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Modal operators on compact regular frames and de Vries algebras
Bezhanishvili, G.; Bezhanishvili, N.; Harding, J.
DOI
10.1007/s10485-013-9332-9
Publication date
2015
Document Version
Submitted manuscript
Published in
Applied Categorical Structures
Link to publication
Citation for published version (APA):
Bezhanishvili, G., Bezhanishvili, N., & Harding, J. (2015). Modal operators on compact regular
frames and de Vries algebras.
Applied Categorical Structures
,
23
(3), 365-379.
https://doi.org/10.1007/s10485-013-9332-9
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Download date:10 Aug 2022
MODAL OPERATORS ON COMPACT REGULAR FRAMES AND DE VRIES
ALGEBRAS
GURAM BEZHANISHVILI, NICK BEZHANISHVILI, JOHN HARDING
Abstract. In [7] we introduced the category MKHaus of modal compact Hausdorff spaces, and showed
these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as
modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in [7] we introduced the categories
MKRFrm and MDV of modal compact regular frames, and modal de Vries algebras as algebraic counterparts
to modal compact Hausdorff spaces, much as modal algebras are algebraic counterparts to modal spaces. In
[7], MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another.
Here we provide a direct, choice-free proof of the equivalence of MKRFrm and MDV. We also detail
connections between modal compact regular frames and the Vietoris construction for frames [19, 20], discuss
a Vietoris construction for de Vries algebras, and how it is linked to modal de Vries algebras. Also described
is an alternative approach to the duality of MKRFrm and MKHaus obtained by using modal de Vries algebras
as an intermediary.
1. Introduction
In [7] we began a program of lifting structures and techniques of modal logic, based fundamentally on
Stone spaces and Boolean algebras, to the setting of compact Hausdorff spaces, de Vries algebras, and
compact regular frames. Here, we consider aspects of this work more closely linked to the study of point-free
topology than to modal logic. While we briefly recall some important facts from [7], the reader would benefit
from having access to this paper when reading this note.
A modal space, or descriptive frame, (X, R) is a Stone space X with binary relation R satisfying certain
properties equivalent to requiring the associated map from X into its Vietoris space V(X) be continuous.
With the so-called p-morphisms between them, the category MS of modal spaces is isomorphic to the category
of coalgebras for the Vietoris functor on Stone spaces. This lies at the heart of the coalgebraic treatment
of modal logic. A modal algebra (B, ◇) is a Boolean algebra with unary operation ◇ that preserves finite
joins. The category MA of modal algebras and the homomorphisms between them is dually equivalent to
MS via a lifting of Stone duality. These equivalences and dual equivalences tie the coalgebraic, algebraic,
and relational treatments of modal logic.
In [7] the situation was lifted from the setting of Stone spaces to compact Hausdorff spaces. We defined
a modal compact Hausdorff space (X, R) to be a compact Hausdorff space with binary relation R satisfying
conditions equivalent to having the associated map from X to its Vietoris space V(X) be continuous. Then
with morphisms again being p-morphisms, we showed the category MKHaus of modal compact Hausdorff
spaces is isomorphic to the category of coalgebras for the Vietoris functor on the category KHaus of compact
Hausdorff spaces. For algebraic counterparts to modal compact Hausdorff spaces, we lifted Isbell duality
between KHaus and compact regular frames, and de Vries duality between KHaus and de Vries algebras,
obtaining categories MKRFrm of modal compact regular frames, and MDV of modal de Vries algebras, each
dually equivalent to MKHaus. For various reasons, the category MDV was a bit poorly behaved. We defined
two full subcategories of MDV, the categories LMDV and UMDV of lower and upper continuous modal de Vries
algebras, that were better behaved, and showed both were equivalent to MDV. The situation is summarized
in Figure 1 below.
