattices don’t have to be isomorphic (in the straightforward

sense) if their associated partial frames of round ideal-ﬁlter pairs are isomorphic.

¥ ¨

Indeed, any -sub-lattice of a given compact regular partial frame that

7 7

a±%V a E is dense, in the sense that for any E in , there is with , will

produce a partial frame of round ideal-ﬁlter pairs that is isomorphic to . This is actually a good thing because it allows us not only to drop the information order

but also to restrict to a “basis” of a given compact regular partial frame.

N N

If è§^õØ is a homomorphism of partial frames and if and are

dense sub-lattices of and , respectively, then the restriction of to has no N

reason to return results in . However, because of density (and regularity), an

N N

¥§78 À ¨7À ¨ ¥§78 À

¿ Ü "za¿Û (za %¯ a è +

image è in can still be written as ,

7 %

so this suggests to replace è by a relation that associates with a given the

N ¥§78 N ¥§78

0 è 0 è sets I and F . From previous work, [MJ02, JKM01], we know

that the situation is well captured by consequence relations:

N N

Å îÐ

Deﬁnition 8.4 Let Å be strong proximity lattices. A subset is G

called a consequence relation if the following conditions are satisﬁed:

î¯! áî î¯!rî4

( î – ) and

î n`nsîVE

(n`n – )

fgf îWfgf

( î – )

7 ¨û7N ÆqÇ 7 7dN

î î¿E î2E S îVE

( S – )

7 ¨û7 N ÆqÇ 7 N

î¿E î¿E îVEH¿E

( î – )

7 À Ç ÉÉÀüN N ÀûN NÀ 7 ÀüN

S î¿EÑS ! %V Ê îVEÑS

( î – ) and

7 Ç É N N 7 N

î gahî2E ! a %¿¯Êa a 4a î¿E

( – ) and

N N

¿Ð î Ð

A pair of consequence relations îï , is called adjoint if

G G

oî æî î 3î ï

ï and , where “ ” denotes relational composition.

ï ï

The category Prox has strong proximity lattices as objects and adjoint pairs of

¥ ¨ î

consequence relations as morphisms. We choose the direction of a pair î ï as

Å î

going from Å to , that is, with the direction of and opposite to the direction

¥ ¨

î of ï . Identities are given by the pairs , and composition is component-

wise relational product.

¨ N N

èÑ^£Ø

Proposition 8.5 Let be compact regular partial frames and a

N N

¿ ¿ î ¿ ¿

homomorphism. Deﬁne relations î ð and by

G G

7 ¥§78

Eîð ^ x E è

7 ¥§78 N

î E ^ x è E ð

This is an adjoint pair of consequence relations between the associated strong

Ð Å#Ð

proximity lattices Å and .

¥ ¨

×Ø ÅñÐ èÔ×Ø î î Å

The assignments , ð constitute a functor from the ð category cr-pFrm of compact regular partial frames to Prox. Proof. The conditions for consequence relations follow straightforwardly from

the analogous properties of strong implication . Of some interest, perhaps, is

Eî ð

the argument for the interpolation part of ( î – ): In , or equivalently,

¥§78 7 À 7 À

è ÜÎ÷«"za¿Û (¢a +

E , we are allowed to replace by because is

N regular. Then Scott-continuity of è and compactness of allow us to conclude

N N

¥ ¥2À 7 À ¥

è a Ûè a E è a

that E for some , and consequently or

Eî a I

ð by ( – ). N

The adjointness conditions follow from transitivity and interpolativity of ,

and the assumption that è preserve .

§ N N N N N N §8¥§78

è§^8 Ø a è

For functoriality assume ^O*Ø and . Then is

§:¥978 N§8¥97:

aoî î arî E ð

equivalent to ð . Using ( – ) this is equivalent to for

N 7

EÎ%Á¿ ahî æî=ò some , which is equivalent to ð as required.

For a functor from Prox to cr-pFrm we assign to a strong proximity lat-

óê

tice Å the compact regular partial frame of round ideal-ﬁlter pairs, following

¥ ¨ î

Proposition 8.2. If îï is an adjoint pair of consequence relations between

N ¥ ¨

Å Å î î è

strong proximity lattices and then we let ï be the function which

¥ ¨

e Å

assigns to a round ideal-ﬁlter pair Ó on the pair

¥ ÉÉÀ À ¨ É 7 7

Ó î

in other words, the inverse image of under the relation ï , and the forward image î

of e under . From the properties of consequence relations one readily derives ï

that the result consists of a round ideal and a round ﬁlter. Regarding the connection

À À 7 7

%Ó î E %e

between the two, let a½îï for some , and for some . Since

À 7 7

î E a E

, we get aî=ï and hence by the second adjointness condition. ï

The conditions for a homomorphism of partial frames are easily checked, given

¥ ¨ ¥ ¨

e !@CÐ! n`n fgf

the characterisations in Proposition 8.2. For example, if Ó I F

ÉÉÀ À

¤ê "za1( n`nõÊamî +2! n`n in one shows that ï I : containment of the latter in

the former is clear because In`n is the smallest round ideal. For the other inclusion

aªî n`n aî n`nPî nn a n`n ï

one uses that ï implies , hence by the second ï

adjointness condition. The analogous fact about F fgf is proved similarly. Another

¥ ¨ ¥ N ¨ Ng

e Ó e

case of some interest is the preservation of ; assume Ó , that is,

N N

À 7 À

!X43 a e*0¿Ó aî îï

e 02Ó . By roundness one ﬁnds in which implies

N N N

7 ¥ ¨ ¥ ¨

è Ó e è Ó e

for some %V . This element is the witness that .

Ø ^ Ø Theorem 8.6 The functors Åm^ cr-pFrm Prox and Prox cr-pFrm constitute an equivalence of categories.

82 Proof. The natural isomorphism between a compact regular partial frame

and its collection of round ideal-ﬁlter pairs was presented in 8.3. For a strong

7 ¥ 7õ¨ 78

èÑ^£ÅjØ ÅñÐô ×Ø

proximity lattice Å , consider the map , I F , and deﬁne

7 ¥ ¨

î î õî æî î Ó e î E

ð ð

ð and as before. We already have , so let ,

ð ð ð

¥ 7õ¨ 78 ¥97: ¥ ¨ ¥ ¥ ¨

Ó e è E ! E E

which by deﬁnition reduces to I F !Ïè I F .

78N 7 N

E %e*0 E

This in turn is equivalent to having elements %PÓ0 F , I , and since

7N N 7

E is guaranteed, we get by transitivity. On the other side we know

¥ ¨ ¥ N9¨ Ng 7 N

î æî Ó e Ó e %ce*0¯Ó

ð , so consider which means that there is .

