# Coalgebraic Fuzzy geometric logic

Litan Kumar Das
Department of Mathematics
Jadavpur University<EMAIL_ADDRESS>Kolkata  India  Kumar Sankar Ray
ECSU<EMAIL_ADDRESS>Indian Statistical Institute  Kolkata  India
Prakash Chandra Mali
Department of Mathematics
Jadavpur University<EMAIL_ADDRESS>Kolkata  India  Litan Kumar Das,
Department of Mathematics, Jadavpur University

###### Abstract

The paper aims to develop a framework for coalgebraic fuzzy geometric logic by
adding modalities to the language of fuzzy geometric logic. Using the methods
of coalgebra, the modal operators are introduced in the language of fuzzy
geometric logic. To define the modal operators, we introduce a notion of
fuzzy-open predicate lifting. Based on coalgebras for an endofunctor $T$ on
the category Fuzzy-Top of fuzzy topological spaces and fuzzy continuous maps,
we build models for the coalgebraic fuzzy geometric logic. Bisimulations for
the defined models are discussed in this work.

###### keywords:

Fuzzy geometric logic, Fuzzy topological spaces, Coalgebra, Final coalgebra,
bisimulation

Coalgebraic Fuzzy geometric logic

##  1 Introduction

Fuzzy geometric logic has been introduced in [9] which is a natural
generalization of the propositional geometric logic [30]. Vicker in [30]
developed geometric logic based on point-free topology, logic, and the logic
of finite observations [3]. It has been pointed out in several works (see[4,
5, 6, 31, 32]). The language of geometric logic is formed on a set of
propositional variables by applying propositional connectives: finite
conjunction$(\wedge)$ and arbitrary disjunction$(\bigvee)$. The property of
finite observability is preserved by these connectives. Vickers in [30]
explored the relationship between topological spaces, topological systems and
geometric logic. A topological system is defined by a triple $(X,\models,A)$
in which $X$ occurs as a non-empty set of objects, $A$ defines a frame and
$\models$ is given by a satisfaction relation from $X$ to $A$. The authors in
[9] have generalized geometric logic to the many-valued context by extending
the notion of satisfiability relation. They observed that if the satisfaction
relation is fuzzy then the corresponding consequence relation has two
possibilities: crisp or fuzzy. Consequently, they introduced general fuzzy
geometric logic and fuzzy geometric logic with graded consequence. The
relationship among fuzzy geometric logic, fuzzy topology and fuzzy topological
systems has also been shown in their work. The concept of Fuzzy topological
spaces introduced in [10] and it was taken into account in various
works(e.g.,[11, 12, 13, 14, 15]). Graded consequence and its related aspects
can be found in [16]. Different kinds of modal logics were developed based on
the theory of coalgebras. In [17], A. Kurz discussed coalgebraic logic in more
detail. Coalgebraic logic for functors on the category of sets extensively
studied in [18, 19], where they used the notion of predicate lifting [21] or
relation lifting [20] to define modal operators. In [22], Enqvist and Sourabh
developed coalgebraic modal logic in the category of Stone coalgebras.
Coalgebraic geometric logic has been developed in [23, 24], where modal
operators are defined to the language of geometric logic based on the theory
of coalgebraic logic. Motivated by their work, we study modalities in fuzzy
geometric logic using the coalgebraic machinery of predicate liftings. The
structures, named fuzzy geometric models for $T$, provide the semantics for
coalgebraic fuzzy geometric logic. The notion of bisimulation [26] is
extensively applied in various fields of computer science as well as in
mathematics. The bisimulation between coalgebras is a fundamental concept in
“state-based systems” which associates the states in systems with the same
behaviour. In [27], Staton developed different notions of coalgebraic
bisimulation and studied the relationship between these notions. Bisimulations
for the Stone coalgebras studied in [22]. In our work, we have discussed
bisimulations for the proposed fuzzy geometric models. It is expected that the
proposed coalgebraic logic and the results developed in this work will have
implications in computer science areas such as knowledge representations,
logic programming, formal verification, and fuzzy reasoning. The paper is
organized as follows. In Section 2, we have mentioned the ideas that are
required for the work. We define coalgebraic logic for Fuzzy-Top-coalgebras in
Section 3, and then introduce fuzzy geometric models for an endofunctor $T$.
In Section 4, we construct a final model in the category $FMOD(T)$ of fuzzy
geometric models for $T$, where $T$ is an endofunctor on the category SFuzzy-
Top of sober fuzzy topological spaces. In Section 5, we discuss bisimulations
for fuzzy geometric models. We conclude the paper by mentioning future
directions of work in Section 6.

##  2 Preliminaries

The reader is referred to [1] for category theory basics. For clarity and
understanding of our work, we mention some useful ideas.
In 1965, Zadeh investigated fuzzy set theory [38]. We go through some
fundamental ideas in fuzzy set theory.

###### Definition 1

A fuzzy set $\tilde{f}$ on a set $S$ is defined by the membership function
$\tilde{f}:S\to[0,1]$.