The functors in Figure 1 are described in [7]. Those between MKRFrm and MKHaus lift the usual point
and open set functors between compact regular frames and compact Hausdorff spaces, and those between
MDV and MKHaus lift the usual end and regular open set functors between de Vries algebras and compact
Hausdorff spaces. As such, they require the axiom of choice. The composite of these functors then gives
2000 Mathematics Subject Classification. 06D22; 54E05; 54B20.
Key words and phrases. Compact Hausdorff space, Vietoris space, compact regular frame, de Vries algebra, modal operator.
1
2 GURAM BEZHANISHVILI, NICK BEZHANISHVILI, JOHN HARDING
+
6
?
-
Q
Q
Q
Q
Q
Q
Qs
UMDVLMDV MKHaus
MDV
MKRFrm
(−)
L
L
Ω
p
(−)
∗
(−)
∗
(−)
U
U
6
?
Figure 1
an equivalence between MKRFrm and MDV, but again, this requires the axiom of choice. The equivalences
between MDV and its subcategories LMDV and UMDV are choice-free.
A primary purpose here is to give a direct, choice-free proof of the equivalence of MKRFrm and each of
MDV, LMDV and UMDV. To do so, we construct functors L ∶ MKRFrm → LMDV and U ∶ MKRFrm → UMDV
that lift the Booleanization functor in two ways, and a functor R ∶ MDV → MKRFrm that lifts the round
ideal functor. After the preliminaries in Section 2, this equivalence is established in Section 3.
The definition of modal compact regular frames involves identities for the modal operators that appear
in Johnstone’s construction of Vietoris frames [19, 20]. This is not surprising as modal compact regular
frames arise as algebraic counterparts of coalgebras for the Vietoris functor on compact Hausdorff spaces.
The details of this connection are given in Section 4. In this section we also discuss a counterpart of the
Vietoris construction for de Vries algebras.
The equivalence of MKRFrm and MDV of Section 3 composed with the dual equivalence of MDV and
MKHaus of [7] provides an alternative approach to the duality of MKRFrm and MKHaus. Restricted to KRFrm
and KHaus, this composite is a particular case of Hofmann-Lawson duality [17], and closely resembles Stone
duality. In the modal setting, it resembles the familiar duality between modal algebras and modal spaces.
Details of this alternative approach are given in Section 5.
2. Preliminaries
We briefly recall the primary definitions. The reader should consult [7] for complete details.
Definition 2.1. A frame is a complete lattice L where finite meets distribute over infinite joins, and a frame
homomorphism is a map between frames preserving finite meets and infinite joins. A frame is compact if
S = 1 implies there is a finite subset T ⊆ S with
T = 1. Using ¬a for the pseudocomplement of an element
a, we say a is well inside b, and write a ≺ b, if ¬a ∨ b = 1. A frame L is a regular frame if for each b ∈ L we
have b =
{a ∶ a ≺ b}. The category of compact regular frames and the frame homomorphisms between them
is denoted KRFrm.
For more about compact regular frames see, e.g., [18, 4, 19, 22].
Definition 2.2. A modal compact regular frame (abbreviated: MKR-frame) is a triple L = (L, ◻, ◇) where
L is a compact regular frame, and ◻, ◇ are unary operations on L satisfying the following conditions.
(1) ◻ preserves finite meets, so ◻1 = 1 and ◻(a ∧ b) = ◻a ∧ ◻b.
(2) ◇ preserves finite joins, so ◇0 = 0 and ◇(a ∨ b) = ◇a ∨ ◇b.
(3) ◻(a ∨ b) ≤ ◻a ∨ ◇b and ◻a ∧ ◇b ≤ ◇(a ∧ b).
(4) ◻, ◇ preserve directed joins, so ◻
S =
{◻s ∶ s ∈ S}, ◇
S =
{◇s ∶ s ∈ S} for any up-directed S.
An MKR-morphism is a frame homomorphism h that satisfies h(◻a) = ◻h(a) and h(◇a) = ◇h(a). The
category of modal compact regular frames and their morphisms, composed by ordinary function composition,
is denoted MKRFrm.
