[1]Litan Kumar Das

[1]Department of Mathematics, Jadavpur University, Kolkata, India

2]ECSU, Indian Statistical Institute, Kolkata, India

3]Department of Mathematics, Jadavpur University, Kolkata, India

# Bi-coalgebraic view of Fitting’s Heyting-valued modal logic

<EMAIL_ADDRESS>Kumar Sankar Ray<EMAIL_ADDRESS>Prakash Chandra Mali
<EMAIL_ADDRESS>*  [  [

###### Abstract

Fitting’s Heyting-valued logic and Heyting-valued modal logic have already
been studied from an algebraic viewpoint. In addition to algebraic
axiomatizations with the completeness of Fitting’s logic and modal logic,
topological duality theorems have also been developed. Bitopological methods
have recently been employed to investigate duality for Heyting-valued logic.
However, bitopology and biVietoris-coalgebra techniques are conspicuously
absent from the development of duality for Heyting-valued modal logic. In this
paper, we are attempting to close this gap. We develop a bitopological duality
for algebras of Fitting’s Heyting-valued modal logic. We construct a bi-
Vietoris functor on the category $PBS_{\mathcal{L}}$ of Heyting-valued version
of pairwise Boolean spaces. Finally, we obtain a dual equivalence between
categories of bi-Vietoris coalgebras and algebras of Fitting’s Heyting-valued
modal logic. As a result, we have that Fitting’s many-valued modal logic is
sound and complete with respect to coalgebras of a bi-Vietoris functor.

###### keywords:

Bitopology, Fitting’s modal logic, Duality, Coalgebra

##  1 Introduction

Algebraic axiomatization of a modified version of Fitting’s Heyting-valued
modal logic has already been addressed in maruyama2009algebraic . Maruyama
maruyama2011dualities  proposed Jónsson-Tarski topological duality (see
blackburn2001modal ; hansoul1983duality ; jonsson1951boolean ) for
$\mathcal{L}$-$\mathcal{ML}$-algebras (algebras of Fitting’s Heyting-valued
modal logic). Jónsson-Tarski duality for $\mathcal{L}$-$\mathcal{ML}$-algebras
is essentially a $\mathcal{L}$-valued version of Jónsson-Tarski duality for
modal algebras.
We aim to construct a bitopological duality for algebras of Fitting’s Heyting-
valued modal logic by setting up a notion of $PRBS_{\mathcal{L}}$ as the
category of $\mathcal{L}$-valued version of pairwise Boolean spaces with a
relation. As a result, natural duality theory for modal algebras is extended
in the context of bitopological languages. The main results are bitopological
and coalgebraic dualities for $\mathcal{L}$-$\mathcal{ML}$-algebras, where
$\mathcal{L}$ is a semi-primal algebra having a bounded lattice reduct. Our
general theory extends both the Jónsson-Tarski duality and Abramsky-Kupke-
Kurz-Venema coalgebraic duality abramsky2011cook ; kupke2004stone  in the
setting of bitopological language. Furthermore, it introduces a novel
coalgebraic duality for $\mathcal{L}$-$\mathcal{ML}$-algebras.
An exemplary story in coalgebraic logic can be found in kurz2006coalgebras .
The Stone duality stone1938representation  between Boolean algebras and sets
represents syntax and semantics of propositional logic. The algebras and
coalgebras of an endofunctor on a category define the syntax and semantics of
the modal propositional logic. As an illustration, the modal logic $K$ and
Kripke semantics derive from the Stone duality by taking an endofunctor on
sets. So, in acceptable circumstances, we can achieve duality between the
relevant algebras and coalgebras. In addition to demonstrating the fact that
the widely recognized Stone duality could be articulated in coalgebraic terms,
Abramsky abramsky2011cook  also showed that a coalgebraic formulation could be
provided for the Jónsson-Tarski duality between descriptive general Kripke
frames and modal algebras(see also kupke2004stone  for further information).
In particular, the category of descriptive general Kripke frames is isomorphic
to the category of Boolean spaces i.e., zero-dimensional, compact and
Hausdorff spaces. Esakia esakia1974topological  also noticed this connection.
Therefore, coalgebras for the Vietoris functor on the category of Boolean
spaces can represent sound and complete semantics for modal logic. In
palmigiano2004coalgebraic , the author showed that coalgebras of a Vietoris
functor on the category of Priestley spaces i.e., compact, totally ordered
disconnected spaces, provide sound and complete semantics for positive modal
logic. The objective of this work is to combine the idea that the semantics of
Fitting’s many-valued modal logic can be understood as coalgebras for the bi-
Vietoris functor on the category $PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued
pairwise Boolean spaces and pairwise continuous maps. The structure of the
paper is as follows. In Section 2, we review the fundamental notions of
bitopological spaces and algebras of Fitting’s Heyting-valued logic, denoted
as $\mathcal{L}$-$\mathcal{VL}$-algebras, and we also discuss the
bitopological duality theorem for $\mathcal{L}$-$\mathcal{VL}$-algebras.
Section 3 contains our first result i.e., bitopological duality for Fitting’s
Heyting-valued modal logic. In Section 4, we introduce the concept of pairwise
Vietoris space and construct an endofunctor $V_{\mathcal{L}}^{bi}$ on the
category $PBS_{\mathcal{L}}$. Finally, we show the coalgebraic duality for
Fitting’s Heyting-valued modal logic. In Section 5, we conclude the paper by
outlining several potential future research directions.

##  2 Preliminaries

We assume that the readers are familiar with the basic concepts of topology
and category theory. We refer the reader to burris1981course ;
davey2002introduction  for information on universal algebra and lattice
theory. For category theory, we refer to adamek1990abstract .

###  2.1 Bitopological spaces

A bitopological space is defined as a triple $(X,\tau_{1},\tau_{2})$ in which
$(X,\tau_{1})$ and $(X,\tau_{2})$ are topological spaces. Consider
$\delta_{1}$ and $\delta_{2}$ represent, respectively, the collections of
$\tau_{1}$-closed sets and $\tau_{2}$-closed sets. We set
$\beta_{1}=\tau_{1}\cap\delta_{2}$ and $\beta_{2}=\tau_{2}\cap\delta_{1}$.

######  Definition 1.

salbany1974bitopological

  1. (i)

A bitopoological space $(X,\tau_{1},\tau_{2})$ is said to be pairwise
Hausdorff space if for every pair $(x,y)$ of distinct points $x,y\in X$ there
exists disjoint open sets $U_{x}\in\tau_{1}$ and $U_{y}\in\tau_{2}$ containing
$x$ and $y$, respectively.

  2. (ii)

A bitopoological space $(X,\tau_{1},\tau_{2})$ is said to be pairwise zero-
dimensional if $\beta_{1}$ is a basis for $\tau_{1}$ and $\beta_{2}$ is a
basis for $\tau_{2}$.

  3. (iii)

A bitopoological space $(X,\tau_{1},\tau_{2})$ is said to be pairwise compact
if the topological space $(X,\tau)$, where $\tau=\tau_{1}\vee\tau_{2}$, is
compact.

According to Alexander’s Lemma( a classical result in general topology), the
idea of pairwise compact described in Definition 1 is equivalent to the
condition that every cover $\\{U:U\in\tau_{1}\cup\tau_{2}\\}$ of X has a
finite subcover. A pairwise Boolean space is a bitopological space that is
pairwise Hausdorff, pairwise zero-dimensional, and pairwise compact. A map
$f:(P,\tau_{1},\tau_{2})\to(P_{1},\tau_{1}^{1},\tau_{2}^{1})$ is said to be
pairwise continuous if the map $f:(P,\tau_{i})\to(P_{1},\tau_{i}^{1})$ is
continuous for $I\in\\{1,2\\}$. Pairwise Boolean spaces and pairwise
continuous maps form a category, denoted by $PBS$.

######  Proposition 1.

lauridsen2015bitopological  If $T_{1}$ and $T_{2}$ are subbasis for the
topologies $\tau_{1}$ and $\tau_{2}$, respectively, then $T_{1}\cup T_{2}$ is
a subbasis for the topology $\tau_{1}\vee\tau_{2}$.

######  Proposition 2.

lauridsen2015bitopological  Let $(X,\tau_{1},\tau_{2})$ be a pairwise compact
bitopological space. Consider a finite collection
$\\{C_{i}:C_{i}\in\delta_{1}\cup\delta_{2},i=1,2,\cdots,n\\}$ of subsets of
$X$. Then $\displaystyle\bigcap_{i=1}^{n}C_{i}$ is pairwise compact.

It is clear from the above proposition that any $\tau_{1}$-closed or
$\tau_{2}$-closed subset of a pairwise compact space $X$ is pairwise compact.

###  2.2 Fitting’s Heyting-valued logics

Fitting fitting1991many  proposed $\mathcal{L}$-valued logics and
$\mathcal{L}$-valued modal logics for a finite distributive lattice
$\mathcal{L}$(i.e., $\mathcal{L}$ is a Heyting algebra) in 1991. Maruyama
maruyama2009algebraic  introduced algebraic axiomatization of Fitting’s
logics. In maruyama2009algebraic  the author studied Fitting’s Heyting-valued
logic and Heyting-valued modal logic without regard for fuzzy truth constants
other than $0$ and $1$, and added a new operation $T_{\ell}(-)$,
$\ell\in\mathcal{L}$. From a logical perspective, $T_{\ell}(p)$ infers that
the truth value of a proposition $p$ is $\ell$. The operations of
$\mathcal{L}$-valued logic, denoted by $\mathcal{L}$-$\mathcal{VL}$, are
$\vee,\wedge,\rightarrow,0,1$ and $T_{\ell}(-)$, $\ell\in\mathcal{L}$, where
$\vee,\wedge,\rightarrow$ are binary operations, $0$ and $1$ are nullary
operations and $T_{\ell}$ is a unary operation. For
$\ell_{1},\ell_{2}\in\mathcal{L}$, $\ell_{1}\rightarrow\ell_{2}$ means the
pseudo-complement of $\ell_{1}$ relative to $\ell_{2}$.
The following lemmas describe some term functions.

######  Lemma 1.

Define a function $f:\mathcal{L}^{4}\to\mathcal{L}$ by

$f(\ell_{1},\ell_{2},\ell_{3},\ell_{4})=\left\\{\begin{array}[]{ll}\ell_{3}&(\ell_{1}=\ell_{2})\\\
\ell_{4}&(\ell_{1}\neq\ell_{2})\\\ \end{array}\right.$

Then, $f$ is a term function of $\mathcal{L}$.

######  Lemma 2.

For every $\ell\in\mathcal{L}$, define $T_{\ell}:\mathcal{L}\to\mathcal{L}$ by

$T_{\ell}(\ell^{\prime})=\left\\{\begin{array}[]{ll}1&(\ell^{\prime}=\ell)\\\
0&(\ell^{\prime}\neq\ell)\\\ \end{array}\right.$

Then, $T_{\ell}$ is a term function of $\mathcal{L}$.

######  Lemma 3.

Let $\ell\in\mathcal{L}$. Then the function
$U_{\ell}:\mathcal{L}\to\mathcal{L}$ defined by

$U_{\ell}(\ell^{\prime})=\left\\{\begin{array}[]{ll}1&(\ell^{\prime}\geq\ell)\\\
0&(\ell^{\prime}\ngeq\ell)\\\ \end{array}\right.$

, is a term function of $\mathcal{L}$.

We recall the idea of $\mathcal{L}$-$\mathcal{VL}$-algebras, which provides
sound and complete semantics of $\mathcal{L}$-valued logic
$\mathcal{L}$-$\mathcal{VL}$.