7 À

eÔ0cÓ

By roundness we can expand this to a within which proves

N N N N

¥ ¨ ¥§78 ¥ 7õ¨ 78 ¥ ¨ ¥ ¨ ¥ ¨

e è ! Ó e Ó e îð æî Ó e

Ó I F or .

lûæ^ Ø

We note that the composition Å Prox Prox has a “normalising” effect

Å Å J

on a strong proximity lattice ; while is not required to hold in , it is

ô #Ð true in Å . Nonetheless, our experience in working with strong proximity lattices for describing stably compact spaces has been that the extra freedom afforded by Deﬁnition 2.18 is essential. For more detail see [Keg02]. Let us also have a look at how d-points manifest themselves in the cate-

gory Prox. The dualising partial frame (depicted in Figure 4 on page 72) has

n`n ´

a dense sub-lattice consisting of fgf and only; we denote it with . For a

¥ ¨

î î Å ´

Prox-morphism ï from a strong proximity lattice to , we consider

e ! "zac%V (fgfóî aû+ Ó*! "za%V (¢aî n`n£+

ï and . Although this looks su-

¥ ¨ ¥ ¨

î î " n`n£+ "fgf¶+

perﬁcially similar to the action of ï on the ideal-ﬁlter pair , in

¥ ¨ "fgf¶+

fact we are transporting " nnõ+ in the opposite direction, and while it is still

e Ó e true that Ó is a round ideal and a round ﬁlter, can not be shown. Instead

we have the following properties, generalising Proposition 2.2:

Ó Å Proposition 8.7 Let e and be subsets of a strong proximity lattice . The

following are equivalent:

¥ ¨

î î Å ´ e !

1. There is a Prox-morphism ï from to such that

"zac%V (fgfóî aû+ ÓÎ!"zac%V (zaî n`n£+

ï and ;

Ó e a

2. e is a round ﬁlter and is a round ideal disjoint from such that

%¯Ó %'e implies a or ;

¥ ¨ Ó 3. among disjoint pairs of a round ﬁlter and a round ideal, e is maximal

with respect to component-wise inclusion;

ÉÉÀ À

Ó!&"zak( %¯e§Ê3 a +

4. e is a round prime ﬁlter and ;

À É À

e/!" ( a 3%'ÓÊea +

5. Ó is a round prime ideal and .

Ç À À À

a amî æî aî n`n fgfóî ï

Proof. (1) (2): implies ï , so either or , hence

ï ï

À 7 7

a1%PÓ %e %ÓT0¯e fgfñî î n`n

or . For disjointness assume . This implies ï

n`n

and hence fgf by the second adjointness condition, but this is not valid in the 7

dualising proximity lattice ´ , and so no such exists.

Ó e

(2) (3): Assume Ó is a round ideal containing and disjoint from . If

N Nè N

c%¯Ó YõÓ a a Ó YõÓ

a then there is also in by roundness. By disjointness, neither

c%¯Ó a %¯e

a , nor , contradicting (2).

N N

Ç À 7 7

mS %re e !" (zamS %'e+ eòRe

(3) (4): Assume a , and let . Then

À N N 7 7

e a¿S %ce E a¿S

and %e . Furthermore, is a round ﬁlter: implies for

7dN 7dN 7 7ON N

S E amS %We some E%We , and by ( – ), for some . By deﬁnition, ,

too. Closure under meets is shown by a simple application of the distributivity

N N À

Ó e !Ze %/e

law. If e is disjoint from then we can conclude , and hence

7 7

S %e %&Ó

as desired. If, on the other hand, a for some , then repeat the

N 7 À N

a a e

construction of e with replacing , and replacing . This is guaranteed

a%'e

to be disjoint from Ó , and is shown.

ÉÉÀ À

zak( %'eÑÊ3 a + e

The set " is a round ideal because is prime. It is also Ó

disjoint from e , and so it must be contained in because the latter is maxi- Ó mal. On the other hand, every element of Ó is below some other element of

by roundness, which by disjointness does not belong to e . This shows that

ÉÉÀ À

za ( %'eÑÊ3 a +

ÓX" . Ç

(4) x (5) is Proposition 2.20. The argument for (5) (2) is straightforward.

fgf}î a aò%&e n`noî a ï

For (2) (1) we let ï if and only if (and always), and

n`n aR%±Ó aöî fgf

aàî if and only if (and always). The axioms for consequence

ï ï

relations are easy to check, so let us focus on the adjointness conditions. If a

À À À

a½%Ó %±e aöî fgfjî aàî n`nmî î æî )

ï ï

then or , hence either or . If ï

ï ï ï

)P!knn then it can not be that !òfgf and as otherwise there would be an element

e ) ´ in the intersection of Ó and . In the three other cases holds in .

Since Prox-morphisms into ´ are in one-to-one correspondence to pFrm- Ù

morphisms into ´ , and since the latter are in one-to-one correspondence to d- Êå frame homomorphisms into å , we see that the duality between strong proximity lattices and stably compact spaces, Theorem 2.22, is a special case of the duality of d-frames and bitopological spaces.

It may also be helpful to identify the topology on the spectrum in the current

¥ ¨

e Ó

setting. If È*! is a point of a strong proximity lattice according to the

N N

7 ¥ ¨

Ó e

proposition, and if ! is a round ideal-ﬁlter pair encoding a partial

¥97:

e/0WÓ !-43 È±%

predicate on the spectrum, then È±%6 if and only if , and

¥978 N

Ó0Ve !R43 6=< if and only if .

8.2 Reﬂexivity: Distributive Lattices

Deﬁnition 8.8 An element a of a partial frame is called a total predicate or

a ÷éÐ

reﬂexive element if a . The set of all reﬂexive elements is denoted by .

This terminology is justiﬁed because a in the partial frame is equivalent

% 0 to a con tot in the associated d-frame, so these are elements which always return an answer, and never give conﬂicting information. Yet another way of putting this is to say that they are classical Boolean predicates. It follows from Axiom (con–tot) that with respect to the information order a

total predicate is always a maximal element in a partial frame, but not all maximal

S S

elements need to be total. Axioms (con– ), (con– ), (tot– ), and (tot– ) imply

¥ ¨ ¨ ¨

¿,¶ S fgf nn ÷Ð

that ÷jÐ is a sub-lattice of . Strong implication restricted to

is the same as ordinary implication: The inclusion ÏJ always holds for a

À À

ÔJ a a/!©aW

partial frame, and for the reverse we use that a implies ,

which implies a by ( – ). A partial frame homomorphism maps total predicates to total predicates because it is required to respect . Hence, restriction

to reﬂexive elements yields a functor ø from pFrm to dLat.

Deﬁnition 8.9 A Stone partial frame is a compact regular partial frame for

which the set of reﬂexive elements is dense. In other words, for every a in

7 7 À

4÷TÐ a there is % such that .