MODAL OPERATORS ON COMPACT REGULAR FRAMES AND DE VRIES ALGEBRAS 3
We next describe de Vries algebras. Here, as is common, we use the symbol ≺ for a certain type of relation
on a Boolean frame (complete Boolean algebra). This is a different usage than in Definition 2.1, although
there are many connections. For further details see [11, 5, 7, 8], as well as Section 3 below.
Definition 2.3. A de Vries algebra is a pair (A, ≺) where A is a Boolean frame and ≺ is a binary relation
on A, called a proximity, satisfying
(1) 1 ≺ 1.
(2) a ≺ b implies a ≤ b.
(3) a ≤ b ≺ c ≤ d implies a ≺ d.
(4) a ≺ b, c implies a ≺ b ∧ c.
(5) a ≺ b implies ¬b ≺ ¬a.
(6) a ≺ b implies there exists c with a ≺ c ≺ b.
(7) a ≠ 0 implies there exists b ≠ 0 with b ≺ a.
A morphism between de Vries algebras is a function α that satisfies (i) α(0) = 0, (ii) α(a ∧ b) = α(a) ∧ α(b),
(iii) a ≺ b implies ¬α(¬a) ≺ α(b), and (iv) α(a) =
{α(b) ∶ b ≺ a}.
The motivating example of a de Vries algebra is the complete Boolean algebra ROX of regular open
sets of a compact Hausdorff space X with relation ≺ on ROX defined by S ≺ T if CS ⊆ T where C is
usual topological closure. A continuous map f ∶ X → Y between compact Hausdorff spaces gives a de Vries
morphism ICf
−1
[−] from ROY to ROX where I is usual topological interior. In this setting, one can see
that the ordinary function composite of de Vries morphisms need not be a de Vries morphism.
Definition 2.4. For de Vries morphisms α and β, define their composite to be β ⋆ α where
(β ⋆ α)(a) =
{βα(b) ∶ b ≺ a}.
Let DeV be the category of de Vries algebras and their morphisms under this ∗ composition.
Remark 2.5. The idea of a proximity has a long history, see [21] for details. A number of authors have
considered structures closely related to de Vries algebras; see, e.g., [23, 12, 3, 13, 24]. The crucial notion of a
de Vries morphism essential for obtaining categorical duality appears to originate in [11]. Further discussion
can be found in [5, 8].
Definition 2.6. A modal de Vries algebra (abbreviated: MDV-algebra) is a triple A = (A, ≺, ◇) where (A, ≺)
is a de Vries algebra and ◇ is a unary operation on A that satisfies the following conditions.
(1) ◇0 = 0.
(2) a
1
≺ b
1
and a
2
≺ b
2
imply ◇(a
1
∨ a
2
) ≺ ◇b
1
∨ ◇b
2
.
A morphism between modal de Vries algebras is a de Vries morphism α for which a ≺ b implies both
α(◇a) ≺ ◇α(b) and ◇α(a) ≺ α(◇b). Let MDV be the category of modal de Vries algebras and morphisms
with composition being the ∗ composition of Definition 2.4.
Two full subcategories of MDV play an important role in [7], and also in our considerations here.
Definition 2.7. An MDV-algebra (A, ≺, ◇) is called lower continuous if ◇a =
{◇b ∶ b ≺ a} and upper
continuous if ◇a =
{◇b ∶ a ≺ b}. Let LMDV and UMDV be the full subcategories of MDV consisting of all
lower, respectively upper, continuous MDV-algebras.
We recall that in [7, Sec. 4.3] it was shown that each member of MDV is isomorphic to a member of LMDV
and to a member of UMDV, this despite the fact that a modal de Vries algebra need be neither lower nor
upper continuous. This somewhat counterintuitive situation is due to the fact that composition in MDV is
not function composition, and isomorphisms are not structure preserving bijections.
3. Equivalence of MKRFrm, MDV, LMDV, and UMDV
In this section we provide direct equivalences between MKRFrm and each of MDV, LMDV, and UMDV.
These proofs do not rely on the axiom of choice, as did ones in [7].