Let $\tilde{f}^{c}$ denote the complement of $\tilde{f}$. Define
$\tilde{f}^{c}:S\to[0,1]$ by $\tilde{f}^{c}(s)=1-\tilde{f}(s)$, $\forall s\in
S$. $\tilde{f}^{c}$ is a fuzzy set on $S$.

###### Note 1

If $\tilde{f_{1}}$ and $\tilde{f_{2}}$ are fuzzy sets on $S$, then
$\tilde{f_{1}}\vee\tilde{f_{2}}$ and $\tilde{f_{1}}\wedge\tilde{f_{2}}$ are
fuzzy sets on $S$, where the fuzzy sets $\tilde{f_{1}}\vee\tilde{f_{2}}$ and
$\tilde{f_{1}}\wedge\tilde{f_{2}}$ are defined by
$(\tilde{f_{1}}\vee\tilde{f_{2}})(s)=\tilde{f_{1}}(s)\vee\tilde{f_{2}}(s)$ and
$(\tilde{f_{1}}\wedge\tilde{f_{2}})(s)=\tilde{f_{1}}(s)\wedge\tilde{f_{2}}(s)$,
respectively.

###### Remark 1

For each $s\in S$, the grade of membership of $s$ in the fuzzy set $\tilde{f}$
is given by the value $\tilde{f}(s)$. It is represented by the symbol
$gr(s\in\tilde{f})$.

###### Definition 2

Let $S_{1}$ and $S_{2}$ be two sets and $f:S_{1}\to S_{2}$ be a given
function. For a fuzzy set $\tilde{s_{1}}$ on $S_{1}$, the direct image
$f(\tilde{s_{1}}):S_{2}\to[0,1]$ of the fuzzy set $\tilde{s_{1}}$ under the
function $f$ is defined by
$f(\tilde{s_{1}})(s)=\bigvee\\{\tilde{s_{1}}(t):t\in f^{-1}(\\{s\\})\\}$,
where $s\in S_{2}$.

###### Definition 3

Let $S_{1}$ and $S_{2}$ be two sets and $f:S_{1}\to S_{2}$ be a given
function. For a fuzzy set $\tilde{s_{2}}$ on $S_{2}$, the inverse image
$f^{-1}(\tilde{s_{2}}):S_{1}\to[0,1]$ of the fuzzy set $\tilde{s_{2}}$ under
the function $f$ is defined by $f^{-1}(\tilde{s_{2}})=\tilde{s_{2}}\circ f$.

###### Definition 4

Let $\mu$ and $\eta$ be fuzzy sets on $S$. Then, $\mu$ is a fuzzy subset of
$\eta$, denoted by $\mu\leq\eta$, $\iff$ $\mu(s)\leq\eta(s)$, $\forall s\in
S$.

###### Definition 5

Consider a mapping $f:S\to T$ and a collection
$\\{\mu_{i}:i\in\mathcal{I},\mathcal{I}\text{ is an index set }\\}$ of fuzzy
sets on $T$. Then

  1. (i)

$f^{-1}(\displaystyle\bigvee_{i\in\mathcal{I}}\mu_{i})=\bigvee_{i\in\mathcal{I}}f^{-1}(\mu_{i})$;

  2. (ii)

$f^{-1}(\displaystyle\bigwedge_{i\in\mathcal{I}}\mu_{i})=\bigwedge_{i\in\mathcal{I}}f^{-1}(\mu_{i})$.

We recall the definition of fuzzy topological spaces from [10].

###### Definition 6

Let $S$ be a set. A collection $\tau_{S}$ of fuzzy sets on $S$ is said to be
fuzzy topology on $S$ if the following conditions hold:

  1. (i)

$\tilde{\emptyset},\tilde{S}\in\tau_{S}$, where $\tilde{\emptyset}(s)=0$,
$\forall s\in S$ and $\tilde{S}(s)=1$, $\forall s\in S$;

  2. (ii)

if $\tilde{g_{1}},\tilde{g_{2}}\in\tau_{s}$ then
$\tilde{g_{1}}\wedge\tilde{g_{2}}\in\tau_{S}$, where
$(\tilde{g_{1}}\wedge\tilde{g_{2}})(s)=\tilde{g_{1}}(s)\wedge\tilde{g_{2}}(s)$;

  3. (iii)

if $\tilde{g_{j}}\in\tau_{S}$ for $j\in\Lambda$, $\Lambda$ is an index set,
then $\displaystyle\bigvee_{j\in\Lambda}\tilde{g_{j}}\in\tau_{S}$, where
$\displaystyle\bigvee_{j\in\Lambda}\tilde{g_{j}}(s)=\sup_{j\in\Lambda}\\{\tilde{g_{j}}(s)\\}$.

Then, the pair $(S,\tau_{S})$ is referred to as a fuzzy topological space and
members of $\tau_{S}$ are said to be fuzzy open sets on $(S,\tau_{S})$.