######  Definition 2.

maruyama2009algebraic  An algebraic structure
$(\mathcal{A},\wedge,\vee,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1)$ is
said to be a $\mathcal{L}$-$\mathcal{VL}$-algebra iff the following conditions
hold for any $\ell_{1},\ell_{2}\in\mathcal{L}$, and $a,b\in\mathcal{A}$:

  1. (i)

$(\mathcal{A},\wedge,\vee,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1)$ is a
Heyting algebra;

  2. (ii)

$T_{\ell_{1}}(a)\wedge T_{\ell_{2}}(b)\leq
T_{\ell_{1}\rightarrow\ell_{2}}(a\rightarrow b)\wedge
T_{\ell_{1}\wedge\ell_{2}}(a\wedge b)\wedge T_{\ell_{1}\vee\ell_{2}}(a\vee
b)$;
$T_{\ell_{2}}(a)\leq T_{T_{\ell_{1}}(\ell_{2})}(T_{\ell_{1}}(a))$;

  3. (iii)

$T_{0}(0)=1$; $T_{\ell}(0)=0$ ($\ell\neq 0$); $T_{1}(1)=1$; $T_{\ell}(1)=0$,
if $\ell\neq 1$;

  4. (iv)

$\bigvee\\{T_{\ell}(a):\ell\in\mathcal{L}\\}=1$;
$T_{\ell_{1}}(a)\vee(T_{\ell_{2}}(a)\rightarrow 0)=1$;
$T_{\ell_{1}}(a)\wedge T_{\ell_{2}}(a)=0$, $(\ell_{1}\neq\ell_{2})$;

  5. (v)

$T_{1}(T_{\ell}(a))=T_{\ell}(a)$, $T_{0}(T_{\ell}(a))=T_{\ell}(a)\rightarrow
0$, $T_{\ell_{2}}(T_{\ell_{1}}(a))=0$, $(\ell_{2}\neq 0,1)$;

  6. (vi)

$T_{1}(a)\leq a$, $T_{1}(a\wedge b)=T_{1}(a)\wedge T_{1}(b)$;

  7. (vii)

$\displaystyle\bigwedge_{\ell\in\mathcal{L}}(T_{\ell}(a)\leftrightarrow
T_{\ell}(b))\leq(a\leftrightarrow b)$.

######  Note 1.

The class of all $\mathcal{L}$-$\mathcal{VL}$-algebras forms a variety (in the
sense of universal algebra). If $\mathcal{L}=\\{0,1\\}$, then
$\mathcal{L}$-$\mathcal{VL}$-algebras becomes Boolean algebras.

######  Definition 3.

A function between $\mathcal{L}$-$\mathcal{VL}$-algebras is said to be
homomorphism if it preserves the operations
$\vee,\wedge,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1$.

Let $\mathcal{VA}_{\mathcal{L}}$ denote the category of
$\mathcal{L}$-$\mathcal{VL}$-algebras.
$\mathcal{L}$-valued modal logic denoted by $\mathcal{L}$-$\mathcal{ML}$, is
defined by $\mathcal{L}$-valued Kripke semantics. The idea of
$\mathcal{L}$-valued Kripke semantics can be found in maruyama2011dualities .
The operations of $\mathcal{L}$-valued modal logic
$\mathcal{L}$-$\mathcal{ML}$ are the operations of
$\mathcal{L}$-$\mathcal{VL}$ and a unary operation $\Box$, called modal
operation. We now recall the concept of $\mathcal{L}$-$\mathcal{ML}$-algebras,
which defines a sound and complete semantics for $\mathcal{L}$-$\mathcal{ML}$.

######  Definition 4.

maruyama2009algebraic  An algebraic structure
$(\mathcal{A},\wedge,\vee,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),\Box,0,1)$
is said to be a $\mathcal{L}$-$\mathcal{ML}$-algebra iff it satisfies the
following conditions:

  1. (i)

$(\mathcal{A},\wedge,\vee,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1)$ is a
$\mathcal{L}$-$\mathcal{VL}$-algebra;

  2. (ii)

$\Box(a\wedge b)=\Box a\wedge\Box b$;

  3. (iii)

$\Box U_{\ell}(a)=U_{\ell}(\Box a)$, $\forall\ell\in\mathcal{L}$, where the
unary operation $U_{\ell}(\ell\in\mathcal{L})$ is defined by
$U_{\ell}(a)=\bigvee\\{T_{\ell^{\prime}}(a):\ell\leq\ell^{\prime},\ell^{\prime}\in\mathcal{L}\\}$,
$a\in\mathcal{A}$. Logically it means that truth value of $a$ is greater than
or equal to $\ell$.

A homomorphism of $\mathcal{L}$-$\mathcal{ML}$-algebras is a function that
preserves all the operations of $\mathcal{L}$-$\mathcal{VL}$-algebras and the
modal operation $\Box$. Let $\mathcal{MA}_{\mathcal{L}}$ denote the category
of $\mathcal{L}$-$\mathcal{ML}$-algebras and homomorphisms of
$\mathcal{L}$-$\mathcal{ML}$-algebras.
For a Kripke frame $(P,\mathcal{R})$, $\mathcal{R}[x]=\\{y\in
P:x\mathcal{R}y\\}$, where $x\in P$, and $\mathcal{R}^{-1}[P^{\prime}]=\\{y\in
P:\exists x\in P^{\prime},y\mathcal{R}x\\}$, where $P^{\prime}\subseteq P$. We
recall a modal operation $\Box_{\mathcal{R}}$ on $\mathcal{L}$-valued powerset
algebra $\mathcal{L}^{P}$ of $P$.

######  Definition 5.

maruyama2009algebraic  Let $(P,\mathcal{R})$ be a Kripke frame and
$f\in\mathcal{L}^{P}$. Then $\Box_{\mathcal{R}}f:P\to\mathcal{L}$ is defined
by $(\Box_{\mathcal{R}}f)(x)=\bigwedge\\{f(y):y\in\mathcal{R}[x]\\}$.

######  Definition 6.

maruyama2011dualities  Let $\mathcal{A}$ be an object in
$\mathcal{MA}_{\mathcal{L}}$. A binary relation $\mathcal{R}_{\Box}$ on
$HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$ is defined as
follows:
$\psi\mathcal{R}_{\Box}\phi\iff\forall\ell\in\mathcal{L},\forall
a\in\mathcal{A},\psi(\Box a)\geq\ell\Rightarrow\phi(a)\geq\ell$.

A $\mathcal{L}$-valued map
$\mathcal{D}:HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})\times\mathcal{A}\to\mathcal{L}$
is defined by $\mathcal{D}(\psi,a)=\psi(a),\psi\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$.

######  Lemma 4.

maruyama2011dualities  The $\mathcal{L}$-valued canonical model
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),\mathcal{R}_{\Box},\mathcal{D})$
of $\mathcal{A}$ is a $\mathcal{L}$-valued Kripke model. Then,
$\mathcal{D}(\psi,\Box a)=\psi(\Box
a)=\bigwedge\\{\phi(a):\phi\in\mathcal{R}_{\Box}[\psi]\\}$.

###  2.3 Bitopological duality for Fitting’s Heyting-valued logic

We will introduce the key ideas and findings from the bitopological duality
theory for Fitting’s Heyting-valued logic. We refer to das2021bitopological
for a more thorough explanation of the bitopological duality for Fitting’s
Heyting-valued logic. Let $\mathcal{S}_{\mathcal{L}}$ denote the collection of
subalgebras of $\mathcal{L}$. For a pairwise Boolean space $\mathcal{B}$,
$\Lambda_{\mathcal{B}}$ denotes the collection of pairwise closed subspaces of
$\mathcal{B}$. It is shown in adnadzhevich1987bicompactness  that a pairwise
closed subset of a pairwise compact space is also pairwise compact. Hence,
each member of $\Lambda_{\mathcal{B}}$ is pairwise Boolean space. A finite
distributive lattice $\mathcal{L}$ endowed with unary operation
$T_{\ell}(\ell\in\mathcal{L})$ forms a semi-primal algebra. We have expanded
the theory of natural duality clark1998natural  by creating a bitopological
duality for $\mathcal{L}$-$\mathcal{VL}$-algebras das2021bitopological .
We now recall the category $PBS_{\mathcal{L}}$ from das2021bitopological .

######  Definition 7 (das2021bitopological ).

The category $PBS_{\mathcal{L}}$ is defined as follows:

  1. (1)

Objects$\colon$ An object in $PBS_{\mathcal{L}}$ is a tuple
$(\mathcal{B},\alpha_{\mathcal{B}})$ where $\mathcal{B}$ is a pairwise Boolean
space and a mapping
$\alpha_{\mathcal{B}}:\mathcal{S}_{\mathcal{L}}\to\Lambda_{\mathcal{B}}$
satisfies the following conditions:

    1. (i)

$\alpha_{\mathcal{B}}(\mathcal{L})=\mathcal{B}$;

    2. (ii)

if
$\mathcal{L}_{1}=\mathcal{L}_{2}\wedge\mathcal{L}_{3}(\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}\in\mathcal{L})$,
then
$\alpha_{\mathcal{B}}(\mathcal{L}_{1})=\alpha_{\mathcal{B}}(\mathcal{L}_{2})\cap\alpha_{\mathcal{B}}(\mathcal{L}_{3})$.

  2. (2)

Arrows$\colon$ An arrow
$\psi:(\mathcal{B}_{1},\alpha_{\mathcal{B}_{1}})\to(\mathcal{B}_{2},\alpha_{\mathcal{B}_{2}})$
in $PBS_{\mathcal{L}}$ is a pairwise continuous map
$\psi:\mathcal{B}_{1}\to\mathcal{B}_{2}$ that satisfies the criterion that if
$x\in\alpha_{\mathcal{B}_{1}}(\mathcal{L}_{1})(\mathcal{L}_{1}\in\mathcal{S}_{\mathcal{L}})$,
then $\psi(x)\in\alpha_{\mathcal{B}_{2}}(\mathcal{L}_{1})$ i.e., $\psi$ is a
subspace preserving map.

######  Note 2.

  1. (1)

The bitopological space $(\mathcal{L},\tau,\tau)$, where $\tau$ is the
discrete topology on $\mathcal{L}$, is a pairwise Boolean space. Hence,
$(\mathcal{L},\tau,\tau,\alpha_{\mathcal{L}})$, where $\alpha_{\mathcal{L}}$
is a mapping from $\mathcal{S}_{\mathcal{L}}$ to $\Lambda_{\mathcal{L}}$ that
is defined by
$\alpha_{\mathcal{L}}(\mathcal{L}^{\prime})=\mathcal{L}^{\prime}$, is an
object in $PBS_{\mathcal{L}}$.

  2. (2)

For an object $\mathcal{A}$ in $\mathcal{VA}_{\mathcal{L}}$ , consider a
bitopological space
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),\tau_{1},\tau_{2})$,
where the topologies $\tau_{1}$ and $\tau_{2}$ are generated by the bases
$B^{\tau_{1}}=\\{\langle a\rangle:a\in\mathcal{A}\\}$, where $\langle
a\rangle=\\{h\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}):h(a)=1\\}$, and
$B^{\tau_{2}}=\\{B^{c}:B\in B^{\tau_{1}}\\}$, respectively. Here $B^{c}$
denotes the complement of $B$.

######  Fact 1 (das2021bitopological ).

The bitopological space
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),tau_{1},\tau_{2})$
is a pairwise Boolean space.

The duality between the categories $\mathcal{VA}_{\mathcal{L}}$ and
$PBS_{\mathcal{L}}$ is obtained via the following functors.

######  Definition 8 (das2021bitopological ).

A contravariant functor
$\mathfrak{F}:PBS_{\mathcal{L}}\to\mathcal{VA}_{\mathcal{L}}$ is defned as
follows:

  1. (i)

For an object $(\mathcal{B},\alpha_{\mathcal{B}})$ in $PBS_{\mathcal{L}}$,
define
$\mathfrak{F}(\mathcal{B},\alpha_{\mathcal{B}})=(HOM_{PBS_{\mathcal{L}}}((\mathcal{B},\alpha_{\mathcal{B}}),(\mathcal{L},\alpha_{\mathcal{L}})),\vee,\wedge,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1)$,
where $\vee,\wedge,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1$ are pointwise
operations on the set
$HOM_{PBS_{\mathcal{L}}}((\mathcal{B},\alpha_{\mathcal{B}}),(\mathcal{L},\alpha_{\mathcal{L}}))$.
The operations $0$ and $1$ are regarded as constant functions, with $0$ and
$1$ being their respective values.