Definition 3.1. For a de Vries algebra (A, ≺) and S ⊆ A, define ↓S = {a ∶ a ≤ s for some s ∈ S}, and
↡S = {a ∶ a ≺ s for some s ∈ S}. An ideal I of A is called round if I = ↡I.
4 GURAM BEZHANISHVILI, NICK BEZHANISHVILI, JOHN HARDING
It is known (see, e.g., [3, Lem. 2] or [8, Prop. 4.6]) that the collection RA of all round ideals of A is a
subframe of the frame of all ideals of A.
Definition 3.2. For A = (A, ≺, ◇) an MDV-algebra, define ◻ on A by setting ◻a = ¬ ◇ ¬a for all a ∈ A.
Lemma 3.3. Let A = (A, ≺, ◇) be an MDV-algebra and a ≺ b, a
1
≺ b
1
, a
2
≺ b
2
. Then
(1) ◇a ≺ ◇b and ◻a ≺ ◻b.
(2) ◇(a
1
∨ a
2
) ≺ ◇b
1
∨ ◇b
2
and ◻a
1
∧ ◻a
2
≺ ◻(b
1
∧ b
2
).
(3) ◻(a
1
∨ a
2
) ≺ ◻b
1
∨ ◇b
2
and ◻a
1
∧ ◇a
2
≺ ◇(b
1
∧ b
2
).
Proof. The definition of an MDV-algebra gives ◇a ≺ ◇b and ◇(a
1
∨a
2
) ≺ ◇b
1
∨◇b
2
. In any de Vries algebra
we have a ≺ b iff ¬b ≺ ¬a. This gives ◻a ≺ ◻b and ◻a
1
∧◻a
2
≺ ◻(b
1
∧b
2
). So (1) and (2) are established. For (3)
use interpolation to find a
1
≺ c
1
≺ d
1
≺ b
1
and a
2
≺ c
2
≺ d
2
≺ b
2
. Then a
1
∨a
2
≺ c
1
∨c
2
and ¬d
2
≺ ¬c
2
, so by (2),
◻(a
1
∨ a
2
) ∧◻¬d
2
≺ ◻((c
1
∨ c
2
) ∧¬c
2
). As (c
1
∨ c
2
) ∧¬c
2
≤ c
1
≺ d
1
, applying (1) gives ◻(a
1
∨ a
2
) ∧◻¬d
2
≺ ◻d
1
,
hence ◻(a
1
∨ a
2
) ≤ ◻d
1
∨ ¬ ◻ ¬d
2
= ◻d
1
∨ ◇d
2
. Finally use (1) once again to obtain ◻d
1
∨ ◇d
2
≺ ◻b
1
∨ ◇b
2
.
This gives the first statement in (3). Using that x ≺ y iff ¬y ≺ ¬x, the second statement in (3) is equivalent
to ◻(¬b
1
∨ ¬b
2
) ≺ ◇¬a
1
∨ ◻¬a
2
, which is equivalent to the first.
Definition 3.4. For A = (A, ≺, ◇) an MDV-algebra, define R A = (RA,
◻, ◇) where RA is the frame of
round ideals of A and ◻, ◇ are given by ◻(I) = ↡◻[I] and ◇(I) = ↡◇[I].
Proposition 3.5. If A is an MDV-algebra, then R A is an MKR-frame.
Proof. It is well-known that RA is a subframe of the ideal frame of A that is compact regular (see, e.g., [3]
or [6]). It is easy to see that ◻(I) and ◇(I) are round ideals so ◻, ◇ are well defined. By Lemma 3.3.1, both
◻, ◇ are proximity preserving on A, so we can alternately describe ◻(I) = ↓◻[I] and ◇(I) = ↓◇[I].