The density condition for Stone partial frames can be translated to d-frames 0 using the fact that ÷jÐ corresponds to con tot in the associated d-frame, but the result is cumbersome and does not seem to shed any additional light on the

situation. N

Theorem 8.10 Deﬁne a functor from dLat to pFrm by setting

N§¥ ¥ ¨ ¥ ¨

ö ^`! " Ó e ( Ó e ö¿+

is an ideal-ﬁlter pair in for objects, and

N N N N N

¥ ¥ É ¥ ¨ É ¥

Then

N ø

1. is left adjoint to ;

øØ\

2. ! id;

N N

\ø 3. The image of is contained in SpFrm and restricted to SpFrm, ! id.

Proof. A distributive lattice can be viewed as a strong proximity lattice where

!ôJ . The deﬁnition of on objects, then, coincides with that of in Propo-

N§¥ ö

sition 8.2, and we obtain that is a compact regular partial frame. Given the

N9¥ N§¥

ö è concrete description of the operations on in 8.2, it is also clear that is

a partial frame homomorphism.

¥ ¨ ¥

È [È ö For any ÈË%Wö , the pair is a reﬂexive element of and there are

no others. Thus we have shown (2).

Nþ¥ ¥ ¨ ¥ N ¨ N£

ö Ó e Ó e

The reﬂexive elements of are dense, because means

N N N

¥ ¨ ¥ ¨ ¥ ¨

Ó e È [È Ó e

by deﬁnition that there is È2%We*0VÓ , so . Hence the N

image of is contained in SpFrm. Part (3) now follows from Proposition 8.3.

ö F

For Part (1), let è be a distributive lattice homomorphism from to è

where is any partial frame. We get a homomorphism of partial frames from

N§¥ ¥ ¨ ¥ ¥ ¨

ö è Ó e ^! "¾è È Û2è R (¾Èq%¯Ó R=%reæ+

to by setting Üh÷ . Restricting

N§¥ ¥ ¨

ö È [È

è to the reﬂexive elements of , that is, to the ideal-ﬁlter pairs , we $

N N

¥ ¥ ¨ ¥

recover è , as in this case. Vice versa,

N§¥

ö Ø

starting with a partial frame homomorphism Þ^d , denote the restriction $

to the reﬂexive elements by Þ . We already know that this is a distributive lattice

F Þ$

homomorphism from ö to . Extending to ideal-ﬁlter pairs we obtain

¥ ¥ ¨ ¥ ¥ ¨

¥ ¨ ¥ ¨ ¨

Í Í

Üh÷

¥ ¥ ¨ ¥ ¨ ¨

Í Í

¥ ¨

! Þ Ó e

N§¥ ö

where the last step uses the fact that is a Stone partial frame. Thus

¥ N§¥ ;¨ ¥ ¨ ¥ ~

ö ö F pFrm and dLat are (naturally) isomorphic.

Corollary 8.11 The categories dLat and SpFrm are equivalent.

An alternate view of a distributive lattice is as a special proximity lattice,

J èÑ^õÐØ ù

namely, one in which ! . To a partial frame map one would

¥ ¨

î î

then associate the adjoint pair ð . There is no real difference between this ð

and a distributive lattice homomorphism, however, as we can show that the graph

<àß

÷TÐ ÷éú îð 0éî ac%m÷jÐ

of è (restricted to and co-restricted to ) is equal to : For

¥ ¥ ¥ ¥

è a J è a arî è a è a î a

we have , so and ð . For the reverse inclusion

À À ¥ À ¥

îða è a J J½è a

one assumes ahî and and obtains by deﬁnition and

the fact that !J . HÙ

The dualising object ´ in pFrm is a Stone partial frame, and its set of re- n`n ﬂexive elements consists of fgf and . Adapting Proposition 8.7 to the reﬂexive setting, we see that Stone’s duality of bounded distributive lattices is a special

case of our bitopological duality.

¥ ¨ ©<

Deﬁnition 8.12 Say that a bitopological space is totally order discon-

! < >*3J Q

nected if J and whenever , there is an upper open neighbourhood

¤ ¤

> ú Q ®¯0æú!½4 ® ú!* ® of and lower open neighbourhood of so that and .

Corollary 8.13 The category dLat is dually equivalent to the category of compact totally order disconnected bitopological spaces.

87 Proof. Stone partial frames are spatial and the category SpFrm is equivalent to

dLat. So it sufﬁces to show that the spectra of Stone partial frames are precisely

¥ ¨

eû< J 3

the compact totally order disconnected spaces. Consider points e

¤ ¤

¥28 ¨78

< of a Stone partial frame. By (partial frame) compact regularity, the ¤

order is inclusion in the ﬁrst component and (equivalently) reverse inclusion in

Z%òe Y e

the second. So there is a token a . Because is round and the

¤ ¤ ¤

a% <

reﬂexive elements are dense, a can be chosen to be reﬂexive, from which

¥ ¥

a 6\< a

follows. Thus 6 and are the desired opens.

¥ ¨ $<

Now consider a compact totally order disconnected space . We

ü û

show that the reﬂexive elements of con û are dense. For two consistent pairs of

N N N N N

¥ ¨ ¶¨$¥ ¨ ¥ ¨ ¥ ¨

ú ® ú %V $< ® ú ® ú ® ú&!

opens ® , holds if and only if

¤ G

¥ ¨ ¥ ¥ N9¨ N£ ¥ ¨

ú&!½ ® ú 4 ® ú 4

. If , then and is reﬂexive. Similarly,

N ¥ ¨ ¥ ¨ ¥ N9¨ Ng

!½ ® ú 4 ® ú YMú

if ® , then . If neither of these holds, then

N N

>'%VkY8ú Qs%¿ Y:® >c3J½Q

and kY¡® are non-empty, and for each and , we have .

¥2E=ý ®$¨3þ¡ý ® $<

For each such pair of points, choose a disjoint pair %V according

¤HG

þóý ®

to total order disconnectedness. For each >%-Yú , the sets together with

N þ N þ=N

YM®

® cover . By compactness, ﬁnitely many ’s sufﬁce to cover . Let be

þ¡ý ® EhN

the ﬁnite union of these ’s and let be the intersection of the corresponding

N N N

E£ý ® E þ E

>°%

’s. Then and are disjoint and cover . Moreover, . So the

sets together with ú cover . Again by compactness, ﬁnitely many of them

E þ $

sufﬁce. So we can let $ be the union of these and the intersection of the

N N N N

þ þ þ

® !½

corresponding sets . Since each covers òYÑ® , we have . And

E ¥ ¨ ¥6E ¨3þ ¥ N9¨ N£

úZ!a ® ú $ $ ® ú

clearly $ . Putting this together, and

¥2E ¨æþ $ $ is reﬂexive. Of course, from Stone’s original duality theorem we know that the upper topology of a compact totally order disconnected bitopological space is a spectral space. Notice, however, that Stone’s characterisation requires that we restrict to perfect maps, so that the topological category Spec is not a full sub-category of Top. In contrast, the corollary establishes a duality between the category of distributive lattices and a full sub-category of BiTop.