  2. (ii)

For an arrow
$\phi:(\mathcal{B},\alpha_{\mathcal{B}})\to(\mathcal{B}^{\prime},\alpha_{\mathcal{B}^{\prime}})$
in $PBS_{\mathcal{L}}$, define
$\mathfrak{F}(\phi):\mathfrak{F}((\mathcal{B}^{\prime},\alpha_{\mathcal{B}^{\prime}}))\to\mathfrak{F}((\mathcal{B},\alpha_{\mathcal{B}}))$
by $\mathfrak{F}(\phi)(\zeta)=\zeta\circ\phi$, where $\zeta\in
HOM_{PBS_{\mathcal{L}}}((\mathcal{B}^{\prime},\alpha_{\mathcal{B}^{\prime}}),(\mathcal{L},\alpha_{\mathcal{L}}))$.

######  Definition 9 (das2021bitopological ).

A contravariant functor $\mathfrak{G}:\mathcal{VA}_{\mathcal{L}}\to
PBS_{\mathcal{L}}$ is defined as follows:

  1. (i)

$\mathfrak{G}$ acts on an object $\mathcal{A}$ in $\mathcal{VA}_{\mathcal{L}}$
as
$\mathfrak{G}(\mathcal{A})=(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),\tau_{1},\tau_{2},\alpha_{\mathcal{A}})$,
where $\alpha_{\mathcal{A}}$ is a mapping from $\mathcal{S}_{\mathcal{L}}$ to
$\Lambda_{HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})}$ which is
defined by
$\alpha_{\mathcal{A}}(\mathcal{L}^{*})=HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}^{*})$,
$\mathcal{L}^{*}\in\mathcal{S}_{\mathcal{L}}$.

  2. (ii)

$\mathfrak{G}$ acts on an arrow $\psi:\mathcal{A}\to\mathcal{A}^{*}$ in
$\mathcal{VA}_{\mathcal{L}}$ as follows:
$\mathfrak{G}(\psi):\mathfrak{G}(\mathcal{A}^{*})\to\mathfrak{G}(\mathcal{A})$
is defined by $\mathfrak{G}(\psi)(\phi)=\phi\circ\psi$,
$\phi\in\mathfrak{G}(\mathcal{A}^{*})$.

In das2021bitopological  the following duality result is proved for
$\mathcal{L}$-$\mathcal{VL}$-algebras.

######  Theorem 1.

The categories $\mathcal{VA}_{\mathcal{L}}$ and $PBS_{\mathcal{L}}$ are dually
equivalent.

##  3 Bitopological duality for Fitting’s modal logic

###  3.1 Category

######  Definition 10.

We define a category $PRBS_{\mathcal{L}}$ as follows:

  1. (1)

Objects: An object in $PRBS_{\mathcal{L}}$ is a triple
$(P,\alpha_{P},\mathcal{R})$ such that $(P,\alpha_{P})$ is an object in
$PBS_{\mathcal{L}}$ and $\mathcal{R}$ is a binary relation on $P$ that
satisfies the following conditions:

    1. (i)

for each $p$ in $P$, $\mathcal{R}[p]$ is a pairwise compact subset of $P$;

    2. (ii)

$\forall\mathcal{C}\in\beta_{1}$,
$[\mathcal{R}]\mathcal{C},\langle\mathcal{R}\rangle\mathcal{C}\in\beta_{1}$;

    3. (iii)

for any $\mathcal{L^{\prime}}\in\mathfrak{S}_{\mathcal{L}}$, if
$m\in\alpha_{P}(\mathcal{L^{\prime}})$ then
$\mathcal{R}[m]\subseteq\alpha_{P}(\mathcal{L^{\prime}})$.

  2. (2)

Arrows: An arrow
$f:(P,\alpha_{P},\mathcal{R})\to(P^{\prime},\alpha_{P^{\prime}},\mathcal{R}^{\prime})$
in $PRBS_{\mathcal{R}}$ is an arrow in $PBS_{\mathcal{L}}$ which additionally
satisfies the following conditions:

    1. (i)

if $p_{1}\mathcal{R}p_{2}$ then $f(p_{1})\mathcal{R}^{\prime}f(p_{2})$;

    2. (ii)

if $f(p)\mathcal{R}^{\prime}p^{\prime}$ then $\exists p^{*}\in P$ such that
$p\mathcal{R}p^{*}$ and $f(p^{*})=p^{\prime}$.

In the next sequel, we introduce functors $\mathcal{F}$ and $\mathcal{G}$ to
establish the dual equivalence between the categories
$\mathcal{MA}_{\mathcal{L}}$ and $PRBS_{\mathcal{L}}$.

###  3.2 Functors

######  Definition 11.

We define a functor $\mathcal{G}:\mathcal{MA}_{\mathcal{L}}\to
PRBS_{\mathcal{L}}$.

  1. (i)

$\mathcal{G}$ acts on an object $(\mathcal{A},\Box)$ in
$\mathcal{MA}_{\mathcal{L}}$ as
$\mathcal{G}(\mathcal{A})=(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),tau_{1},\tau_{2},\alpha_{\mathcal{A}},\mathcal{R}_{\Box})$,
where $\alpha_{\mathcal{A}}$ is a mapping from $\mathfrak{S}_{\mathcal{L}}$ to
$\Lambda_{HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})}$ defined
by
$\alpha_{\mathcal{A}}(\mathcal{L}_{1})=HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}_{1})$,
and $\mathcal{R}_{\Box}$ is a binary relation on
$HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$ that is described
in Definition 6.

  2. (ii)

$\mathcal{G}$ acts on an arrow $\psi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ in
$\mathcal{MA}_{\mathcal{L}}$ as follows:
Define
$\mathcal{G}(\psi):\mathcal{G}(\mathcal{A}_{2})\to\mathcal{G}(\mathcal{A}_{1})$
by $\mathcal{G}(\psi)(\phi)=\phi\circ\psi$, where $\phi\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A}_{2},\mathcal{L})$.

Lemma 5 and Lemma 6 demonstrate the well-definedness of $\mathcal{G}$.

######  Lemma 5.

For an object $(\mathcal{A},\Box)$ in $\mathcal{MA}_{\mathcal{L}}$,
$\mathcal{G}(\mathcal{A})$ is an object in $PRBS{\mathcal{L}}$.

Proof: Definition 9 shows that
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),\tau_{1},\tau_{2},\alpha_{\mathcal{A}})$
is an object in $PBS_{\mathcal{L}}$. So it is enough to show that
$\mathcal{R}_{\Box}$ meets the conditions specified in the object part of
Definition 10. We first show that for $\mathcal{W}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$,
$\mathcal{R}_{\Box}[\mathcal{W}]\in\delta_{1}\cup\delta_{2}$. Let
$\mathcal{U}\not\in\mathcal{R}_{\Box}[\mathcal{W}]$. Then by Definition 6,
there is an element $a\in\mathcal{A}$ such that there is
$L_{1}\in\mathcal{L}$, for which $\mathcal{W}(\Box a)\geq L_{1}$ but
$\mathcal{U}(a)\not\geq L_{1}$. It follows that $\mathcal{U}\in\langle\neg
U_{L_{1}}(a)\rangle\in\tau_{2}$ and
$\mathcal{R}_{\Box}[\mathcal{W}]\cap\langle\neg U_{L_{1}}(a)\rangle=\emptyset$
i.e., $\langle\neg
U_{L_{1}}(a)\rangle\subseteq(\mathcal{R}_{\Box}[\mathcal{W}])^{c}$. Hence,
$\mathcal{R}_{\Box}[\mathcal{W}]\not\subset\bar{\mathcal{R}_{\Box}[\mathcal{W}]}^{\tau}_{2}$.
Equivalently,
$\bar{\mathcal{R}_{\Box}[\mathcal{W}]}^{\tau}_{2}\subset\mathcal{R}_{\Box}[\mathcal{W}]$.
Therefore, $\mathcal{R}_{\Box}[\mathcal{W}]$ is $\tau_{2}$-closed. Since,
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),\tau_{1},\tau_{2})$
is pairwise compact, by Proposition 2, we have
$\mathcal{R}_{\Box}[\mathcal{W}]$ is pairwise compact.
Now we verify the condition $(ii)$ in the object part of Definition 10. Since
$\\{\langle a\rangle:a\in\mathcal{A}\\}\in\beta_{1}$ and $\\{\langle
T_{1}(a)\rightarrow 0\rangle:a\in\mathcal{A}\\}\in\beta_{2}$ are the basis for
the topologies $\tau_{1}$ and $\tau_{2}$, respectively, so we show that for
each $a\in\mathcal{A}$, $\langle\mathcal{R}_{\Box}\rangle(\langle
a\rangle)\in\beta_{1}$ and $[\mathcal{R}_{\Box}]\langle a\rangle\in\beta_{1}$.
We see that

$\begin{split}\langle\mathcal{R}_{\Box}\rangle\langle
a\rangle&=\\{\mathcal{W}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}):\mathcal{R}_{\Box}[\mathcal{W}]\cap\langle
a\rangle\neq\emptyset\\}\\\ &=([\mathcal{R}_{\Box}]\langle T_{1}(a)\rightarrow
0\rangle)^{c}\\\ &=\\{\mathcal{W}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}):\mathcal{R}_{\Box}[\mathcal{W}]\not\subset\langle
T_{1}(a)\rightarrow 0\rangle\\}\end{split}$

We show that $([\mathcal{R}_{\Box}]\langle T_{1}(a)\rightarrow 0\rangle)^{c}$
is $\tau_{1}$-open and $\tau_{2}$-closed. Let
$\mathcal{U}\in([\mathcal{R}_{\Box}]\langle T_{1}(a)\rightarrow
0\rangle)^{c}$. Then $\mathcal{R}_{\Box}[\mathcal{U}]\not\subset\langle
T_{1}(a)\rightarrow 0\rangle$. It is easy to see that $\exists$
$\tau_{1}$-open set $\langle\Box T_{1}(a)\rangle$ such that
$\mathcal{U}\in\langle\Box T_{1}(a)\rangle$. Let $\mathcal{E}\in\langle\Box
T_{1}(a)\rangle$. Then $\mathcal{E}(\Box T_{1}(a))=1$. Using Kripke condition
we have $1=\mathcal{E}(\Box
T_{1}(a))=\bigwedge\\{\mathcal{U}(T_{1}(a)):\mathcal{E}\mathcal{R}_{\Box}\mathcal{U}\\}$.
According to Lemma 2, $\mathcal{U}(T_{1}(a))$ is either $0$ or $1$.
Henceforth, for all $\mathcal{U}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$ with
$\mathcal{E}\mathcal{R}_{\Box}\mathcal{U}$ we have $\mathcal{U}(T_{1}(a))=1$.
As a result, $\mathcal{R}_{\Box}[\mathcal{E}]\not\subset\langle
T_{1}(a)\rightarrow 0\rangle$ i.e.,
$\mathcal{E}\in([\mathcal{R}_{\Box}]\langle T_{1}(a)\rightarrow
0\rangle)^{c}$. Henceforth, $\mathcal{U}\in\langle\Box
T_{1}(a)\rangle\subset([\mathcal{R}_{\Box}]\langle T_{1}(a)\rightarrow
0\rangle)^{c}$. Therefore, $[\mathcal{R}_{\Box}]\langle T_{1}(a)\rightarrow
0\rangle)^{c}$ is $\tau_{1}$-open i.e.,
$\langle\mathcal{R}_{\Box}\rangle\langle a\rangle$ is $\tau_{1}$-open.
Let $\mathcal{W}\in(\langle\mathcal{R}_{\Box}\rangle\langle a\rangle)^{c}$.
Then $\mathcal{R}_{\Box}[\mathcal{W}]\cap\langle a\rangle=\emptyset$. It is
easy to see that there is $\tau_{1}$-open set $\langle\Box(T_{1}(a)\rightarrow
0)\rangle$ such that $\mathcal{W}\in\langle\Box(T_{1}(a)\rightarrow
0)\rangle$. Also, by applying Kripke condition, we have
$\langle\Box(T_{1}(a)\rightarrow
0)\rangle\subset(\langle\mathcal{R}_{\Box}\rangle\langle a\rangle)^{c}$.
Therefore, $\mathcal{W}\in\langle\Box(T_{1}(a)\rightarrow
0)\rangle\subset(\langle\mathcal{R}_{\Box}\rangle\langle a\rangle)^{c}$. It
shows that $(\langle\mathcal{R}_{\Box}\rangle\langle a\rangle)^{c}$ is
$\tau_{1}$-open i.e., $\langle\mathcal{R}_{\Box}\rangle\langle a\rangle$ is
$\tau_{1}$-closed. It follows from Lemma 2 that
$\langle\mathcal{R}_{\Box}\rangle\langle a\rangle$ is pairwise compact. Since,
the topological space
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}),\tau_{2})$ with
basis $\\{\langle T_{1}(a)\rightarrow 0\rangle:a\in\mathcal{A}\\}$ is a
Hausdorff space, so $\langle\mathcal{R}_{\Box}\rangle\langle a\rangle$ is
$\tau_{2}$-closed. Hence, $\langle\mathcal{R}_{\Box}\rangle\langle
a\rangle\in\beta_{1}$.
Next we show that $[\mathcal{R}_{\Box}]\langle a\rangle\in\beta_{1}$. We see
that