######  Definition 7 ([34])

Let $(S,\tau_{S})$ be a fuzzy topological space. Then a subset $\mathfrak{B}$
of $\tau_{S}$ is called a basis for $(S,\tau_{S})$ if it satisfies the
following conditions:

  1. (i)

if $\tilde{b}_{1},\tilde{b}_{2}\in\mathfrak{B}$ then
$\tilde{b}_{1}\wedge\tilde{b}_{2}\in\mathfrak{B}$;

  2. (ii)

for each member $\tilde{t}\in\tau_{S}$, there exists a subcollection
$\mathcal{C}=\\{\tilde{t}_{j}\in\mathfrak{B}:j\in\Lambda\\}$ such that
$\tilde{t}=\displaystyle\bigvee_{j\in\Lambda}\tilde{t}_{j}$.

######  Definition 8 ([34])

Let $\tau_{S}$ be a fuzzy topology on $S$ and $\mathfrak{S}\subset\tau_{S}$.
Then, $\mathfrak{S}$ is a subbasis for a fuzzy space $(S,\tau_{S})$ $\iff$ the
collection of all finite meets of members of $S$ is a basis for
$(S,\tau_{S})$.

###### Definition 9

A fuzzy topological space $(S,\tau_{S})$ is said to be Kolmogorov space or
$T_{0}$-space if for any pair $(x,y)$ of distinct points in $S$, there is a
fuzzy open set $\tilde{g}$ on $S$ such that $\tilde{g}(x)\neq\tilde{g}(y)$.

###### Definition 10

Let $(F,\tau_{F})$ and $(G,\tau_{G})$ be fuzzy topological spaces. A mapping
$f:F\to G$ is fuzzy continuous if and only if, for every fuzzy open set
$\tilde{g}$ on $(G,\tau_{G})$, $f^{-1}(\tilde{g})$ is a fuzzy open set on
$(F,\tau_{F})$.

Definitions of category, opposite category and functors can be found in [1].
We recall the concept of natural transformations.

######  Definition 11 ([1])

Let $\mathcal{F}$ and $\mathcal{G}$ be functors from a category $\mathcal{A}$
to a category $\mathcal{B}$. A natural transformation
$\mathcal{T}:\mathcal{F}\to\mathcal{G}$ is a mapping that carries each object
$A$ in $\mathcal{A}$ to a morphism
$\mathcal{T}_{A}:\mathcal{F}(A)\to\mathcal{G}(A)$ in $\mathcal{B}$ such that
for any morphism $\zeta:A\to\bar{A}$ in $\mathcal{A}$ we have
$\mathcal{G}(\zeta)\circ\mathcal{T}_{A}=\mathcal{T}_{\bar{A}}\circ\mathcal{F}(\zeta)$.
As a result, the Fig. 1 commutes.

${\mathcal{F}(A)}$${\mathcal{G}(A)}$${\mathcal{F}(\bar{A})}$${\mathcal{G}(\bar{A})}$$\scriptstyle{\mathcal{T}_{A}}$$\scriptstyle{\mathcal{F}(\zeta)}$$\scriptstyle{\mathcal{G}(\zeta)}$$\scriptstyle{\mathcal{T}_{\bar{A}}}$
Figure 1: Illustration of natural transformation

Goguen first considered the category of fuzzy sets in [33]. Several authors
studied on the category of fuzzy sets (e.g.,[35, 36, 37]). Let FS denote the
category of fuzzy sets.

######  Definition 12 ([35])

The category FS is defined as follows:

  1. (i)

An object in FS is a pair $(S,\tilde{g})$, where $S$ is a set and
$\tilde{g}:S\to[0,1]$ is a membership function;

  2. (ii)

A morphism $f:(S,\tilde{g})\to(T,\tilde{h})$ in FS is a function $f:S\to T$
such that $\tilde{g}(s)\leq f^{-1}(\tilde{h})(s)$.

Let Fuzzy-Top denote the category of fuzzy topological spaces.

###### Definition 13

The category Fuzzy-Top is defined as follows:

  1. (i)

Objects in Fuzzy-Top are fuzzy topological spaces $(S,\tau_{S})$;

  2. (ii)

Morphisms $f:(S,\tau_{S})\to(T,\tau_{T})$ in Fuzzy-Top are fuzzy continuous
mappings.

###### Definition 14

A functor $\mathcal{Q}$ from the category Fuzzy-Top to the category FS of
fuzzy sets can be defined as follows:

  1. (i)

For an object $(S,\tau_{S})$ in Fuzzy-Top, define $\mathcal{Q}(S)=\text{set of
fuzzy open sets on }(S,\tau_{S})$;

  2. (ii)

For a morphism $\phi:(S,\tau_{S})\to(T,\tau_{T})$ in Fuzzy-Top, define
$\mathcal{Q}(\phi)=\phi^{-1}:\mathcal{Q}(T)\to\mathcal{Q}(S)$ by
$\phi^{-1}(\mu)=\mu\circ\phi$, $\mu\in\mathcal{Q}(T)$.