88 9 Negation

9.1 Negation as additional structure

9.1.1 Symmetric d-frames

Negation exchanges true for false and false for true. Where the truth value is unknown it remains unknown, and where a contradiction occurs it remains a con-

tradiction. This is the approach taken by Belnap [Bel77], and it deﬁnes an op-

å Ê`å eration on the four-element lattice of truth values (depicted in Figure 1 on page 29).

The effect on the bitopological space of models is that the positive ex-

¥ ¥

tent 6 of a formula is exchanged with the negative extent , in other

¥ ¥ ¥ ¥

!X6=< 6=< !X6

words, 6 and . Since the positive extents com-

¤ ¤

, <

prise the topology , , and the negative extents the topology , we see that in ¤

the presence of negation the two topologies must coincide, or, in the language of

¥ ¨

¨, ,û<

Section 4.1, that the bitopological space spec KT is symmetric.

F F <

In order to add negation to the abstract setting of d-frames F! ,

¤ÑG F <

we therefore require that the frames F and are isomorphic via an explicit

^:FH

bijection . The associated negation operation maps !Öý

¤ ¤

¥ ¶¨ ¥

<àß

ÿ 6 <

to ý , and will be looked at below. For the moment, we tentatively

¥ ¨ ¨

F F <£ 7

deﬁne a symmetric d-frame to be a structure K*! con tot and require ¤

that homomorphisms preserve the symmetry operation . The latter requirement F

is necessary as there may be many different isomorphisms between F=< and , ¤ and we must make sure that homomorphisms respect the chosen exchange of true and false.

In general, an abstract point of a d-frame K is given as a homomorphism from

å Êå K to ; since now we require that it additionally preserve the symmetry, there

will in general be fewer “symmetric points” than general ones. We denote the ¦ resulting bitopological space as spec K . One shows without difﬁculty that this is a symmetric bitopological space. However, there is little else one can show about symmetric d-frames and we

89 doubt that there is much use for the concept at this level of generality. The problem is that there is no link between the symmetry operation and the predicates con and tot. Indeed, from a semantic point of view it seems natural further to require

the laws

¥ ¨ ÆqÇ ¥

Q % >Û Q !A

(icon) > con

¥ ¨ ÆqÇ ¥

Q % >ó Q !Ý (itot) > tot We take this as our ofﬁcial deﬁnition.

Deﬁnition 9.1 A symmetric d-frame is a d-frame equipped with an isomorphism

satisfying (icon) and (itot). The isomorphism will also be referred to ¤ as the symmetry operation. Homomorphisms of symmetric d-frames are required to preserve the symmetry operation.

At ﬁrst glance, symmetric d-frames appear to be quite rich structures, but in

actual fact, they are nothing else but ordinary frames; the ﬁrst component F com- ¤ pletely speciﬁes (up to isomorphism) the whole structure. This correspondence allows us to compare the concepts introduced in this paper with their classical frame-theoretic counterparts. For example, the following observations are easily

checked:

¥ ¨ ¨ ¨

F F <£Âfgf n`n£ 7

Proposition 9.2 Let K*! con tot be a symmetric d-frame.

¥ ¨ ¥ ~

<àß

F e e

1. e is a completely prime ﬁlter of iff is a symmetric d-point ¤

of K .

2. K is reasonable and satisﬁes all cut rules. F

3. K is spatial (regular, compact) if and only if is a spatial (regular, com- ¤ pact) frame.

We can also show that Lemma 6.5 specialises to the usual Hofmann-Mislove »

Theorem for frames, in the sense that any Scott-open ﬁlter on a frame F has a

partner b which satisﬁes conditions (hmcon) and (hmtot) of 6.5. As shown there,

É »

Íe

!/Üh÷Â".R ( Èq% Ê ÈÛlRH!RA +

we should examine F§Y where . By construction,

» ¨

!ôÝq% È R

(hmcon) holds, and we check (hmtot): Assume >só , then there are

» » »

È ÛgR!A >qógR %

with È¿% and such that , because is Scott-open. Now we

¥ ¥ ¥ » »

ÈTÛË> ó ÈÛlR !ÈÛ >,ólR %

have ÈTÛË>2! because is a ﬁlter. It follows »

that > belongs to as required by (hmtot).

From a semantic point of view, the universal way for obtaining a symmetric

¥ ¨ $<

space from a general bitopological space >£~ is to construct the common

¤ SP©<

reﬁnement . The d-frame analogue of this is the free biframe over a d- ¤ frame, for which we gave two constructions in Section 5.2. This yields a left

adjoint to the forgetful functor from symmetric d-frames to dFrm as follows:

¥ ¨ ¨ ¥ ¨

F F <£ ès! è è8<

Let K*! con tot be a d-frame and a dFrm morphism

¤ ¤

¥ ¨ N ¨ N Ng

> >

ÿ ! <£ 7 Fû¢

from K to a symmetric d-frame con tot . Let be the frame ¤

deﬁned by generators and relations as in the proof of Theorem 5.9. We interpret

¥ ¥ ¥

y¾>{¢¡ !Xè > >¯%'F yQ¢{£¡Ñ!o èO< Q

the generators in by setting for , and

¤ ¤ ¤

F < ýþ> Q5ÿò%

for QÏ% , and check the relations. For example, if con then

¥ ;¨ ¥ N ¥ N|¥ ¥

> èO< Q ÿ % è > ÛA è8< Q !A

ýyè con which by symmetry is equivalent to

¤ ¤

¥ Nþ¥ ¥ ~

y¾>{ÛgyQ¢{£¡!¤ y¾>{¥¡ Û¦ yQÉ{§¡T!òè > ÛI èO< Q !°A

and from this we read .

> èd¢à^dFû¢ Ø

¥ ¨ ¨

K\¢Ñ! FM¢ FM¢Â ¢ ¢Â

From Fû¢ we obtain the symmetric d-frame con tot id , where

¥ ¥

<àß <àß

¢ Û A ó Ý èà¢

con ¢ and tot are equal to and , respectively, and we extend to

¥ ¨ N

<àß

Tè¢ K§¢ ÿ è¢# !

the symmetric d-frame homomorphism è¢ from to : id

N N

¥ ¥ ¨

<àß

èà¢ © è èO<

è¢Ë! . It extends the dFrm morphism we started with

¥ ¥ ¥ N ¥

<àß

because è and

¤ ¤

N ¥ N ¥

<àß <àß

where and are

¤ ¤ the frame homomorphisms that map an element to (the equivalence class of) the

corresponding generator of Fü¢ .