$\begin{split}[\mathcal{R}_{\Box}]\langle a\rangle&=\\{\mathcal{W}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}):\mathcal{R}_{\Box}[\mathcal{W}]\subseteq\langle
a\rangle\\}\\\ &=(\langle\mathcal{R}_{\Box}\rangle\langle T_{1}(a)\rightarrow
0\rangle)^{c}\end{split}$

We claim that $\langle\mathcal{R}_{\Box}\rangle\langle T_{1}(a)\rightarrow
0\rangle=\langle\Box T_{1}(a)\rightarrow 0\rangle$. Let
$\mathcal{W}\in\langle\Box T_{1}(a)\rightarrow 0\rangle$. Then
$\mathcal{W}(\Box T_{1}(a)\rightarrow 0)=1$. Hence, $\mathcal{W}(\Box
T_{1}(a))=0$. Using Kripke condition we have, $0=\mathcal{W}(\Box
T_{1}(a))=\bigwedge\\{\mathcal{U}(T_{1}(a)):\mathcal{W}\mathcal{R}_{\Box}\mathcal{U}\\}$.
Since $\mathcal{U}(T_{1}(a))=0\text{ or }1$, hence $\exists$ $\mathcal{U}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$ with
$\mathcal{W}\mathcal{R}_{\Box}\mathcal{U}$ such that
$\mathcal{U}(T_{1}(a))=0$. Then $\mathcal{U}\in\langle T_{1}(a)\rightarrow
0\rangle$. Therefore, $\mathcal{R}_{\Box}[\mathcal{W}]\cap\langle
T_{1}(a)\rightarrow 0\rangle\neq\emptyset$. Thus,
$\mathcal{W}\in\langle\mathcal{R}_{\Box}\rangle\langle T_{1}(a)\rightarrow
0\rangle$. Similarly, by employing Kripke condition we can show that if
$\mathcal{W}\in\langle\mathcal{R}_{\Box}\rangle\langle T_{1}(a)\rightarrow
0\rangle$ then $\mathcal{W}\in\langle\Box T_{1}(a)\rightarrow 0\rangle$. since
$\langle\Box T_{1}(a)\rightarrow 0\rangle\in\beta_{2}$, we have
$\langle\mathcal{R}_{\Box}\rangle\langle T_{1}(a)\rightarrow
0\rangle\in\beta_{2}$. As a result,$[\mathcal{R}_{\Box}]\langle
a\rangle\in\beta_{1}$.
Finally, we demonstrate that $\mathcal{G}(\mathcal{A})$ meets condition
$(iii)$ in the object part of Definition 10. Let
$u\in\alpha_{\mathcal{A}}(\mathcal{L}^{\prime})=HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L}^{\prime})$.
Suppose
$\mathcal{R}_{\Box}[u]\not\subset\alpha_{\mathcal{A}}(\mathcal{L}^{\prime}$.
Then $\exists v\in\mathcal{R}_{\Box}[u]$ such that
$v\notin\alpha_{\mathcal{A}}(\mathcal{L}^{\prime})$. Hence, $\exists
a^{*}\in\mathcal{A}$ such that $v(a^{*})\notin\mathcal{L}^{\prime}$. Let
$v(a^{*})=\ell^{*}$. Now for any element
$\psi\in\alpha_{\mathcal{A}}(\mathcal{L}^{\prime})$,

$\psi(T_{\ell^{*}}(a^{*}\rightarrow
a^{*})=\left\\{\begin{array}[]{ll}a^{*}&\text{ if }\psi(a^{*})=\ell^{*}\\\
1&\text{ if}\psi(a^{*})\neq\ell^{*}\\\ \end{array}\right.$

Using Kripke condition, we have $u(\Box(T_{\ell^{*}}(a^{*})\rightarrow
a^{*}))=\bigwedge\\{\psi(T_{\ell^{*}}(a^{*})\rightarrow
a^{*}):\psi\in\mathcal{R}_{\Box}[u]\\}$. This shows that
$u(\Box(T_{\ell^{*}}(a^{*})\rightarrow
a^{*}))=\ell^{*}\notin\mathcal{L}^{\prime}$. But this contradicts the fact
that $u\in\alpha_{\mathcal{A}}(\mathcal{L}^{\prime})$. As a result,
$\mathcal{G}(\mathcal{A})$ satisfies condition $(iii)$.

######  Lemma 6.

Let $(\mathcal{A}_{1},\Box_{1})$, $(\mathcal{A}_{2},\Box_{2})$ be the objects
in $\mathcal{MA}_{\mathcal{L}}$ and $\psi:\mathcal{A}_{1}\to\mathcal{A}_{2}$
be an arrow in $\mathcal{MA}_{\mathcal{L}}$. Then, $\mathcal{G}(\psi)$ is an
arrow in $PRBS_{\mathcal{L}}$.

Proof: Here
$\mathcal{G}(\psi):\mathcal{G}(\mathcal{A}_{2})\to\mathcal{G}(\mathcal{A}_{1})$
is defined by $\mathcal{G}(\psi)(\phi)=\phi\circ\psi$, $\phi\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A}_{1},\mathcal{L})$. It follows
from Definition 9 that $\mathcal{G}(\psi)$ is an arrow in $PBS_{\mathcal{L}}$.
Therefore, it is still necessary to demonstrate that $\mathcal{G}(\psi)$
satisfies conditions $(i)$ and $(ii)$ listed in the arrow portion of
Definition 10. We first check condition $(i)$. Let
$v_{1}\mathcal{R}_{\Box_{2}}v_{2}$, where
$v_{1},v_{2}\in\mathcal{G}(\mathcal{A}_{2})$. We are to show that
$\mathcal{G}(\psi)(v_{1})\mathcal{R}_{\Box_{1}}\mathcal{G}(\psi)(v_{2})$. Now,
if $v_{1}\circ\psi(\Box_{1}a_{1})\geq\ell$ for $a_{1}\in\mathcal{A}_{1}$ and
$\ell\in\mathcal{L}$, then we have $v_{1}(\Box_{2}\psi(a_{1}))\geq\ell$. As
$v_{1}\mathcal{R}_{\Box_{2}}v_{2}$, so we get $v_{2}(\psi(a_{1}))\geq\ell$.
Hence,
$\mathcal{G}(\psi)(v_{1})\mathcal{R}_{\Box_{1}}\mathcal{G}(\psi)(v_{2})$. We
then check condition $(ii)$, which is mentioned in the arrow part of
Definition 10. This is equivalent to verifying
$\mathcal{R}_{\Box_{1}}[\mathcal{G}(\psi)(v_{1})]=\mathcal{G}(\psi)(\mathcal{R}_{\Box_{2}}[v_{1}])$.
Let $\mathcal{W}\in\mathcal{R}_{\Box_{1}}[v_{1}\circ\psi]$, where
$\mathcal{W}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A}_{1},\mathcal{L})$. Then
$(v_{1}\circ\psi)\mathcal{R}_{\Box_{1}}\mathcal{W}$. Suppose
$\mathcal{W}\notin\mathcal{G}(\psi)(\mathcal{R}_{\Box_{2}}[v_{1}])$. Then
$\mathcal{W}\neq\mathcal{G}(\psi)(v^{*})$, $\forall v^{*}\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A}_{2},\mathcal{L})$ such that
$v_{1}\mathcal{R}_{\Box_{2}}v^{*}$. As
$(HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A}_{1},\mathcal{L}),\tau_{1}^{\mathcal{A}_{1}},\tau_{2}^{\mathcal{A}_{1}})$
is a pairwise Hausdorff space, so we can consider $\mathcal{W}\in\langle
a_{1}\rangle$ and $\mathcal{G}(\psi)(v^{*})=v^{*}\circ\psi\in\langle
T_{1}(a_{1})\rightarrow 0\rangle$. Since
$\mathcal{W}\in\mathcal{R}_{\Box_{1}}[\mathcal{G}(\psi)(v_{1})]$ and
$\mathcal{W}(a_{1})=1$, we have $\mathcal{G}(\psi)(v_{1})(\Box_{1}a_{1})=1$
i.e., $(v_{1}\circ\psi)(\Box_{1}a_{1})=1$. Since
$v_{1}\mathcal{R}_{\Box_{2}}v^{*}$, we have
$\mathcal{G}(\psi)(v_{1})\mathcal{R}_{\Box_{1}}\mathcal{G}(\psi)(v^{*})$ using
the condition $(i)$ specified in the arrow part of Definition 10. As
$\mathcal{G}(\psi)(v_{1})(\Box_{1}a_{1})=1$, Lemma 4 shows that
$\mathcal{G}(\psi)(v^{*})(a_{1})=1$, i.e., $v^{*}\circ\psi\in\langle
a_{1}\rangle$. This contradicts the fact that
$\mathcal{G}(\psi)(v^{*})\in\langle T_{1}(a_{1})\rightarrow 0\rangle$.
Therefore,
$\mathcal{R}_{\Box_{1}}[\mathcal{G}(\psi)(v_{1})]\subseteq\mathcal{G}(\psi)(\mathcal{R}_{\Box_{2}}[v_{1}])$.
Similarly, we can show the reverse direction.

######  Definition 12.

We define a functor
$\mathcal{F}:PRBS_{\mathcal{L}}\to\mathcal{MA}_{\mathcal{L}}$.

  1. (i)

Define
$\mathcal{F}(P,\alpha_{P},\mathcal{R})=(HOM_{{PBS}_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}})),\wedge,\vee,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1,\Box_{\mathcal{R}})$
for an object $(P,\alpha_{P},\mathcal{R})$ in $PRBS_{\mathcal{L}}$. Definition
5 describes the modal operation $\Box_{\mathcal{R}}$. Here
$\wedge,\vee,\rightarrow,T_{\ell}$ are pointwise operations defined on the set
$HOM_{{PBS}_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}}))$.

  2. (ii)

Let
$\psi:(P_{1},\alpha_{P_{1}},\mathcal{R}_{1})\to(P_{2},\alpha_{P_{2}},\mathcal{R}_{2})$
be an arrow in $PRBS_{\mathcal{L}}$. Define
$\mathcal{F}(\psi):\mathcal{F}(P_{2},\alpha_{P_{2}},\mathcal{R}_{2})\to\mathcal{F}(P_{1},\alpha_{P_{1}},\mathcal{R}_{1})$
by $\mathcal{F}(\psi)(\phi)=\phi\circ\psi$ for
$\phi\in\mathcal{F}(P_{2},\alpha_{P_{2}},\mathcal{R}_{2})$.

######  Note 3.

If
$\psi,\phi:(P,\tau_{1}^{P},\tau_{2}^{P},\alpha_{P})\to(\mathcal{L},\tau,\tau,\alpha_{\mathcal{L}})$
are pairwise continuous maps then
$\psi\wedge\phi,\psi\vee\phi,\psi\rightarrow\phi,T_{\ell}(\psi)$ are pairwise
continuous maps.
$(HOM_{{PBS}_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}})),\wedge,\vee,\rightarrow,T_{\ell}(\ell\in\mathcal{L}),0,1)$
is a $\mathcal{L}$-$\mathcal{VL}$-algebra.

The following lemmas (Lemma 7 and Lemma 8) show that the functor $\mathcal{F}$
is well-defined.

######  Lemma 7.

Let $(P,\alpha_{P},\mathcal{R})$ be an object in $PRBS_{\mathcal{L}}$. Then,
$\mathcal{F}(P,\alpha_{P},\mathcal{R})$ is an object in
$\mathcal{MA}_{\mathcal{L}}$.