######  Definition 15 ([30])

A frame is a partially ordered set which is closed under finite meet
$(\wedge)$ and arbitrary join $(\bigvee)$ such that $\wedge$ distributes over
$\bigvee$.

###### Note 2

Let $(S,\tau_{S})$ be a fuzzy topological space. Then, the fuzzy topology
$\tau_{S}$ on $S$ can be considered as a frame.

######  Definition 16 ([30])

A function $f$ from a frame $F_{1}$ to a frame $F_{2}$ is said to be a frame
homomorphism if the function $f$ preserves finite meets and arbitrary joins.

The collection of frames and frame homomorphisms forms a category, denoted by
FRM.
Similar to other algebraic structures, frames may be presented by generators
and relations $\langle G\vline R\rangle$, where $G$ denotes the set of
generators, and $R$ is the set of relations between expressions generated by
$G$. One can find a detailed description of frame presentations in [32].

######  Note 3 ([24])

Consider a frame $F_{1}$. Now, $\langle G\vline R\rangle$ presents the frame
$F_{1}$ if $\exists$ an assignment $h:G\to F_{1}^{*}$, where $F_{1}^{*}$
denotes the underlying set of $F_{1}$, such that the following properties
hold:

  1. (i)

$F_{1}$ is generated by the set $\\{h(s):s\in G\\}$;

  2. $h$ can be extended to an assignment $\hat{h}$ for any expression $r$ that is generated by $G$.

  3. (ii)

If $r*=r^{\prime}*$ is a relation in $R$, then
$\hat{h}(r*)=\hat{h}(r^{\prime}*)$ in $F_{1}$;

  4. (iii)

For a frame $F_{2}$ and an assignment $h^{\prime}:G\to F_{2}^{*}$ that
satisfies $(ii)$ there is a unique frame homomorphism $g:F_{1}\to F_{2}$ such
that $g^{*}\circ h=h^{\prime}$, where $g^{*}$ is a mapping from $F_{1}^{*}$ to
$F_{2}^{*}$. So, the diagram shown in Fig. 2 commutes.

${G}$${F_{1}^{*}}$${F_{2}^{*}}$$\scriptstyle{h^{\prime}}$$\scriptstyle{h}$$\scriptstyle{g^{*}}$
Figure 2: Illustration of frame presentation

###### Remark 2

We can define a frame homomorphism $f:F_{1}\to F_{2}$ from a frame $F_{1}$ to
a frame $F_{2}$, where $\langle G\vline R\rangle$ presents the frame $F_{1}$.
According to Note 3, it is sufficient to provide an assignment $\hat{f}:G\to
F_{2}$ that satisfies the condition that if $r*=r^{\prime}*$ is a relation in
$R$, then $\hat{f}(r*)=\hat{f}(r^{\prime}*)$ in $F_{2}$.

Coalgebras are categorical structures that are dual or opposite (in the sense
of category theory) to the notion of algebras. The coalgebraic approach is
abundantly applied in computer science and artificial intelligence (e.g.,
knowledge representation, concurrency, logical reasoning, automata theory,
etc.).

######  Definition 17 ([2])

Assume that $T$ is an endofunctor on a category $\mathcal{S}$. A $T$-coalgebra
is a pair $(A,\zeta)$, where $A$ is an object in $\mathcal{S}$ and $\zeta:A\to
T(A)$ is a morphism in $\mathcal{S}$.

######  Definition 18 ([2])

A morphism between $T$-coalgebras $(A,\delta)$ and $(B,\beta)$ is defined by a
morphism $\psi:A\to B$ in $\mathcal{S}$ satisfying the equation
$T(\psi)\circ\delta=\beta\circ\psi$, i.e., the diagram shown in Fig. 3
commutes.

${A}$${B}$${T(A)}$${T(B)}$$\scriptstyle{\psi}$$\scriptstyle{\delta}$$\scriptstyle{\beta}$$\scriptstyle{T(\psi)}$
Figure 3: Illustration of coalgebra morphism

$T$-coalgebras and morphisms between $T$-coalgebras form a category, denoted
by $\textbf{COALG}(T)$.
A final coalgebra is a final or terminal object in $\textbf{COALG}(T)$. It has
a significant impact on computer science. The final coalgebra is crucial as it
makes sense of behaviourally equivalent states in coalgebras.

###### Definition 19

A final coalgebra in $\textbf{COALG}(T)$ is a $T$-coalgebra $(A,\delta)$ which
satisfies that for each $T$-coalgebra $(B,\beta)$, a unique morphism exists
from $(B,\beta)$ to $(A,\delta)$.

###### Definition 20

Let $(A,\delta)$ and $(B,\beta)$ be objects in $\textbf{COALG}(T)$. We say
that any two states $a\in A$ and $b\in B$ are behaviourally equivalent if
there exists an object $(C,\alpha)$ in $\textbf{COALG}(T)$ and $T$-coalgebra
morphisms $g:(A,\delta)\to(C,\alpha)$ and $h:(B,\beta)\to(C,\alpha)$ such that
$g(a)=h(b)$.