9.1.2 D-frames with negation

Q5ÿp×Ø

Instead of symmetry one can alternatively axiomatise the operation ýþ>

¥ ;¨ ¥

<àß

6 Q > ÿ Fò!ÚF F <

ý on . This, of course, is the d-frame version of Bel- ¤HG

nap’s negation discussed at the beginning of this section; we denote it by . One

sees without difﬁculties that it satisﬁes the following axioms:

7 Ç 7

I*E I½ME

I= ! ì

7 7 7

x Û¿ !XC

% con

7 7 7

% ó¿ !XD

tot x

F°!jF F <

We call a map with these properties on a reasonable d-frame a ¤ G

negation. It is again easy to see that a negation gives rise to a symmetry between

F FH<

F < and when restricted to : It preserves the frame structure because it is

¤ fgf

monotone and its own inverse. Further, the element n`n is mapped to , which is 7

the only element which makes the last two rules true for !nn . It follows that

¨ ¨

¥ ¨ ¨

FÑÂfgf n`nõ ¶

We deﬁne the category dFrm to consist of objects con tot

¥ ¨ ¨

n`nõ where FÑÂfgf con tot is a reasonable d-frame and is a negation on it. The

morphisms are d-frame morphisms that preserve . We stress that it follows from

the discussion above that dFrm is equivalent to Frm, the category of frames. Nonetheless, we are interested how negation manifests itself on partial frames.

Deﬁnition 9.3 A partial frame with negation is a structure

¥ ¨ ¨ ¨ ¨ ¨ ¨

, Û Ü ÷ C¶ S fgf n`nõ ¿ which satisﬁes the conditions for a partial

frame (Deﬁnition 7.2) and also

7 Ç 7

I I*E ! I½ME

( – )

4=c! Ó

( – ) id

7 ÆqÇ 7

E EÑS¯

( – )

7 ÆqÇ 7õ¨

Û ÛmME!XC " E+ I ( – ) is bounded w.r.t. As well as the other operations, homomorphisms must also respect negation.

Theorem 9.4 The category dFrm is equivalent to the category of partial frames with negation. Proof. The constructions are the same as the ones we used to prove Proposi- tion 7.4 and Theorem 7.5, and we only need to check that the two notions of negation agree.

92 Regarding the translation from dFrm to partial frames, note that a nega-

tion on a d-frame can be restricted to the subset of elements that satisfy the con-

7 7 ¥§7 78 7 7

!XC Û\ !X Û !

predicate. This is so because with Û= one also has

C . From the axioms only ( – ) and ( – ) can be in doubt. For the former con-

7 ¥ 78

Î E h <æómE %

sider ; by deﬁnition, this is equivalent to tot, which by ¤

symmetry and the distributivity laws is equivalent to

7 ¥ ¥§7 ¥

:<æó <æó2E ó2 :< ó¿ < ó2 E !XDÊ

¤ ¤

EõS

The right hand side in ( – ), , rewrites to the same equality using the

¥ 78 7

! <

fact that .

7 7

Û Û\ME!XC ó§EÎ%

In order to check that ( – ) is valid, let . To show that con

¥§7 ¥§7

Ûü óûE !XC

we must show óûE . Rewriting the left-hand side by distributivity,

¥97 78 7 7

E\ÛVME Û2ûE E\ÛV

we obtain the join of four terms: ÛV , , and . The ﬁrst

7¨

E,% C

two equal C because con, the second two equal by assumption. For the

7õ¨ 7

±% Û'ME2IsÛ'P!òC

reverse implication assume EËI con. Then since ¿% con.

As a homomorphism of symmetric d-frames “does the same” on the two con- F <

stituent frames F and , the corresponding partial frame homomorphism re-

7 ¨7 7

spects exchange of the two components < of an element . ¤

In the reverse direction we associate with a given partial frame the (product of

¨ ¨

!©C fgf F

the) two frames F and . Negation establishes a bijection

C I üC

between these: First of all, C is the smallest element, so , but then IÔûûCa!òC

ûC by the (information order) monotonicity of negation. Second,

fgf!-fgfSËn`n Hfgf!-fgfMrHfgf n`n n`nqfgf!Xn`n n`n

fgf implies by ( – ), and

n`næSVHfgfí!XHfgf J Hfgfí!_n`n implies n`n . Since , we can conclude , and hence

fgfü!RüHfgfí!8n`n , too.

Next we check that the con and tot predicates, as deﬁned in Proposition 7.4, are

a*Iòfgf I n`n

derived from the frame structure, say of F , alone. Assume , . By

¥ ¨7ÀH ¨7À

% "za + Û

deﬁnition, we have a con if and only if is bounded, and by ( – )

À ¥ ¨7ÀH

WÛr !ôC a %

this happens if and only if a . Regarding tot, we have tot if

À À

fgf a¿S¯

and only if a by deﬁnition. Using ( – ) this is equivalent to ,

ta2SV and this happens if and only if fgfü! .

If è§^õ is a homomorphism of partial frames with negation then the

¨ ¨

as required for a map of d-frames. They respect the symmetry operation because

C n`n è

that is just negation restricted to © , and is required to preserve it. Û It may be worthwhile to point out that both directions of ( – ) are indis- pensable for the equivalence proof. Consider the free partial frame over one

generator depicted in Figure 5 on page 74. Reﬂection at the vertical axis of

I Û

symmetry satisﬁes ( – ), ( – ), and ( – ) but not ( – ), and indeed, this !RF <

structure does not arise from a symmetric d-frame. Next let F be the four- ¤

element Boolean algebra and let the symmetry operation be identity. The subset

/!&" ýþ> Q5ÿ %'F(b>m!A QÎ!A + ¿ or is a partial frame and exchanging the compo-

nents of a pair satisﬁes all conditions of Deﬁnition 9.3 except the direction from Û left to right in ( – ).

9.2 Negation as a structural property

9.2.1 Negation on regular d-frames

Let us compare the treatment of negation in d-frames with the classical situation. There, the existence of a negation operation is a purely structural property of the logic; in the language of bounded distributive lattices, it is expressed by the (ﬁrst-

order) formula

N N N

d> > Ê~>æm> !±nn >æSË> !&fgf

k and

N > One shows that there can be only one > that is related to a given in this way,

and so if the formula holds in a bounded distributive lattice F , then a negation N operation can be deﬁned by setting ¼>W^!> . The laws of Boolean algebras hold and furthermore, homomorphisms of bounded lattices preserve negation.

For general d-frames the situation is quite different. The mere existence of an F <

isomorphism between F and , even when it satisﬁes (icon) and (itot), does not ¤ mean that it is uniquely determined or that it will be preserved by dFrm homo-

morphisms. Consequently, the usual spectrum spec K of such a d-frame need not even be a symmetric bitopological space. An example is shown in the lower right

94 F <

hand corner of Figure 1 on page 29, where both F and are the three-element ¤

chain, con and tot are minimal (while reasonable), and (icon) and (itot) are satisﬁed,

$< å Êå! u5Êvu

but and on spec are quite different.