Proof: It is clear from Definition 8 that $\mathcal{F}(P,\alpha_{P})$ is an
object in $\mathcal{VA}_{\mathcal{L}}$. We need to show that
$\Box_{\mathcal{R}}$ on $\mathcal{F}(P,\alpha_{P},\mathcal{R})$ is well-
defined. Let $\eta\in\mathcal{F}(P,\alpha_{P},\mathcal{R})$. We then verify
$\Box_{\mathcal{R}}\eta\in\mathcal{F}(P,\alpha_{P},\mathcal{R})$. For any
$\ell\in\mathcal{L}$,

$\begin{split}(\Box_{\mathcal{R}}\eta)^{-1}(\\{\ell\\})&=\\{p\in
P:\bigwedge\\{\eta(p^{\prime}):p^{\prime}\in\mathcal{R}[p]=\ell\\}\\\
&=\langle\mathcal{R}\rangle((T_{\ell}(\eta))^{-1}(\\{1\\}))\cap(\langle\mathcal{R}\rangle((U_{\ell}(\eta))^{-1}(\\{0\\})))^{c}\end{split}$

As both $T_{\ell}(\eta)$ and $U_{\ell}(\eta)$ are pairwise continuous maps,
henceforth $(T_{\ell}(\eta))^{-1}(\\{1\\})\in\beta_{1}\cap\beta_{2}$ and
$(U_{\ell}(\eta))^{-1}(\\{0\\})\in\beta_{1}\cap\beta_{2}$.
$(\Box_{\mathcal{R}}\eta)^{-1}\\{\ell\\}\in\tau_{2}^{P}$. As a result,
$\Box_{\mathcal{R}}\eta$ is a pairwise continuous map from $P$ to
$\mathcal{L}$. Furthermore, by applying condition $(iii)$ that is stated in
the object part of Definition 10, we see that for any subalgebra
$\mathcal{M}\in\mathfrak{S}_{\mathcal{L}}$, and if
$m\in\alpha_{\mathcal{L}}(\mathcal{M})$ then
$(\Box_{\mathcal{R}}\eta)(m)=\bigwedge\\{\eta(m^{\prime}):m^{\prime}\in\mathcal{R}[m]\\}\in\alpha_{\mathcal{L}}(\mathcal{M})$.
Thus $\Box_{\mathcal{R}}\eta$ is a subspace preserving map. Hence,
$\Box_{\mathcal{R}}\eta\in\mathcal{F}(P,\alpha_{P},\mathcal{R})$.

######  Lemma 8.

Let
$\psi:(P_{1},\alpha_{P_{1}},\mathcal{R}_{1})\to(P_{2},\alpha_{P_{2}},\mathcal{R}_{2})$
be an arrow in $PRBS_{\mathcal{L}}$. Then, $\mathcal{F}(\psi)$ is an arrow in
$\mathcal{MA}_{\mathcal{L}}$.

Proof: According to Definition 8, $\mathcal{F}(\psi)$ is an arrow in
$\mathcal{VA}_{\mathcal{L}}$. Therefore, it is sufficient to demonstrate that
$\mathcal{F}(\psi)(\Box_{\mathcal{R}_{2}}\phi_{2})=\Box_{\mathcal{R}_{1}}(\mathcal{F}(\psi)\phi_{2})$,
where $\phi_{2}\in
HOM_{{PBS}_{\mathcal{L}}}((P_{2},\alpha_{P_{2}}),(\mathcal{L},\alpha_{\mathcal{L}}))$.
For any $p_{1}\in P_{1}$, we have
$\mathcal{F}(\psi)(\Box_{\mathcal{R}_{2}}\phi_{2})(p_{1})=\Box_{\mathcal{R}_{2}}\phi_{2}\circ\psi(p_{1})=\bigwedge\\{\phi_{2}(p_{2}):p_{2}\in\mathcal{R}_{2}[\psi(p_{1})]\\}$,
and
$\Box_{\mathcal{R}_{1}}(\mathcal{F}(\psi)\phi_{2})(p_{1})=\Box_{\mathcal{R}_{1}}(\phi_{2}\circ\psi)(p_{1})=\bigwedge\\{\phi_{2}\circ\psi(p):p\in\mathcal{R}_{1}[p_{1}]\\}$.
As $\psi$ satisfies conditions $(i)$ and $(ii)$ listed in item 2 of Definition
10, it is easy to show that
$(\mathcal{F}(\psi)(\Box_{\mathcal{R}_{2}}\phi_{2}))(p_{1})\leq\Box_{\mathcal{R}_{1}}(\mathcal{F}(\psi)\phi_{2})(p_{1})$
and
$\Box_{\mathcal{R}_{1}}(\mathcal{F}(\psi)\phi_{2})(p_{1})\leq(\mathcal{F}(\psi)(\Box_{\mathcal{R}_{2}}\phi_{2}))(p_{1})$.
As a result,
$\mathcal{F}(\psi)(\Box_{\mathcal{R}_{2}}\phi_{2})=\Box_{\mathcal{R}_{1}}(\mathcal{F}(\psi)\phi_{2})$.

###  3.3 Bitopological Duality for $\mathcal{L}$-$\mathcal{ML}$ Algebras

In this subsection, we develop bitopological duality for algebras Fitting’s
Heyting-valued modal logic.

######  Theorem 2.

Let $\mathcal{A}$ be a $\mathcal{L}$-$\mathcal{ML}$ algebra. Then
$\mathcal{A}$ is isomorphic to $\mathcal{F}\circ\mathcal{G}(\mathcal{A})$ in
$\mathcal{MA}_{\mathcal{L}}$.

Proof: We define
$\beta^{\mathcal{A}}:\mathcal{A}\to\mathcal{F}\circ\mathcal{G}(\mathcal{A})$
by $\beta^{\mathcal{A}}(a)(g)=g(a)$, where $a\in\mathcal{A}$ and $g\in
HOM_{\mathcal{VA}_{\mathcal{L}}}(\mathcal{A},\mathcal{L})$. It is known from
Theorem 1 that $\beta^{\mathcal{A}}$ is an isomorphism in the category
$\mathcal{VA}_{\mathcal{L}}$. The only thing left to prove is that
$\beta^{\mathcal{A}}$ preserves the modal operator $\Box$, i.e.,
$\beta^{\mathcal{A}}(\Box a)=\Box_{\mathcal{R}_{\Box}}\beta^{\mathcal{A}}(a)$,
$a\in\mathcal{A}$. Let $g\in\mathcal{G}(\mathcal{A})$. Then

$\begin{split}(\Box_{\mathcal{R}_{\Box}}\beta^{\mathcal{A}}(a))(g)&=\bigwedge\\{\beta^{\mathcal{A}}(g^{*}):g^{*}\in\mathcal{R}_{\Box}[g]\\}\\\
&=\bigwedge\\{g^{*}(a):g^{*}\in\mathcal{R}_{\Box}[g]\\}\\\ &=g(\Box a)\text{
(by Lemma \ref{KM})}\\\ &=\beta^{\mathcal{A}}(\Box a)(g)\end{split}$

Hence the result follows.

######  Theorem 3.

Consider an object $(P,\alpha_{P},\mathcal{R})$ in $PRBS_{\mathcal{L}}$. Then,
$(P,\alpha_{P},\mathcal{R})$ is isomorphic to
$\mathcal{G}\circ\mathcal{F}(P,\alpha_{P},\mathcal{R})$ in the category
$PRBS_{\mathcal{L}}$.

Proof: Define
$\zeta_{(P,\alpha_{P},\mathcal{R})}:(P,\alpha_{P},\mathcal{R})\to\mathcal{G}\circ\mathcal{F}(P,\alpha_{P},\mathcal{R})$
by $\zeta_{(P,\alpha_{P},\mathcal{R})}(p)(\psi)=\psi(p)$, where $p\in P$ and
$\psi\in
HOM_{{PBS}_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}}))$.
Theorem 1 shows that $\zeta_{(P,\alpha_{P},\mathcal{R})}$ is a bi-
homeomorphism in the category $PBS_{\mathcal{L}}$. We show that
$\zeta_{(P,\alpha_{P},\mathcal{R})}$ and
$\zeta^{-1}_{(P,\alpha_{P},\mathcal{R})}$ satisfy the conditions given in item
2 of Definition 10. We claim that for any $p,p^{\prime}\in P$,
$p^{\prime}\in\mathcal{R}[p]\iff\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})\in\mathcal{R}_{\Box_{\mathcal{R}}}[\zeta_{(P,\alpha_{P},\mathcal{R})}(p)]$.
Let $p^{\prime}\in\mathcal{R}[p]$. Suppose
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p)(\Box_{\mathcal{R}}\psi)\geq\ell$, where
$\ell\in\mathcal{L}$ and $\psi\in
HOM_{{PBS}_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}}))$.
Then
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p)(\Box_{\mathcal{R}}\psi)=(\Box_{\mathcal{R}}\psi)(p)=\bigwedge\\{\psi(p^{*}):p*\in\mathcal{R}[p]\\}$.
Since $p^{\prime}\in\mathcal{R}[p]$ and
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p)(\Box_{\mathcal{R}}\psi)\geq\ell$, we
have $\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})(\psi)\geq\ell$. Hence,
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p)\mathcal{R}_{\Box_{\mathcal{R}}}\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})$,
i.e.,
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})\in\mathcal{R}_{\Box_{\mathcal{R}}}[\zeta_{(P,\alpha_{P},\mathcal{R})}(p)]$.
Now we verify if
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})\in\mathcal{R}_{\Box_{\mathcal{R}}}[\zeta_{(P,\alpha_{P},\mathcal{R})}(p)]$
then $p^{\prime}\in\mathcal{R}[p]$. We verify its contrapositive statement.
Suppose $p^{\prime}\notin\mathcal{R}[p]$. By Definition 10, $\mathcal{R}[p]$
is a pairwise compact subset of pairwise Boolean space $P$. Then it is easy to
show that $\mathcal{R}[p]$ is pairwise closed. Therefore we can get a
$\tau_{1}^{P}$-basis open set $\mathcal{O}\in\beta_{1}^{P}$ such that
$p^{\prime}\in\mathcal{O}$ and $\mathcal{O}\subseteq P-\mathcal{R}[p]$, i.e.,
$\mathcal{O}\cap\mathcal{R}[p]=\emptyset$. Define a mapping
$f:P\to\mathcal{L}$ by

$f(p)=\left\\{\begin{array}[]{ll}0&\text{ if}p\in\mathcal{O}\\\ 1&\text{
if}p\in\mathcal{O}^{c}\\\ \end{array}\right.$

Then $f$ is a pairwise continuous map from $(P,\tau_{1}^{P},\tau_{2}^{P})$ to
$(\mathcal{L},\tau,\tau)$. As a result, it can be shown that $f\in
HOM_{PBS_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}}))$.
Now, $\Box_{\mathcal{R}}f(p)=\bigwedge\\{f(z):z\in\mathcal{R}[p]\\}=1$ and
$f(p^{\prime})=0$. Hence,
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p)(\Box_{\mathcal{R}}f)=1$ but
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})(f)\neq 1$. Therefore,
$\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})\notin\mathcal{R}_{\Box_{\mathcal{R}}}[\zeta_{(P,\alpha_{P},\mathcal{R})}(p)]$.
Hence, we have for any $p,p^{\prime}\in P$,
$p^{\prime}\in\mathcal{R}[p]\iff\zeta_{(P,\alpha_{P},\mathcal{R})}(p^{\prime})\in\mathcal{R}_{\Box_{\mathcal{R}}}[\zeta_{(P,\alpha_{P},\mathcal{R})}(p)]$.
As a result, $\zeta_{(P,\alpha_{P},\mathcal{R})}$ and
$\zeta^{-1}_{(P,\alpha_{P},\mathcal{R})}$ satisfy conditions $(i)$ and $(ii)$
mentioned in item 2 of Definition 10.

Finally, we obtain the bitopological duality for Fitting’s Heyting-valued
modal logic.

######  Theorem 4.

The categories $\mathcal{MA}_{\mathcal{L}}$ and $PRBS_{\mathcal{L}}$ are
dually equivalent.