######  Definition 21 ([28])

Let $(A,\delta)$ and $(B,\beta)$ be two $T$-coalgebras. Then a relation
$\mathcal{R}\subseteq A\times B$ is said to be a bisimulation between
$(A,\delta)$ and $(B,\beta)$ if there exists a $T$-coalgebra
$(\mathcal{R},\gamma)$ such that the projection maps $\pi_{1}:\mathcal{R}\to
A$ and $\pi_{2}:\mathcal{R}\to B$ are coalgebra morphisms and satisfy the
relations $\delta\circ\pi_{1}=T(\pi_{1})\circ\gamma$,
$\beta\circ\pi_{2}=T(\pi_{2})\circ\gamma$. So, the diagram shown in Fig. 4 is
commutative.

${A}$${\mathcal{R}}$${B}$${T(A)}$${T(\mathcal{R})}$${T(B)}$$\scriptstyle{\delta}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\gamma}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\beta}$$\scriptstyle{T(\pi_{1})}$$\scriptstyle{T(\pi_{2})}$
Figure 4: Illustration of coalgebraic bisimulation

###### Definition 22

We define a functor $\mathcal{P}:\textbf{Fuzzy-Top}\to\textbf{FRM}$ as
follows:

  1. (i)

For an object $S$ in Fuzzy-top, define $\mathcal{P}(S)=\tau_{S}$;

  2. (ii)

For an arrow $\eta:S_{1}\to S_{2}$ in Fuzzy-top, define
$\mathcal{P}(\eta):\mathcal{P}(S_{2})\to\mathcal{P}(S_{1})$ by
$\mathcal{P}(\eta)(\xi)=\xi\circ\eta$, where $\xi\in\mathcal{P}(S_{2})$.

###### Note 4

Let $F$ be a frame and $(S,\tau_{S})$ be a fuzzy topological space. Then, the
frame $F$ is spatial if there is an isomorphism from $F$ to $\mathcal{P}(S)$.

Consider that S-FRM is the category of spatial frames and homomorphisms
between frames. Let $F$ denote a frame and $\textit{PT}F$ denote the
collection of frame homomorphisms $h$ from $F$ to $[0,1]$. Then the collection
$\\{\Psi(a):a\in F\\}$ is a fuzzy topology on $\textit{PT}F$, where for each
$a\in F$, $\Psi(a)$ is a membership function from $\textit{PT}F$ to $[0,1]$
which is defined by $\Psi(a)(h)=h(a)$.

######  Definition 23 ([29])

Let $\tau$ be a fuzzy topology on $S$. Assume that $\tilde{T}\in\tau$. A
membership function $\Psi(\tilde{T}):\textit{PT}\tau\to[0,1]$ can be defined
as $\Psi(\tilde{T})(h)=h(\tilde{T})$. Then, $\Psi(\tilde{T})$ is a fuzzy set
on $\textit{PT}\tau$.

The collection $\\{\Psi(\tilde{T}):\tilde{T}\in\tau\\}$ is a fuzzy topology on
$\textit{PT}\tau$.

######  Corollary 1 ([29])

Consider a fuzzy topological space $(S,\tau)$. A mapping
$f:S\to\textit{PT}\tau$ is defined by $f(s)(\tilde{\phi})=\tilde{\phi}(s)$,
where $\tilde{\phi}\in\tau$. Hence, $(S,\tau)$ becomes sober $\iff$ $f$ is
bijective.

The category of sober fuzzy topological spaces and fuzzy continuous maps is
denoted by SFuzzy-Top.

###### Definition 24

We define a functor $\textit{PT}:\textbf{FRM}\to\textbf{Fuzzy-Top}$ as
follows:

  1. (i)

For an object $F$ in FRM, define $\textit{PT}(F)=\textit{PT}F$;

  2. (ii)

For an arrow $f:F\to F^{\prime}$ in FRM, define
$\textit{PT}(f):\textit{PT}F^{\prime}\to\textit{PT}F$ by
$\textit{PT}(f)(h)=h\circ f$, where $h\in\textit{PT}F^{\prime}$.

###### Theorem 1

The category S-FRM is dually equivalent to the category SFuzzy-Top.