N A

Indeed, any frame F gives rise to an example of this nature: add new bottom

N N

F F

and top Ý to to obtain , and deﬁne con and tot minimally:

¥ ¨ N N

Q % ^ x >m!RA QÎ!A

> con or

¥ ¨ N N

Q % ^ x >m!Ý QÎ!Ý

> tot or

N N N

¥ ¨ ¨

^! F F This yields a (reasonable) d-frame K con tot that is symmetric. This

example also shows that the symmetry operation is not uniquely determined by

N K

(icon) and (itot): any automorphism of F gives rise to a symmetry on . N

The example works because of the paucity of con and tot in K . Regularity is the exact opposite of this situation, and so the following result should not be too much of a surprise:

Proposition 9.5 Let K be a symmetric regular d-frame. Then, FH<

1. F and are regular as individual frames; ¤

2. the symmetry operation is uniquely determined;

3. the spectrum of K is a symmetric bitopological space.

Furthermore, dFrm homomorphisms between symmetric regular d-frames pre-

serve the symmetry operation. Á

Proof. In the context of this proof, we write for the well-inside relation of the Á

d-frame, and for the well-inside relation of the individual frame F .

¤ ¤

N9¨ NÁ É5¦ ¥ N ¨¦

> F > W> %-F <£Ê > %

(1) For two elements > of we have iff

¨b¥ ¨¦ É5¦ N ¥§¦

% %½FH<£Ê> Û4 !

con > tot which by (icon) and (itot) is equivalent to

¨ ¥þ¦ NÁ ¥§¦

>,ó !Ý > > F

A and this just says with witness (in ).

¤ ¤

N Á >

(2) Let be a symmetry operation satisfying (icon) and (itot). We have

N N

¥ É5¦ ¦ ¨ ¥ ¦ É5¦ ¦ ¨$¥þ¦$¨

Q %PF Ê>,Û !/A Q ó !°Ý %WF Ê~>Û !/A Q %

iff iff tot iff

¤ ¤

É5¦ ¦ ¨ ¥ ¦ ¥

Ê~>Û !&A Q ó !°Ý >WIo Q

%F . The last formula implies that and by

N9¥ ¥

Q IL Q regularity follows.

¥ ¨ ¥ ¨ ¥ ¥

eû< K e eû< %6 Q

(3) Let e be an abstract point of . We have

¤ ¤ ¤

¥ É NüÁ ¥ N ¦ NüÁ ¥

Q %&e > Q Ê> %&e %&F > Q

iff iff . If is a witness for

¤ ¤ ¤ ¤ ¤

¦ N ¦

!XA %Pe3 Qm%eû<

then because of ÛË> we must have and so follows because

¦ ¥ ¥þ¦$¨

# Q !Ý Q %

ó is equivalent to tot. In other words, we obtain that the abstract

¥ ¨ ¥

eM< 6\< Q

point e belongs to . ¤

In order to show that in the presence of regularity, homomorphisms preserve

¥ ¨ ¥ ¨ ¨ ¥ ¨ N ¨ N Nv

> >

è èO< ^ F F <£ 7 Ø <¡ ¨

symmetry, let ès! con tot con tot be a

¤ ¤ ¤

d-frame homomorphism. We use an argument similar to that in (2) to show that

Á ¥

ñ!r \èO< Qs%'F < Èq%'F È Q

è . Let be arbitrary. For any such that we

¤ ¤ ¤

¦ ¦ ¥ ¦

È:Û !½A Q ó !&Ý

have a witness %'F such that and . The latter is equivalent

¥þ¦$¨ ¥ ¥þ¦ ¥ ¥þ¦ ;¨ ¥ ~

% è È Û2è !A è èO< Q % ÿ

to Q tot, so we get and tot in . The

¤ ¤ ¤

N§¥ ¥ ¥þ¦ ¥ Nþ¥ ¥

è8< Q óqè !Ý è È IL è8< Q

latter is equivalent to and we conclude .

¤ ¤

Taking the supremum of all such Èq%'F we obtain

¥ ¥ ¥ ¥

è Q IL èO< Q Ê ¤

By exploiting the regularity of FH< , one shows in exactly the same way that

¥ ¥ ~ N ¥ ¥

<àß <àß

> I+ è > > %*F

èO< for all . This allows us to compute the other

¤ ¤

inequality

N N N N

¥ ¥ ¥ ¥ ¥ ¥ ~ ¥ ¥ ¥ ¥ ~ ¥ ¥

<àß <àß

èO< Q ! èO< Q IL è Q !Xè Q

¤ ¤ and thus complete the proof. On partial frames, the assumption of regularity renders negation a purely log-

ical concept:

æ^ ¿ Ø ¿

Proposition 9.6 If is a regular partial frame and is a map that

I Û

satisﬁes ( – ) and ( – ), then ( – ) and ( – ) also hold. Furthermore, the

¥§7 7

S2E !X ¯ME de Morgan rule is valid. Proof. Throughout this proof we make heavy use of the fact (shown in Propo- sition 7.4) that every partial frame can be seen as the set of pairs satisfying the con-predicate in a concrete d-frame, and with the partial frame structure given concretely as described at the beginning of Section 7.

7 7 7

I I*E E E

( – ): In a regular partial frame, is equivalent to I I and F F .

7 7

E ûE

In any partial frame satisfying ( – )and ( – ), is equivalent to .

7 7

E ME

So I I is equivalent to F F .

Û sÛ'P!ôC

Regarding ( – ) we begin by showing that is always true in a

aW n`n aPcXJ°n`n

regular partial frame. From a we get , so and

¥ ¥ À

qm !ta Û !RA

hence a by Lemma 7.3(9). Likewise, implies

¤ ¤ ¤

À ¥0À À ¥

S A¿! SW <!

fgf , so . Using the characterisation of

¥ À

mÛW±!ZÜ aWÛ ÛW !

regularity in Proposition 7.10, we thus get ÷

¥ ¥ ¨7À ¥ ¥ ¨ 7õ¨

a Û <ËÛ < ! A A !ôC EVI

Ü . Now if then we just use

¤ ¤

I Û2MEÎIæÛ¿Ë!RC

( – ) and obtain .

7 À

a E For the reverse direction assume ÛVMEÎ!C , and also . We have

the following relationships:

7 ¥ ¨

ME !½A ©C fgf¯

1. Û in the frame by assumption;

¤ ¤

7 ¥ ¨

2. <Û in the frame by assumption;

À À À

I a aJ

3. a because implies ;

¤ ¤

£oÀ

û< <

4. a for the same reason;

À ¥ ¨ À

5. ó in the frame because by ( – );

¤ ¤

¥ ¨

6. a in the frame because by ( – );

7 7 7 ¥2À ¥ ¥§7 À ¥§7 ¥ 7 À

ÛæÝ! Û ó ME ! Û ó Û ME ! Û

7. ! ,

¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤

using (5) and (1);

7 7 7 ¥ ¥ ¥§7 ¥§7 ¥ 7

<§ÛæÝ! <\Û aû<=ó ME < ! <\Û%aû< ó <§Û ûE < ! <=Ûáaû< 8.

using (6) and (2);

¥97 À ¥§7 À ¥~¥§7 À ¥¥97 À

SlaôÛ Sa ! Û óa Û óa Û !