We must verify the conditions of Definition 2.2. As ◇0 = 0 and ◻1 = 1, we have ◇0 = 0 and ◻1 = 1. Clearly
◻ and ◇ are order-preserving, so ◇(I)∨◇(J) ⊆ ◇(I ∨J) and ◻(I ∧J ) ⊆ ◻(I)∧◻(J). If a
1
∈ I and a
2
∈ J , then
roundness gives b
1
∈ I and b
2
∈ J with a
1
≺ b
1
and a
2
≺ b
2
. Then Lemma 3.3.2 gives ◇(a
1
∨ a
2
) ≺ ◇b
1
∨ ◇b
2
,
showing ◇(I ∨ J) ⊆ ◇(I) ∨ ◇(J ), and ◻a
1
∧ ◻a
2
≺ ◻(b
1
∧ b
2
), showing ◻(I ∧ J ) ⊆ ◻(I) ∧ ◻(J). Thus ◇
is finitely additive and ◻ is finitely multiplicative. Also, Lemma 3.3.3 gives ◻(a
1
∨ a
2
) ≺ ◻b
1
∨ ◇b
2
and
◻a
1
∧ ◇a
2
≺ ◇(b
1
∧ b
2
), showing
◻(I ∨ J) ⊆ ◻(I) ∨ ◇(J ) and ◻(I) ∧ ◇(J) ⊆ ◇(I ∧ J). Finally, directed joins
in RA are given by unions, and it follows easily that both ◻ and ◇ preserve directed joins.
Theorem 3.6. The assignment A ↦ R A can be extended to a functor R ∶ MDV → MKRFrm by setting
Rα = ↡α[⋅] for an MDV-morphism α ∶ A → B.
Proof. It is known [6, Rem. 3.10] that the “restriction” of R gives a functor R ∶ DeV → KRFrm, so it
remains only to show that the frame homomorphism Rα is an MKR-morphism. This means we must show
(Rα)(◇I) = ◇((Rα)I) and (Rα)(◻I) = ◻((Rα)I) for each round ideal I of A. This follows directly once
we show a ≺ b implies (i) α(◇a) ≺ ◇α(b), (ii) ◇α(a) ≺ α(◇b), (iii) α(◻a) ≺ ◻α(b), and (iv) ◻α(a) ≺ α(◻b).
Items (i) and (ii) are part of the definition of an MDV-morphism. For (iii), use interpolation to find
a ≺ c ≺ d ≺ b and recall that an MDV-morphism also satisfies x ≺ y implies α(¬y) ≺ ¬α(x) and ¬α(y) ≺ α(¬x).
Then as ◇¬c ≺ ◇¬a we have α(◻a) = α(¬◇¬a) ≺ ¬α(◇¬c), and as ¬d ≺ ¬c we have ◇α(¬d) ≺ α(◇¬c), hence
α(◻a) ≺ ¬α(◇¬c) ≺ ¬◇α(¬d). But d ≺ b gives ¬α(b) ≺ α(¬d), hence α(◻a) ≺ ¬◇α(¬d) ≺ ¬◇¬α(b) = ◻α(b).
This gives (iii), and a similar calculation provides (iv).
Next we construct a functor from MKRFrm to MDV. In fact, we will construct two functors, one will have
image in LMDV and the other in UMDV.
Lemma 3.7 ([7, Lem. 3.6]). Let L = (L, ◻, ◇) be an MKR-frame and a, b ∈ L. Then
(1) ◇a ≤ ¬ ◻ ¬a and ◻a ≤ ¬ ◇ ¬a.
(2) If a ≺ b, then ◇a ≺ ◇b and ◻a ≺ ◻b.
(3) If a ≺ b, then ¬ ◻ ¬a ≺ ◇b and ¬ ◇ ¬a ≺ ◻b.
(4) If a ≺ b, then ◻a ≺ ¬ ◇ ¬b and ◇a ≺ ¬ ◻ ¬b.
Recall that for a compact regular frame L, the operation ¬¬ is a closure operator on L whose fixed points
BL are a de Vries algebra with proximity given by the restriction of the well inside relation ≺ on L [6,
Lem. 3.1]. Meets in BL agree with those in L, joins are given by applying the closure operator ¬¬ to the
join in L. We use ⊔ for finite joins in BL and
for infinite joins.