###### Proof.

Let $id_{1}$ and $id_{2}$ be the identity functors on S-FRM and SFuzzy-Top,
respectively. We define two natural transformations
$\zeta:id_{1}\to\mathcal{P}\circ\textit{PT}$ and
$\eta:id_{2}\to\textit{PT}\circ\mathcal{P}$. For a spatial frame $F$, we
define $\zeta_{F}:F\to\mathcal{P}\circ\textit{PT}(F)$ by
$\zeta_{F}(u)(h)=h(u)$, where $h\in\textit{PT}F$. Since $F$ is a spatial
frame, we have $\zeta_{F}$ is an isomorphism. It becomes easy to observe that
$\zeta$ is a natural transformation. As a result, $\zeta$ is a natural
isomorphism.
For an object $S$ in SFuzzy-Top, define
$\eta_{S}:S\to\textit{PT}\circ\mathcal{P}S$ by
$\eta_{S}(s)(\tilde{g})=\tilde{g}(s)$, $\forall s\in S$ and
$\tilde{g}\in\mathcal{P}(S)$. As $S$ is sober, so $\eta_{S}$ is bijective. We
observe that, for $\tilde{g}\in\mathcal{P}(S)$,
$\eta_{S}^{-1}(\Psi(\tilde{g}))(s)=\Psi(\tilde{g})(\eta_{S}(s))=\eta_{S}(s)(\tilde{g})=\tilde{g}(s)$.
Therefore, $\eta_{S}^{-1}(\Psi(\tilde{g}))=\tilde{g}$. Moreover, $\eta_{S}$ is
an open map because
$\eta_{S}(\tilde{g})(h)=\bigvee\\{\tilde{g}(s):s\in\eta_{S}^{-1}(h)\\}=h(\tilde{g})=\Psi(\tilde{g})(h)$.
Therefore, $\eta_{S}(\tilde{g})=\Psi(\tilde{g})$. Consequently, $\eta_{S}$ is
a fuzzy homeomorphism. It can be shown that $\eta$ is a natural
transformation. Hence, $\eta$ is a natural isomorphism. ∎

##  3 Coalgebraic logic

For this section, $T$ is assumed to be an arbitrary endofunctor on the
category $\mathcal{C}=\textbf{Fuzzy-Top}$. We define coalgebraic logic for
Fuzzy-Top-coalgebras. First, we introduce a notion of a predicate lifting for
the endofunctor $T$, called fuzzy-open predicate lifting.

###### Definition 25

A fuzyy-open predicate lifting is defined by a natural transformation

$\lambda:\mathcal{Q}^{n}\to\mathcal{Q}\circ T.$

The dual of $\lambda$ is denoted by $\breve{\lambda}$ and defined as
$\breve{\lambda}(\mu_{1},\mu_{2},\cdots,\mu_{n})=1-\lambda(1-\mu_{1},1-\mu_{2},\cdots,1-\mu_{n})$,
where $\mu_{i}\in\mathcal{Q}(S)$, $i=1,2,\cdots,n$ and $S$ is an object in
Fuzzy-Top.

###### Definition 26

We call the fuzzy-open predicate lifting $\lambda$ is

  1. (i)

monotone if for every object $S$ in Fuzzy-Top and
$\mu_{i},\eta_{i}\in\mathcal{Q}(S)$, $i=1,2,\cdots,n$ such that
$\mu_{1}\leq\eta_{1},\cdots,\mu_{n}\leq\eta_{n}\Rightarrow\lambda_{S}(\mu_{1},\cdots,\mu_{n})\leq\lambda_{S}(\eta_{1},\cdots,\eta_{n})$.

Let $\Sigma$ be a collection of fuzzy-open predicate liftings for $T$. Then,
the collection $\Sigma$ is said to be a fuzzy geometric modal signature for
$T$. If every member of $\Sigma$ is monotone, then the fuzzy geometric modal
signature for $T$ is said to be monotone.

###### Definition 27

A fuzzy geometric modal signature for an endofunctor $T:\textbf{Fuzzy-
Top}\to\textbf{Fuzzy-Top}$ is said to be characteristic for $T$ if for each
object $S$ in Fuzzy-Top, the collection
$\\{\lambda_{S}(\mu_{1},\cdots,\mu_{n}):\lambda\in\Sigma,\mu_{i}\in\mathcal{Q}(S)\\}$
is a sub-basis for the fuzzy topology on $TS$.

We now introduce the modal language for the fuzzy geometric modal signature
$\Sigma$.

###### Definition 28

For a given fuzzy geometric modal signature $\Sigma$ for $T:\textbf{Fuzzy-
Top}\to\textbf{Fuzzy-Top}$, we define the modal language
$\textbf{FGML}(\Sigma)$ by the grammar
$\beta::=\top\lvert
p\rvert\beta_{1}\wedge\beta_{2}\lvert\displaystyle\bigvee_{j\in
J}\beta_{j}\rvert\heartsuit^{\lambda}(\beta_{1},\beta_{2},\cdots,\beta_{n})$,
with $\lambda\in\Sigma$, $\Phi$ denotes the set of propositional variables $p$
and $J$ is an index set.

###### Definition 29

A fuzzy geometric model for the functor $T$ is defined by
$\mathcal{S}=(S,\sigma,\mathcal{V})$ in which $(S,\sigma)$ is a $T$-coalgebra,
and $\mathcal{V}:\Phi\to\mathcal{Q}(S)\subseteq[0,1]^{S}$ is a valuation
function.

###### Definition 30

A category $FMOD(T)$ is defined as follows:

  1. 1.