9. ©

¤ ¤ ¤ ¤ ¤ ¤ ¤

¥97 ¥~¥§7 À ¥ À ~ 7

ga Û Û ó a Û ! ósa

ó , using (3) and (7);

¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤

¥97 ÀH ¥§7 À ¥~¥§7 À ¥¥97 À

SsaÐÛ Sa þ<Ë!

10. ©

¥~¥§7 ¥2À ~ ¥§7 À 7 À

gaû< ó <Ûlaû< Û <æó < ! <,ó <

<Û , using (4) and (8).

¥97 À ¥97 À À

S4ajÛ S4a (za E + !

It follows that the collection "

¥97 ¨7 À À

óla <ó < (za E +

" is directed and that the supremum is above

¤ ¤ 7

and E .

The de Morgan rule is a simple sequence of equivalences, exploiting regular-

7 7 7

rûE x a q a ME x ¤a q E óa x

ity: a

7 ¥97

¤a x a S¿E S¿E .

97 In the compact regular case we would like to replace the partial frame by the

strong proximity lattice structure on a dense subset , as explained in Section 8.1. The negation operation need not map basis elements to basis elements but this

should not disturb us as the idempotency of means that it is enough to add the

image to . Because the de Morgan rules are valid, the result will again be

a sub-lattice of . So we deﬁne:

Deﬁnition 9.7 A unary operation on a strong proximity lattice is called a nega-

tion if it satisﬁes ( – ) and ( – ).

Theorem 9.8 The category of compact regular partial frames with negation is equivalent to the category of strong proximity lattices with negation. Proof. We examine how the constructions that led to Theorem 8.6 interact with

negation. Starting with a strong proximity lattice we deﬁne a negation on the set of

¥ ¨ ¥ ¨

Ó e ^! Me MÓ

round ideal-ﬁlter pairs by setting . Repeated use of ( – )

7¨ ûEc%Me

shows that this is again a round ideal-ﬁlter pair; for example, if ,

7 7

_% e rE E

then by roundness there is such that , so and ; it

S¯ME V%rMe

follows that .

Ó e Ó e Ó e

By Proposition 9.6 it remains to check ( – ):

7 ¥ Ng N N 0PÓ

means that there is an element in eóWe . By the formula for the join

N N N 7

E %òe E _E

of two round ﬁlters there must exist Ek%òe , with , which

N N N N

7 ¥

S1ME E e°0 Ó ó_ûe

is equivalent to E . Thus is an element of , and

N N N N N N

¥ ¨ ¥ ¨ ¥ ¨

e Ó e S Me ¼Ó

Ó is established.

¥ ¨

î î

Proposition 9.9 A Prox morphism ï between strong proximity lattices with

À À

x î=ïÑ¤a

negation satisﬁes amî . ï Proof. We exploit the equivalence between Prox and the category of compact

regular d-frames. The desired result follows from the fact that a dFrm morphism è

between regular d-frames respects negation, as shown in Proposition 9.5. For the

¥ ¨ À À ¥

î î î a x è a ! ð

associated Prox morphism ð we compute

¥ ¥ À À

¤a x è óa x óaî

ûè . ð

98 Corollary 9.10 (Moshier 2004) The category of strong proximity lattices with negation is dually equivalent to the category of compact Hausdorff spaces. Proof. By Proposition 6.4 the spectrum of a strong proximity lattice is order- separated, and since the two topologies are the same in the symmetric setting, this means that it is a Hausdorff space. Everything else remains the same as in the general (i.e., non-symmetric) case.

9.2.2 A proof-theoretic characterisation of negation

The existence of a negation operation can be deduced from a structural property of strong implication. This was ﬁrst discovered for strong proximity lattices by

the second author in [Mos04] but it holds in the more general context of regular 7

partial frames. Recall once again that an element of a regular partial frame is

7 7 7 À 7 À

zak(¢a + !a" ( +

completely determined by the sets I !a" and F , in

7 À 7õ¨7À 7

! Ü "za¿Û (¢ac% % +

the sense that ÷ I F . If there is a negation operation

on and if we want to capture an element in this way, then by ( – ) we

7 7 7 À 7 À

"za1(za2 n`n£+ " (fgf S + can express I and F alternatively by and .

Now, these sets can be deﬁned in any partial frame, whether there is a negation or

7 7

not, so let us call the ﬁrst one I and the second F . If we assume that Gentzen’s

7 À 7 À

±% % a cut rule is valid (cf. Proposition 7.6) then for every a I , F , holds.

We can then form

7 À 7õ¨7À 7

"za¿Û (¢ac% % +

^! I F 7

which is directed by Lemma 7.8. As in the proof of 7.10 we know that !

¨ À

¨ ¨

ÍHa Ü ÷ Í <8ÿ

ýyÜh÷ I F . ¤ As an aside, this provides an alternative proof for the fact that negation on a

regular partial frame is unique if it exists.

7 7 Ø In general, there is not much that we can say about the operation × but with a few more assumptions it moves closer to being a proper negation:

Lemma 9.11 Let be a partial frame satisfying the inﬁnitary Gentzen cut rules

7 7 7

%r I (G-CUT ) and (G-CUT ) (cf. Table 2). Then for all , .

À 7 À 7 7

fgf S

Proof. Suppose % F , that is, . By deﬁnition of as a directed

À ¥ 78

I fgf S ÷

supremum of meets plus ( – ), Ü I holds. Also by deﬁnition, for

7 7 7 À 7 7

@% aP n`n

each a I , holds. So (G-Cut ) implies . Thus F F .

7 7 7 7Éã 7

I I Similarly, I I by (G-Cut ). Hence, .

Next we observe that in the presence of interpolativity of and negation,

7 78N N

EæS1E

Gentzen’s cut rule can be inverted: Whenever then this can

7 7dN N

SE a

be rewritten as ±ME ; by interpolation there is some such that

N N

7 7 7

a ScE ETSµa

ME , and another application of ( – ) yields

7dN N

2 E and a . This motivates our main result on the matter:

Theorem 9.12 Let be a regular partial frame. The following are equivalent: (i) The inﬁnitary Gentzen cut rules (G-CUT ) and (G-CUT ) hold, and

Gentzen’s ﬁnitary cut rule can be inverted.

7 7

(ii) The operation × is a negation on , and .

Ç 7 7

Proof. (i) (ii): We already know I by the preceding lemma. For the

7 7

% fgf«ña Sn`n

converse, consider a I , which is equivalent to . By an application

À 7 À À

<àß

S a¯ n`n

of (G–cut) we obtain an element such that fgf and . By

!½n`n aV n`n

regularity, n`n holds if and only if . Because we have and

¥0À À 7 7

Ën`n n`n am Ûhn`n nn % nn¯% I

a , it follows that . But F and I , so ( – )

7 7 7 7 7

P n`n Ó e

implies a . Thus I , and similarly F . So by regularity,

7 7?ã 7 I

! . Ç (ii) (i): The validity of all cut rules was stated already as Proposition 9.2. Invertibility of Gentzen’s cut rule was presented above. Proposition 6.13 and Lemma 8.1 provide us with the following special case:

Corollary 9.13 A compact regular partial frame carries a negation if and only if Gentzen’s cut rule is invertible.