Proof: Let $ID_{1}$ and $ID_{2}$ be the identity functors on
$\mathcal{MA}_{\mathcal{L}}$ and $PRBS_{\mathcal{L}}$, respectively. This
theorem will be proved by defining two natural isomorphisms,
$\beta:ID_{1}\to\mathcal{F}\circ\mathcal{G}$ and
$\zeta:ID_{2}\to\mathcal{G}\circ\mathcal{F}$. For an object $\mathcal{A}$ in
$\mathcal{MA}_{\mathcal{L}}$ define
$\beta^{\mathcal{A}}:\mathcal{A}\to\mathcal{F}\circ\mathcal{G}(\mathcal{A})$
by $\beta^{\mathcal{A}}(a)(g)=g(a)$, where $a\in\mathcal{A}$ and
$g\in\mathcal{G}(\mathcal{A})$. For an object $(P,\alpha_{P},\mathcal{R})$ in
$PRBS_{\mathcal{L}}$ define
$\zeta_{(P,\alpha_{P},\mathcal{R})}:(P,\alpha_{P},\mathcal{R})\to\mathcal{G}\circ\mathcal{F}(P,\alpha_{P},\mathcal{R})$
by $\zeta_{(P,\alpha_{P},\mathcal{R})}(p)(\psi)=\psi(p)$, where $p\in P$ and
$\psi\in
HOM_{PBS_{\mathcal{L}}}((P,\alpha_{P}),(\mathcal{L},\alpha_{\mathcal{L}}))$.
Then it can be shown that $\beta$ and $\zeta$ are natural transformations.
According to Theorems 2 and 3, $\beta$ and $\zeta$ are natural isomorphisms.

In the next section, we develop a coalgebraic duality theorem for algebras of
Fitting’s Heyting valued modal logic.

##  4 Coalgebraic duality

The primary objective of this section is to define an endofunctor
$V_{\mathcal{L}}^{bi}:PBS_{\mathcal{L}}\to PBS_{\mathcal{L}}$, called
$\mathcal{L}$-biVietoris functor. Then we demonstrate that the category
$COALG(V_{\mathcal{L}}^{bi})$ of coalgebras for the endofunctor
$V_{\mathcal{L}}^{bi}$ is isomorphic to the category $PRBS_{\mathcal{L}}$.
We now recall the definitions of coalgebra and coalgebra morphisms. We refer
the reader to adamek2005introduction  for an overview of coalgebras.

######  Definition 13.

###  4.1 The structure of the endofunctor $V_{\mathcal{L}}^{bi}$

We define the pairwise Vietoris space as follows:

######  Definition 14.

Let $(S,\tau_{1}^{S},\tau_{2}^{S})$ be a pairwise topological space and
$\mathcal{K}(S)$ the set of all pairwise closed subsets of $S$. We define
$\Box U=\\{C\in\mathcal{K}(S):C\subseteq U\\}$ and $\Diamond
U=\\{C\in\mathcal{K}(S):C\cap U\neq\emptyset\\}$, $U\subseteq S$. Let
$\beta_{1}^{S}$ and $\beta_{2}^{S}$ be the basis for the topologies
$\tau_{1}^{S}$ and $\tau_{2}^{S}$, respectively. The pairwise Vietoris space
$V_{P}(S)$ of the pairwise topological space $(S,\tau_{1}^{S},\tau_{2}^{S})$
is defined as a pairwise topological space
$(\mathcal{K}(S),\tau_{1}^{V},\tau_{2}^{V})$, where $\tau_{1}^{V}$ is the
topology on $\mathcal{K}(S)$ generated by subbasis $\\{\Box U,\Diamond
U:U\in\beta_{1}^{S}\\}$ and the topology $\tau_{2}^{V}$ on $\mathcal{K}(S)$ is
generated by subbasis $\\{\Box U,\Diamond U:U\in\beta_{2}^{S}\\}$.

We then show that $V_{P}(S)$ is a pairwise Boolean space whenever $S$ is a
pairwise Boolean space.

######  Lemma 9.

If $(S,\tau_{1}^{S},\tau_{2}^{S})$ is a pairwise Boolean space then
$V_{P}(S)=(\mathcal{K}(S),\tau_{1}^{V},\tau_{2}^{V})$ is pairwise zero-
dimensional.

Proof: We shall show that $\beta_{1}^{V}=\tau_{1}^{V}\cap\delta_{2}^{V}$ is a
basis for $\tau_{!}^{V}$, where $\delta_{2}^{V}$ is the set of
$\tau_{2}^{V}$-closed sets. Let $\mathcal{O}\in\tau_{1}^{V}$. Then
$\mathcal{O}$ can be expressed as
$\mathcal{O}=\displaystyle\bigcup_{\lambda\in\Lambda}(\bigcap_{j=1}^{n_{\lambda}}\Box
U_{j}\cap\bigcap_{k=1}^{m_{\lambda}}\Diamond U_{k})$,
$U_{j},U_{k}\in\beta_{1}^{S}=\tau_{1}^{S}\cap\delta_{2}^{S}$. In order to show
that $\beta_{1}^{V}$ is a basis for $\tau_{1}^{V}$, , it is necessary to show
that $\bigcap_{j=1}^{n_{\lambda}}\Box
U_{j}\cap\bigcap_{k=1}^{m_{\lambda}}\Diamond U_{k}\in\beta_{1}^{V}$. Because
the finite intersection of the members of $\beta_{1}^{V}$ is again in
$\beta_{1}^{V}$, it is sufficient to establish that for $U\in\beta_{1}^{S}$,
$\Box U,\Diamond U\in\beta_{1}^{V}$. As $\tau_{1}^{V}$ is the topology
generated by the subbasis $\\{\Box U,\Diamond U:U\in\beta_{1}^{S}\\}$, hence
$\Box U,\Diamond U\in\tau_{1}^{V}$.Now we see that $(\Box U)^{c}=\Diamond
U^{c}$ and $(\Diamond U)^{c}=\Box U^{c}$. Since $U\in\beta_{1}^{S}$, so
$U^{c}\in\beta_{2}^{S}$. As a result, $\Box U,\Diamond U\in\delta_{2}^{V}$.
Henceforth, $\Box U,\Diamond U\in\beta_{1}^{V}$. Similarly, it can be shown
that $\beta_{2}^{V}=\tau_{2}^{v}\cap\delta_{1}^{V}$, $\delta_{1}^{V}$ is the
set of $\tau_{1}^{V}$-closed sets, is a basis for $\tau_{2}^{V}$.

######  Lemma 10.

If $(S,\tau_{1}^{S},\tau_{2}^{S})$ is a pairwise Boolean space then
$V_{P}(S)=(\mathcal{K}(S),\tau_{1}^{V},\tau_{2}^{V})$ is pairwise Hausdorff.

Proof: Let $C,C^{\prime}\in\mathcal{K}(S)$ and $C\neq C^{\prime}$. Let $c\in
C$ such that $c\neq c^{\prime}$, $\forall c^{\prime}\in C^{\prime}$. For each
point $c^{\prime}\in C^{\prime}$, we choose disjoint open sets
$U_{c^{\prime}}^{c}\beta_{2}^{S}$ and $U_{c^{\prime}}\in\beta_{1}^{S}$ (using
the condition that $(S,\tau_{1}^{S},\tau_{2}^{S})$ is pairwise Hausdorff
space.) containing points $c^{\prime}$ and $c$, respectively. So the
collection $\\{U_{c^{\prime}}^{c}:c^{\prime}\in C^{\prime}\\}$ is
$\tau_{2}^{S}$-open covering of $C^{\prime}$. As $C^{\prime}$ is pairwise
compact, so there is a finite collection
$\\{U_{c^{\prime}_{i}}^{c}:i=1,2,\cdots,n\\}$ such that
$C^{\prime}\subseteq\bigcup_{i=1}^{n}U_{c^{\prime}_{i}}^{c}$.
$V^{\prime}=\displaystyle\bigcup_{i=1}^{n}U_{c^{\prime}_{i}}^{c}$ and
$U=\displaystyle\bigcap_{i=1}^{n}U_{c^{\prime}_{i}}$. As $c\in C\cap U$, hence
$C\cap U\neq\emptyset$. Also, $C^{\prime}\cap U=\emptyset$ because
$C^{\prime}\subseteq U^{c}$. It follows that $C\in\Diamond U\in\tau_{1}^{V}$
and $C^{\prime}\notin\Diamond U$ i.e., $C^{\prime}\in(\Diamond U)^{c}=\Box
U^{c}\in\tau_{2}^{V}$. So we have two disjoint open sets $\Diamond
U\in\tau_{1}^{V}$ and $\Box U^{c}\in\tau_{2}^{V}$ containing $C$ and
$C^{\prime}$, respectively.

######  Lemma 11.

If $(S,\tau_{1}^{S},\tau_{2}^{S})$ is a pairwise Boolean space then
$V_{P}(S)=(\mathcal{K}(S),\tau_{1}^{V},\tau_{2}^{V})$ is pairwise compact.

Proof: It is known from Proposition 1 that $\\{\Box U,\Diamond
U:U\in\beta_{1}^{S}\cup\beta_{2}^{S}\\}$ is a subbasis for the topology
$\tau_{1}^{S}\vee\tau_{2}^{S}$. We shall show that every cover of
$\mathcal{K}(S)$ by subbasis-open sets has a finite subcover. Let
$\mathcal{K}(S)=\bigcup_{\lambda\in\Lambda}\Box U_{\lambda}\cup\bigcup_{i\in
I}\Diamond V_{i}$. Consider $S_{1}=S-\bigcup_{i\in I}V_{i}$. Then $S_{1}$ is a
pairwise closed subset of $S$. Hence, $S_{1}\in\mathcal{K}(S)$. Since,
$S_{1}\notin\Diamond V_{i}$ for each $i\in I$, so that
$S_{1}\in\bigcup_{\lambda\in\Lambda}\Box U_{\lambda}$. Then for some
$\lambda^{\prime}\in\Lambda$, $S_{1}\in\Box U_{{\lambda}^{\prime}}$. As a
result, $S_{1}\subseteq U_{{\lambda}^{\prime}}$ and hence
$S-U_{{\lambda}^{\prime}}\subseteq S-S_{1}=\displaystyle\bigcup_{i\in
I}V_{i}$. Then, $S=U_{{\lambda}^{\prime}}\cup\displaystyle\bigcup_{i\in
I}V_{i}$. As $S$ is pairwise compact, we have
$S=U_{\lambda^{\prime}}\cup\bigcup_{i=1}^{n}V_{i}$. Let $A$ be an arbitrary
element of $\mathcal{K}(S)$. If $A\subseteq U_{{\lambda}^{\prime}}$ then
$A\in\Box U_{{\lambda}^{\prime}}$ otherwise $A\subseteq\bigcup_{i\in I}V_{i}$
i.e., $A\cap V_{i}\neq\emptyset$ for some $i\in\\{1,2,\cdots,n\\}$. As a
result, $A\in\Box U_{\lambda^{\prime}}\cup\bigcup_{i\in I}\Diamond V_{i}$.
Therefore, $V_{P}(S)=(\mathcal{K}(S),\tau_{1}^{V},\tau_{2}^{V})$ is pairwise
compact.

Lemmas 9, 10 and 11 establish the following result:

######  Theorem 5.

If $(S,\tau_{1}^{S},\tau_{2}^{S})$ is a pairwise Boolean space then
$V_{P}(S)=(\mathcal{K}(S),\tau_{1}^{V},\tau_{2}^{V})$ is also a pairwise
Boolean space.

We now construct the $\mathcal{L}$-biVietoris functor $V_{\mathcal{L}}^{bi}$.