Every object in $FMOD(T)$ is a fuzzy geometric model for $T$;

  2. 2.

A morphism
$f:(S,\sigma_{1},\mathcal{V}_{S})\to(S^{\prime},\sigma_{2},\mathcal{V}^{\prime}_{S^{\prime}})$
in $FMOD(T)$ is a coalgebra morphism
$f:(S,\sigma_{1})\to(S^{\prime},\sigma_{2})$ which satisfies the condition:
$f^{-1}\circ\mathcal{V}^{\prime}_{S^{\prime}}=\mathcal{V}_{S}$.

###### Definition 31

The semantics of a formula $\alpha$ in FGML$(\Sigma)$ on a fuzzy geometric
model $\mathcal{S}=(S,\sigma,\mathcal{V})$ is defined as:

  1. (i)

$[[\top]]_{\mathcal{S}}(s)=1$;

  2. (ii)

$[[p]]_{\mathcal{S}}(s)=\mathcal{V}(p)(s)$;

  3. (iii)

$[[\alpha\wedge\beta]]_{\mathcal{S}}(s)=[[\alpha]]_{\mathcal{S}}(s)\wedge[[\beta]]_{\mathcal{S}}(s)$;

  4. (iv)

$[[\displaystyle\bigvee_{i\in
J}\alpha_{i}]]_{\mathcal{S}}(s)=Sup\\{[[\alpha_{i}]]_{\mathcal{S}}(s)\\}$;

  5. (v)

$[[\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})]]_{\mathcal{S}}(s)=\lambda_{S}([[\alpha_{1}]]_{\mathcal{S}},[[\alpha_{2}]]_{\mathcal{S}},\cdots,[[\alpha_{n}]]_{\mathcal{S}})\circ\sigma(s)$.

Grade of a formula $\alpha$ in FGML$(\Sigma)$ satisfied by a state $s$ in $S$
is denoted by $gr(s\models\alpha)$ and defined by
$gr(s\models\alpha)=[[\alpha]]_{\mathcal{S}}(s)$. Two states $s$ and $t$ in
$S$ are modally equivalent if $gr(s\models\alpha)=gr(t\models\alpha)$, for all
formulas $\alpha$ in $\textbf{FGML}(\Sigma)$. We express it by the notation
$s\equiv_{\Sigma}t$.

###### Definition 32

Let $\mathcal{B}=(B,\sigma_{1},\mathcal{V}_{B})$ and
$\mathcal{B}^{\prime}=(B^{\prime},\sigma_{2},\mathcal{V}_{B^{\prime}})$ be
fuzzy geometric models for $T$. States $b\in B$ and $b^{\prime}\in B^{\prime}$
are said to be behaviourally equivalent in $FMOD(T)$ if there exists an object
$\mathcal{C}=(C,\gamma,\mathcal{V}_{C})$ in $FMOD(T)$ and morphisms
$g:\mathcal{B}\to\mathcal{C}$ and $h:\mathcal{B}^{\prime}\to\mathcal{C}$ in
$FMOD(T)$ such that $g(b)=h(b^{\prime})$.

In the following Proposition 1, we shall show that fuzzy geometric model
morphisms preserve truth degrees.

###### Proposition 1

Assume that
$f:\mathcal{S}=(S,\sigma_{1},\mathcal{V}_{S})\to\mathcal{K}=(K,\sigma_{2},\mathcal{V}_{K})$
is a morphism between fuzzy geometric models. Then
$gr(s\models\alpha)=gr(f(s)\models\alpha)$,
$\forall\alpha\in\textbf{FGML}(\Sigma)$ and $s\in S$.

###### Proof.

We have to show that
$[[\alpha]]_{\mathcal{S}}(s)=[[\alpha]]_{\mathcal{K}}(f(s))$ for all formulas
$\alpha$.
If $p$ is a propositional variable then by using the definition of fuzzy
geometric model morphism, we can show that
$[[p]]_{\mathcal{S}}(s)=[[p]]_{\mathcal{K}}(f(s))$ i.e., $gr(s\models
p)=gr(f(s)\models p)$.
It can be easily shown that $gr(s\models\displaystyle\bigvee_{j\in
J}\alpha_{j})=gr(f(s)\models\displaystyle\bigvee_{j\in J}\alpha_{j})$ and
$gr(s\models\alpha_{1}\wedge\alpha_{2})=gr(f(s)\models\alpha_{1}\wedge\alpha_{2})$.
The only part we have to show is that
$gr(s\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})=gr(f(s)\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))$.
Since $f$ is the coalgebra morphism, henceforth
$Tf\circ\sigma_{1}=\sigma_{2}\circ f$. So, the diagram shown in Fig. 5
commutes.

${S}$${K}$${TS}$${TK}$$\scriptstyle{f}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{Tf}$
Figure 5: Coalgebra morphism

Now, the following diagram also commutes which is obtained by applying the
functor $\mathcal{Q}$ to the above diagram (Fig. 5).