Corollary 9.14 The full subcategory of pFrm consisting of compact regular partial frames in which Gentzen’s cut rule is invertible is dually equivalent to the category of compact Hausdorff spaces.

100 9.2.3 Boolean algebras

We are ready to take the ﬁnal step towards Boolean algebras.

Deﬁnition 9.15 A partial frame is called Boolean if in addition to satisfying the conditions of a Stone partial frame, it admits a negation.

Theorem 9.16 The category of Boolean partial frames is equivalent to the category of Boolean algebras.

Proof. We already know that the set of reﬂexive elements forms a sub-lattice;

7 7 7 7

negation restricts to this because implies . We showed in Propo- sition 9.6 that de Morgan’s laws hold and this is sufﬁcient to prove that the set of reﬂexive elements is a Boolean algebra. Everything else is a special case of Theorem 8.10.

10 Discussion

In Figure 7 we have listed the main dualities discussed in the paper. It illustrates our central contention that the classical Stone dualities are best understood as

special cases of the dual adjunction between bitopological spaces and d-frames. Êå

The accompanying Figure 8 displays the dualising objects as sub-structures of å ,

¨ n`n£+

with or without symmetry. It shows quite clearly that the dualiser ´*!Ô"fgf in

¨ Ý¾+ the bottom row is different from the dualiser å!Ô"©A for frames, and thus that the duality of Boolean algebras is not a special case of that of frames. In the traditional understanding of the situation, expounded for example in [Joh82], the connection between the ﬁnitary structures of Boolean algebras and distributive lattices is given by ideal completion of the logical order, and by se- lecting the sub-poset of compact elements, in the other. This works up to a point,

but note that one must restrict frame homomorphisms to those that preserve ù , a condition that is not really justiﬁed when considering a single topology. Indeed, this methodology breaks down when one tries to apply it to the middle layer of strong proximity lattices in Figure 7. While (round) ideal completion

101 with symmetry without symmetry

d-frames, or partial frames vs. bitopological spaces

frames vs. topological spaces inﬁnitary structure: information order and logic

ﬁnitary structure: logic operations and strong implication strong proximity lattices vs. compact regular bispaces

strong proximity lattices with negation vs. compact Hausdorff spaces

algebraic structure: logic operations distributive lattices vs. spectral spaces

Boolean algebras vs. Stone spaces

Figure 7: A hierarchy of Stone-type dualities.

102

¥ ¨

D/! Ý Ý

¥ ¨

Ý A !fgf

n`nË! A Ý

¥ ¨

C/! A A

Figure 8: The dualising d-frames for the dualities of Figure 7. In the second row, solid lines and ﬁlled elements indicate the dualising compact regular partial frame. In the third row, ﬁlled elements indicate the dualising distributive lattice (in the “logical” order). Double arrows indicate negation.

103 can still be applied, there is no way to recover the ﬁnitary structure as a sub- structure of the associated stably continuous frame. We believe that it is the (rather amazing) fact that a compact regular d-frame is completely determined by its ﬁrst

component, Theorem 6.15, that has obscured the bitopological character of Stone Á duality for so long. The coincidence of the well-inside relation with the way-

below relation ù in a compact regular d-frame, Lemma 6.10, also pointed us in the wrong direction. As the bitopological treatment makes clear, the strong implication of partial frames in general captures the former and not the latter. With this paper we have not answered all questions that naturally present themselves when one generalises from frames to d-frames. We consider the following as the most important open problems that remain.

Open Problem 1 Does the Hofmann-Mislove Theorem 6.6 hold for general d- frames, or under more general assumptions than those made in 6.6?

We have invested a considerable amount of energy on this question and now believe the answer to be “no.”

Open Problem 2 Is every reasonable d-frame that satisﬁes the inﬁnitary cut rules derived from a biframe?

We included some thoughts on this question in Section 5, after Proposition 5.8; our conjecture is that this is true.

Open Problem 3 Develop a notion of “locally compact d-frame.”

Not just any notion will do; we would expect that with the right deﬁnition a locally compact d-frame would be spatial. If things work really well, then one could also hope for a link between local compactness and exponentiability in dFrm.

Open Problem 4 Develop a “generators and relations” method of constructing d-frames.

104 In the construction of a free biframe from a d-frame, we employed the tech- nique of covers on a semi-lattice. This same construction yields the symmetri- sation of a d-frame. But as noted in Section 5.2, the resulting elements of the

constructed frame are characterised as sub-lattices with respect to J and as being

Scott closed with respect to I (along with other conditions particular to this construction). This suggests that objects may be constructed from generators and a combination of logical and informational relations. The separate roles that logic and information play in such a construction should help illuminate their relationships more generally. Priestley duality considers the join of the two natural topologies on the spectrum of a distributive lattice. By also keeping the specialisation order as explicit data, it manages to faithfully capture the underlying bitopology. This works equally well for strong proximity lattices (Proposition 2.12) but one wonders whether there is a more general principle available. Ultimately, one will need biframes, which essentially add the two basic topologies explicitly. Since we now understand how free biframes arise from d-frames, a more finitistic construction may be possible. Again, logic and information are likely to play distinct roles in such a construction. In this study, we have defined d-frames and shown that they constitute the Stone duals of bitopological spaces. As a result, we are able to see several known Stone-type duality results as special cases of this. D-frames also open up the study of bitopological spaces to the “point-free” techniques that have become so useful in topology. In particular, studies of d-sobriety for bitopological spaces and spatiality for d-frames show that these concepts are quite rich. Bitopological analogues of the Hofmann-Mislove Theorem are available and find very natural ties to Escardo’´ s approach to quantification over subsets of a topological space. D- frames also yield many “choice-free” proofs of classical results and have helped to further illuminate the relationship between regularity and continuity. Perhaps the most surprising feature of d-frames is the natural split they make between logic and information. This was only partly anticipated when we began this investigation nearly two years ago, and yet has turned out to be an organising

105 theme throughout this paper. Many proofs are significantly simplified, particularly for reasonable d-frames, by maintaining logic and information as separate, but interacting, notions. Indeed, one indication that logical aspects of d-frames are “naturally occurring” is the repeated appearance of Gentzen’s cut rules, and their reflection in d-frames, in unexpected places. Partial frames provide a general topological theory of partial propositional logic that accounts for partiality in terms of information order where accumulation of information is the main operation, and logic where conjunction and disjunction

are the main operations, and strong implication is distinct from J . In this regard, one can regard partial frames as the Lindenbaum algebras for a generalised form of Kleene’s three-valued logic. This generalisation points out the importance of treating information as well as strong implication as distinct concepts.

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