######  Definition 15.

We define a $\mathcal{L}$-biVietoris functor
$V_{\mathcal{L}}^{bi}:PBS_{\mathcal{L}}\to PBS_{\mathcal{L}}$ as follows:

  1. (i)

For an object $(S,\tau_{1}^{S},\tau_{2}^{S},\alpha_{S})$ in
$PBS_{\mathcal{L}}$, we define
$V_{\mathcal{L}}^{bi}(S,\tau_{1}^{S},\tau_{2}^{S},\alpha_{S})=(V_{P}(S),V_{P}\circ\alpha_{S})$
where $\alpha_{S}$ is a mapping from $\mathfrak{S}_{\mathcal{L}}$ to
$\Lambda_{S}$, then for any $\mathcal{L}_{1}\in\mathfrak{S}_{\mathcal{L}}$,
$V_{P}\circ\alpha_{S}(\mathcal{L}_{1})$ is the pairwise Vietoris space of a
pairwise closed subspace (i.e., pairwise Boolean subspace)
$\alpha_{S}(\mathcal{L}_{1})$ of $S$;

  2. (ii)

For an arrow
$f:(S_{1},\tau_{1}^{S_{1}},\tau_{2}^{S_{1}},\alpha_{S_{1}})\to(S_{2},\tau_{1}^{S_{2}},\tau_{2}^{S_{2}},\alpha_{S_{2}})$
in $PBS_{\mathcal{L}}$,
$V_{\mathcal{L}}^{bi}(f):(V_{P}(S_{1}),V_{P}\circ\alpha_{S_{1}})\to(V_{P}(S_{2}),V_{P}\circ\alpha_{S_{2}})$
is defined by $V_{\mathcal{L}}^{bi}(f)(K)=f[K]$, where $K\in V_{P}(S_{1})$.

We now show the well-definedness of the functor $V_{\mathcal{L}}^{bi}$.

######  Lemma 12.

Let $(S,\tau_{1}^{S},\tau_{2}^{S},\alpha_{S})$ be an object in
$PBS_{\mathcal{L}}$. Then
$V_{\mathcal{L}}^{bi}(S,\tau_{1}^{S},\tau_{2}^{S},\alpha_{S})$ is an object in
$PBS_{\mathcal{L}}$.

Proof: Theorem 5 shows that $V_{P}(S)$ is a pairwise Boolean space. Now we
shall show that $V_{P}\circ\alpha_{S}$ is a pairwise closed subspace of
$V_{P}(S)$. For $\mathcal{L}_{1}\in\mathfrak{S}_{\mathcal{L}}$, an element of
$V_{P}(S)\circ\alpha_{S}(\mathcal{L}_{1})$ is a pairwise compact subset of
$\alpha_{S}(\mathcal{L}_{1})$. As $\alpha_{S}(\mathcal{L}_{1})$ is also
pairwise compact subspace of $S$, so that an element of
$V_{P}\circ\alpha_{S}(\mathcal{L}_{1})$ is a pairwise compact subset of $S$.
As a result, $V_{P}\circ\alpha_{S}(\mathcal{L}_{1})$ is a subset of
$V_{P}(S)$. For $U\in\beta_{1}^{S}$, we get $\Box U\cap
V_{P}\circ\alpha_{S}(\mathcal{L}_{1})=\\{C\in
V_{P}\circ\alpha_{S}(\mathcal{L}_{1}):C\subset
U\\}=\Box(U\cap\alpha_{S}(\mathcal{L}_{1}))$ and $\Diamond U\cap
V_{P}\circ\alpha_{S}(\mathcal{L}_{1})=\\{C\in
V_{P}\circ\alpha_{S}(\mathcal{L}_{1}):C\cap
U\neq\emptyset\\}=\Diamond(U\cap\alpha_{S}(\mathcal{L}_{1}))$. Similarly for
$U\in\beta_{2}^{S}$. Hence, $V_{P}\circ\alpha_{S}(\mathcal{L}_{1})$ is a
pairwise subspace of $V_{P}(S)$. Since $\alpha_{S}(\mathcal{L}_{1})$ is a
pairwise Boolean subspace of $S$, by Theorem 5 we have
$V_{P}\circ\alpha_{S}(\mathcal{L}_{1})$ is a pairwise Boolean space.
Henceforth, $V_{P}\circ\alpha_{S}(\mathcal{L}_{1})$ is a pairwise closed
subspace of $V_{P}(S)$.
Now we show that $V_{P}\circ\alpha_{S}$ satisfies the conditions given in the
object part of Definition 7. If $\alpha_{S}(\mathcal{L})=S$ then
$V_{P}\circ\alpha_{S}(\mathcal{L})=V_{P}(S)$.
Let
$\mathcal{L}_{1},\mathcal{L}_{2},\mathcal{L}_{3}\in\mathfrak{S}_{\mathcal{L}}$.
If $\mathcal{L}_{1}=\mathcal{L}_{2}\cap\mathcal{L}_{3}$ then we show that
$V_{P}(\alpha_{S}(\mathcal{L}_{1}))=V_{P}(\alpha_{S}(\mathcal{L}_{2}))\cap
V_{P}(\alpha_{S}(\mathcal{L}_{3}))$. Now
$V_{P}(\alpha_{S}(\mathcal{L}_{1}))=V_{P}(\alpha_{S}(\mathcal{L}_{2}\cap\mathcal{L}_{3}))=V_{P}(\alpha_{S}(\mathcal{L}_{2})\cap\alpha_{S}(\mathcal{L}_{3}))$.
The element structure of
$V_{P}(\alpha_{S}(\mathcal{L}_{2})\cap\alpha_{S}(\mathcal{L}_{3}))$ is of the
form $P\cap(\alpha_{S}(\mathcal{L}_{2})\cap\alpha_{S}(\mathcal{L}_{3}))$ and
$Q\cap(\alpha_{S}(\mathcal{L}_{2})\cap\alpha_{S}(\mathcal{L}_{3}))$, where $P$
and $Q$ are $\tau_{1}^{S}$-closed set and $\tau_{2}^{S}$-closed set,
respectively. The elements of $V_{P}(\alpha_{S}(\mathcal{L}_{2}))\cap
V_{P}(\alpha_{S}(\mathcal{L}_{3}))$ are of the form
$(P_{1}\cap\alpha_{S}(\mathcal{L}_{2}))\cap(P_{2}\cap\alpha_{S}(\mathcal{L}_{3}))$
and
$(Q_{1}\cap\alpha_{S}(\mathcal{L}_{2}))\cap(Q_{2}\cap\alpha_{S}(\mathcal{L}_{3}))$,
where $P_{1},P_{2}$ are $\tau_{1}^{S}$-closed and $Q_{1},Q_{2}$ are
$\tau_{2}^{S}$-closed. Then it is easy to show that
$V_{P}(\alpha_{S}(\mathcal{L}_{2})\cap\alpha_{S}(\mathcal{L}_{3}))\subseteq
V_{P}(\alpha_{S}(\mathcal{L}_{2}))\cap V_{P}(\alpha_{S}(\mathcal{L}_{3}))$ and
$V_{P}(\alpha_{S}(\mathcal{L}_{2}))\cap
V_{P}(\alpha_{S}(\mathcal{L}_{3}))\subseteq
V_{P}(\alpha_{S}(\mathcal{L}_{2})\cap\alpha_{S}(\mathcal{L}_{3}))$.

######  Lemma 13.

Let
$f:(S_{1},\tau_{1}^{S_{1}},\tau_{2}^{S_{1}},\alpha_{S_{1}})\to(S_{2},\tau_{1}^{S_{2}},\tau_{2}^{S_{2}},\alpha_{S_{2}})$
be an arrow in $PBS_{\mathcal{L}}$. Then $V_{\mathcal{L}}^{bi}(f)$ is an arrow
in $PBS_{\mathcal{L}}$.

Proof: Given that $f$ is a pairwise continuous map from a pairwise Boolean
space $S_{1}$ to a pairwise Boolean space $S_{2}$. Let $K\in V_{P}(S_{1})$.
Then $K$ is a pairwise closed subset of $S_{1}$ and hence $K$ is pairwise
compact. Now $V_{\mathcal{L}}^{bi}(f)(K)=f[K]$ is a pairwise compact subset of
$S_{2}$. Since $S_{2}$ is a pairwise Boolean space, $f[K]$ is a pairwise
closed subset of $S_{2}$. As a result, $V_{\mathcal{L}}^{bi}(f)(K)\in
V_{P}(S_{2})$. To show that $V_{\mathcal{L}}^{bi}(f)$ is pairwise continuous,
let $U\in\beta_{1}^{S_{2}}$ and $V\in\beta_{2}^{S_{2}}$. Then

$\begin{split}V_{\mathcal{L}}^{bi}(f)^{-1}(\Box U)&=\\{K\in
V_{P}(S_{1}):V_{\mathcal{L}}^{bi}(f)(K)\in\Box U\\}\\\
&=\\{K\in\mathcal{K}(S_{1}):f(K)\subseteq U\\}\\\
&=\\{K\in\mathcal{K}(S_{1}):K\subseteq f^{-1}(U)\\}\\\ &=\Box
f^{-1}(U)\end{split}$

and

$\begin{split}V_{\mathcal{L}}^{bi}(f)^{-1}(\Diamond U)&=\\{K\in
V_{P}(S_{1}):V_{\mathcal{L}}^{bi}(f)(K)\in\Diamond U\\}\\\
&=\\{K\in\mathcal{K}(S_{1}):f(K)\cap U\neq\emptyset\\}\\\
&=\\{K\in\mathcal{K}(S_{1}):K\cap f^{-1}(U)\neq\emptyset\\}\\\ &=\Diamond
f^{-1}(U)\end{split}$

Similarly, $V_{\mathcal{L}}^{bi}(f)^{-1}(\Box V)=\Box f^{-1}(V)$ and
$V_{\mathcal{L}}^{bi}(f)^{-1}(\Diamond V)=\Diamond f^{-1}(V)$. Therefore,
$V_{\mathcal{L}}^{bi}(f)$ is pairwise continuous. It is still necessary to
demonstrate that $V_{\mathcal{L}}^{bi}(f)$ is subspace preserving. Let $M\in
V_{P}\circ\alpha_{S_{1}}(\mathcal{L}_{1})$,
$\mathcal{L}_{1}\mathfrak{S}_{\mathcal{L}}$. Then
$M\subseteq\alpha_{S_{1}}(\mathcal{L}_{1})$. As $f$ is an arrow in
$PBS_{\mathcal{L}}$, hence $f$ is a subspace preserving. Thus,
$f(M)\subseteq\alpha_{S_{2}}(\mathcal{L}_{1})$. It shows that
$V_{\mathcal{L}}^{bi}(f)(M)\subseteq\alpha_{S_{2}}(\mathcal{L}_{1})$. Thus we
have $V_{\mathcal{L}}^{bi}(f)(M)\in
V_{P}\circ\alpha_{S_{2}}(\mathcal{L}_{1})$.

We introduce two functors $\mathfrak{B}$ and $\mathfrak{C}$ between the
categories $PRBS_{\mathcal{L}}$ and $COALG(V_{\mathcal{L}}^{bi})$ to show that
these two categories are isomorphic.

######  Definition 16.

We define a functor $\mathfrak{B}:PRBS_{\mathcal{L}}\to
COALG(V_{\mathcal{L}}^{bi})$ as follows:

  1. (i)

For an object $(S,\alpha_{S},\mathcal{R})$ in $PRBS_{\mathcal{L}}$, define
$\mathfrak{B}(S,\alpha_{S},\mathcal{R})=(S,\alpha_{S},\mathcal{R}[-])$, where
$\mathcal{R}[-]:(S,\alpha_{S})\to V_{\mathcal{L}}^{bi}(S,\alpha_{S})$ is an
arrow in $PBS_{\mathcal{L}}$ defined by $\mathcal{R}[s]=\\{p\in
S:s\mathcal{R}p\\}$, $s\in S$;

  2. (ii)

For an arrow
$f:(S_{1},\alpha_{S_{1}},\mathcal{R}_{1})\to(S_{2},\alpha_{S_{2}},\mathcal{R}_{2})$
in $PRBS_{\mathcal{L}}$, define
$\mathfrak{B}(f):(S_{1},\alpha_{S_{1}},\mathcal{R}_{1}[-])\to(S_{2},\alpha_{S_{2}},\mathcal{R}_{2}[-])$
by $\mathfrak{B}(f)=f$.

The well-definedness of the functor $\mathfrak{B}$ is shown by the following
two lemmas:

######  Lemma 14.

Let $(S,\alpha_{S},\mathcal{R})$ be an object in $PRBS_{\mathcal{L}}$. Then
$\mathfrak{B}(S,\alpha_{S},\mathcal{R})$ is an object in
$COALG(V_{\mathcal{L}}^{bi})$ .

Proof: We shall show that $\mathcal{R}[-]:(S,\alpha_{S})\to
V_{\mathcal{L}}^{bi}(S,\alpha_{S})$ is an arrow in $PBS_{\mathcal{L}}$. By the
conditions given in the object part of Definition 10, we know that for each
$s\in S$, $\mathcal{R}[s]$ is pairwise compact subset of $S$. As $S$ is
pairwise Boolean space, hence $\mathcal{R}[s]$ is a pairwise closed subset of
$S$. Thus $\mathcal{R}[s]\in V_{P}(S)$. Let $U\in\beta_{1}^{S}$. Then

$\begin{split}\mathcal{R}[-]^{-1}(\Box U)&=\\{s\in S:\mathcal{R}[s]\in\Box
U\\}\\\ &=\\{s\in S:\mathcal{R}[s]\subseteq U\\}\\\
&=[\mathcal{R}]U\in\beta_{1}^{S}\text{ [by Definition
\ref{PRBSL}]}\end{split}$

and

$\begin{split}\mathcal{R}[-]^{-1}(\Diamond U)&=\\{s\in
S:\mathcal{R}[s]\in\Diamond U\\}\\\ &=\\{s\in S:\mathcal{R}[s]\cap
U\neq\emptyset\\}\\\ &=\langle\mathcal{R}\rangle U\in\beta_{1}^{S}\text{ [by
Definition \ref{PRBSL}]}\end{split}$

Similarly, for $U\in\beta_{2}^{S}$, $\mathcal{R}[-]^{-1}(\Box
U)=[\mathcal{R}]U\in\beta_{2}^{S}$ and $\mathcal{R}[-]^{-1}(\Diamond
U)=\langle\mathcal{R}\rangle U\in\beta_{2}^{S}$. Henceforth, $\mathcal{R}[-]$
is a pairwise continuous map. Now we show that $\mathcal{R}[-]$ is subspace
preserving. Let $s\in\alpha_{S}(\mathcal{L}^{\prime})$,
$\mathcal{L}^{\prime}\in\mathfrak{S}_{\mathcal{L}}$. As we know from
Definition 10 that $\mathcal{R}[s]$ is a pairwise compact subset of
$\alpha_{S}(\mathcal{L}^{\prime})$. Since $\alpha_{S}(\mathcal{L}^{\prime})$
is itself a pairwise Boolean space, thus we have $\mathcal{R}[s]\in
V_{P}\circ\alpha_{S}(\mathcal{L}^{\prime})$. Therefore,
$\beta(S,\alpha_{S},\mathcal{R})$ is a $V_{\mathcal{L}}^{bi}$-coalgebra.

######  Lemma 15.

Let
$f:(S_{1},\alpha_{S_{1}},\mathcal{R}_{1})\to(S_{2},\alpha_{S_{2}},\mathcal{R}_{2})$
be an arrow in $PRBS_{\mathcal{L}}$. Then $\mathfrak{B}(f)$ is an arrow in
$COALG(V_{\mathcal{L}}^{bi})$.

Proof: As $f$ is an arrow in $PBS_{\mathcal{L}}$,
$\mathfrak{B}(f)=f:(S_{1},\alpha_{S_{1}},\mathcal{R}_{1}[-])\to(S_{2},\alpha_{S_{2}},\mathcal{R}_{2}[-])$
is a pairwise continuous map. Now using the conditions mentioned in the arrow
part of Definition 10, it is straightforward to verify that
$\mathcal{R}_{2}[-]\circ f=V_{\mathcal{L}}^{bi}\circ\mathcal{R}_{1}[-]$. Thus
$\mathfrak{B}(f)$ is an arrow in $COALG(V_{\mathcal{L}}^{bi})$.

######  Definition 17.

We define a functor $\mathfrak{C}:COALG(V_{\mathcal{L}}^{bi})\to
PRBS_{\mathcal{L}}$ as follows:

  1. (i)

For an object $((C,\alpha_{C}),\zeta)$ in $COALG(V_{\mathcal{L}}^{bi})$,
define
$\mathfrak{C}((C,\alpha_{C}),\zeta)=(C,\alpha_{C},\mathcal{R}_{\zeta})$, where
$\mathcal{R}_{\zeta}$ is a binary relation on $C$ defined by
$d\in\mathcal{R}_{\zeta}[c]\iff d\in\zeta(c)$, $c,d\in C$;

  2. (ii)

For an arrow
$f:((C_{1},\alpha_{C_{1}}),\zeta_{1})\to((C_{2},\alpha_{C_{2}}),\zeta_{2})$ in
$COALG(V_{\mathcal{L}}^{bi})$, define
$\mathfrak{C}(f):(C_{1},\alpha_{C_{1}},\mathcal{R}_{\zeta_{1}})\to(C_{2},\alpha_{C_{2}},\mathcal{R}_{\zeta_{2}})$
by $\mathfrak{C}(f)=f$.

The well-definedness of the functor $\mathfrak{C}$ is shown by Lemma 16 and
Lemma 17.

######  Lemma 16.

For an object $((C,\alpha_{C}),\zeta)$ in $COALG(V_{\mathcal{L}}^{bi})$,
$\mathfrak{C}((C,\alpha_{C}),\zeta)=(C,\alpha_{C},\mathcal{R}_{\zeta})$ is an
object in $PRBS_{\mathcal{L}}$.

Proof: In order to show that $\mathfrak{C}((C,\alpha_{C}),\zeta)$ is an object
in $PRBS_{\mathcal{L}}$, we must verify that
$\mathfrak{C}((C,\alpha_{C}),\zeta)$ satisfies the conditions given in the
object part of Definition 10. For each $c\in C$,
$\mathcal{R}_{\zeta}[c]=\zeta(c)\in V_{P}(C)$. Hence, $\mathcal{R}_{\zeta}[c]$
is a pairwise closed subset of $C$. Thus $\mathcal{R}_{\zeta}[c]$ is pairwise
compact. Let $U\in\beta_{1}^{C}$. Then

$\begin{split}[\mathcal{R}_{\zeta}](U)&=\\{c\in C:\mathcal{R}_{\zeta}\subseteq
U\\}\\\ &=\\{c\in C:\zeta(c)\subseteq U\\}\\\ &=\\{c\in C:\zeta(c)\in\Box
U\\}\\\ &=\zeta^{-1}(\Box U)\in\beta_{1}^{C}\end{split}$

and

$\begin{split}\langle\mathcal{R}_{\zeta}\rangle U&=\\{c\in
C:\mathcal{R}_{\zeta}\cap U\neq\emptyset\\}\\\ &=\\{c\in C:\zeta(c)\cap
U\neq\emptyset\\}\\\ &=\\{c\in C:\zeta(c)\in\Diamond U\\}\\\
&=\zeta^{-1}(\Diamond U)\in\beta_{1}^{C}\end{split}$

Finally, let $m\in\alpha_{C}(\mathcal{L^{\prime}})$ for
$\mathcal{L^{\prime}}\in\mathfrak{S}_{\mathcal{L}}$. As $\zeta$ is a subspace
preserving map from $(C,\alpha_{C})$ to $V_{\mathcal{L}}^{bi}(C,\alpha_{C})$,
we have $\mathcal{R}_{\zeta}[m]=\zeta(m)\in
V_{P}\circ\alpha_{C}(\mathcal{L^{\prime}})$. Henceforth,
$\mathcal{R}_{\zeta}[m]\subset\alpha_{C}(\mathcal{L^{\prime}})$.

######  Lemma 17.

For an arrow
$f:((C_{1},\alpha_{C_{1}}),\zeta_{1})\to((C_{2},\alpha_{C_{2}}),\zeta_{2})$ in
$COALG(V_{\mathcal{L}}^{bi})$,
$\mathfrak{C}(f):(C_{1},\alpha_{C_{1}},\mathcal{R}_{\zeta_{1}})\to(C_{2},\alpha_{C_{2}},\mathcal{R}_{\zeta_{2}})$
is an arrow in $PRBS_{\mathcal{L}}$.

Proof: It is straightforward to prove that $\mathfrak{C}$ is an arrow in
$PRBS_{\mathcal{L}}$.

Now we obtain the following result:

######  Theorem 6.

The categories $PRBS_{\mathcal{L}}$ and $COALG(V_{\mathcal{L}}^{bi})$ are
isomorphic.

Proof: We shall show that the categories $PRBS_{\mathcal{L}}$ and
$COALG(V_{\mathcal{L}}^{bi})$ are isomorphic via the functors $\mathfrak{B}$
and $\mathfrak{C}$. Let $(S,\alpha_{S},\mathcal{R})$ be an object in
$PRBS_{\mathcal{L}}$. Then
$\mathfrak{C}\circ\mathfrak{B}(S,\alpha_{S},\mathcal{R})=\mathfrak{C}(S,\alpha_{S},\mathcal{R}[-])=(S,\alpha_{S},\mathcal{R}_{\mathcal{R}[-]})$.
Now $t\in\mathcal{R}_{\mathcal{R}[-]}(s)\iff t\in\mathcal{R}[s]$. Thus
$(S,\alpha_{S},\mathcal{R})=\mathfrak{C}\circ\mathfrak{B}(S,\alpha_{S},\mathcal{R})$.
Let $((C,\alpha_{C}),\zeta)$ be an object in $COALG(V_{\mathcal{L}}^{bi})$.
$\mathfrak{B}\circ\mathfrak{C}((C,\alpha_{C}),\zeta)=\mathfrak{B}(C,\alpha_{C},\mathcal{R}_{\zeta})=((C,\alpha_{C}),\mathcal{R}_{\zeta}[-])$.
We have $c_{2}\in\mathcal{R}_{\zeta}[c_{1}]\iff c_{2}\in\zeta(c_{1})$. As a
result,
$((C,\alpha_{C}),\zeta)=\mathfrak{B}\circ\mathfrak{C}((C,\alpha_{C}),\zeta)$.
It is clear that for an arrow $f$ in $COALG(V_{\mathcal{L}}^{bi})$,
$\mathfrak{B}\circ\mathfrak{C}(f)=f$ and for an arrow $f$ in
$PRBS_{\mathcal{L}}$, $\mathfrak{C}\circ\mathfrak{B}(f)=f$.

Using Theorems 4 and 6, we arrive at the following duality theorem:

######  Theorem 7.

The categories $\mathcal{MA}_{\mathcal{L}}$ and $COALG(V_{\mathcal{L}}^{bi})$
are dually equivalent.

Thus the modal semi-primal duality for algebras of Fitting’s Heyting-valued
modal logic (for more information, see [maruyama2011dualities ]) can
potentially be represented in terms of the coalgebras of
$\mathcal{L}$-biVietoris functor $V_{\mathcal{L}}^{bi}$.
Finally, based on the preceding theorems, we can conclude:

######  Theorem 8.

Fitting’s Heyting-valued modal logic is sound and complete with respect to
coalgebras of the bi-Vietoris functor $V_{\mathcal{L}}^{bi}$.

##  5 Conclusion

We have introduced the category $PRBS_{\mathcal{L}}$ and found a duality for
the class of all algebras of a version of Fitting’s Heyting-valued modal logic
in bitopological language. This has led to an extension of the natural duality
theory for modal algebras. We have demonstrated how the theory of coalgebras
can be used to characterize the category $PRBS_{\mathcal{L}}$ and thus
obtained a coalgebraic description of the bitopological duality for Fitting’s
Heyting-valued modal logic. In the current work, we have explicitly
constructed the Vietoris functor on the category $PBS_{\mathcal{L}}$ and we
have finally concluded that coalgebras for this functor provide sound and
complete semantics for Fitting’s Heyting-valued modal logic.
In light of our present work, we can suggest some future research directions.
As an application of this coalgebraic duality, we may establish the existence
of a final coalgebra and cofree coalgebras in the category
$COALG(V_{\mathcal{L}}^{bi})$, and we can also develop the coalgebraic duality
theorem for many-valued modal logics in a bitopological scenario.
Another interesting line of research would be to show that coalgebras of an
endofunctor $V$ on the category $BES$ of bi-topological Esakia spaces( the
idea of bitopological Esakia spaces can be found in
bezhanishvili2010bitopological ) can characterise lattice-valued
intuitionistic modal logic. However, it is unclear to us how to characterise
the relation $\mathcal{R}$ on bitopological Esakia spaces in terms of
coalgebras of the functor $V$, and this appears to be an open problem at the
moment.

## Declarations

## References

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