${\mathcal{Q}S}$${\mathcal{Q}K}$${\mathcal{Q}(TS)}$${\mathcal{Q}(TK)}$$\scriptstyle{\mathcal{Q}f=f^{-1}}$$\scriptstyle{\mathcal{Q}\sigma_{1}=\sigma_{1}^{-1}}$$\scriptstyle{\mathcal{Q}(Tf)=(Tf)^{-1}}$$\scriptstyle{\mathcal{Q}\sigma_{2}=\sigma_{2}^{-1}}$

We observe that

$\displaystyle[[\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})]]_{\mathcal{S}}(s)$
$\displaystyle=\lambda_{S}([[\alpha_{1}]]_{\mathcal{S}},\cdots,[[\alpha_{n}]]_{\mathcal{S}})\circ\sigma_{1}(s)$
$\displaystyle=\lambda_{S}([[\alpha_{1}]]_{\mathcal{K}}\circ
f,\cdots,[[\alpha_{n}]]_{\mathcal{K}}\circ f)\circ\sigma_{1}(s)\text{ [ as
$f^{-1}([[\alpha]]_{\mathcal{K}})=[[\alpha]]_{\mathcal{S}}$ ] }$
$\displaystyle=\lambda_{K}([[\alpha_{1}]]_{\mathcal{K}},\cdots,[[\alpha_{n}]]_{\mathcal{K}})\circ
Tf\circ\sigma_{1}(s)\text{ [ by naturality of $\lambda$ ] }$
$\displaystyle=\lambda_{K}([[\alpha_{1}]]_{\mathcal{K}},\cdots,[[\alpha_{n}]]_{\mathcal{K}})\circ\sigma_{2}\circ
f(s)$
$\displaystyle=[[\heartsuit^{\lambda}(\alpha_{1},\cdots,\alpha_{n})]]_{\mathcal{K}}(f(s))$

Therefore,
$gr(s\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))=gr(f(s)\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))$.
∎

Applying Proposition 1, we obtain the following result.

###### Proposition 2

Behaviourally equivalent states are modally equivalent.

###### Proof.

Let $\mathcal{B}=(B,\sigma_{1},\mathcal{V}_{B})$ and
$\mathcal{B}^{\prime}=(B^{\prime},\sigma_{2},\mathcal{V}_{B^{\prime}})$ be
fuzzy geometric models for $T$. Consider $b\in B$ and $b^{\prime}\in
B^{\prime}$ are two states. Suppose, the states $b$ and $b^{\prime}$ are
behaviourally equivalent. We shall show that they are modally equivalent.
Since $b$ and $b^{\prime}$ are behaviourally equivalent in $FMOD(T)$, so there
exists an object $\mathcal{C}=(C,\gamma,\mathcal{V}_{C})$ in $FMOD(T)$ and
morphisms $g:\mathcal{B}\to\mathcal{C}$ and
$h:\mathcal{B}^{\prime}\to\mathcal{C}$ in $FMOD(T)$ such that
$g(b)=h(b^{\prime})$. Now, by Proposition 1, we have
$gr(b\models\alpha)=gr(g(b)\models\alpha)$ and
$gr(b^{\prime}\models\alpha)=gr(h(b^{\prime})\models\alpha)$,
$\forall\alpha\in\textbf{FGML}(\Sigma)$. As $g(b)=h(b^{\prime})$, hence
$gr(b\models\alpha)=gr(b^{\prime}\models\alpha)$,
$\forall\alpha\in\textbf{FGML}(\Sigma)$. Therefore, the states $b$ and
$b^{\prime}$ are modally equivalent. ∎

##  4 Final model

In this section, we assume that $T$ is an endofunctor on SFuzzy-Top, the
category of sober fuzzy topological spaces, and consider a characteristic
fuzzy geometric modal signature $\Sigma$ for the endofunctor $T$. We shall
create a final model in $FMOD(T)$ for the endofunctor $T$. Let
$\mathcal{B}=(B,\gamma,\mathcal{V}_{B})$ be a fuzzy geometric model for $T$.

###### Definition 33

Any two formulas $\alpha$ and $\beta$ are equivalent in $\text{FMOD}(T)$ iff
$gr(b\models\alpha)=gr(b\models\beta)$, $\forall b\in B$. Let
$\alpha\equiv\beta$ denote the formulas $\alpha$ and $\beta$ are equivalent.

Let $[\alpha]$ denote the equivalence class of a formula
$\alpha\in\textbf{FGML}(\Sigma)$. Let $\mathcal{E}$ be the collection of
equivalence classes of formulas in $\textbf{FGML}(\Sigma)$. We define
$gr(b\models[\alpha])=gr(b\models\alpha)$, for any $b\in B$. We shall show
that $\mathcal{E}$ is a frame.

###### Proposition 3

$\mathcal{E}$ is a frame.

###### Proof.

The order relation on $\mathcal{E}$ is defined as: $[\alpha]\leq[\beta]\iff
gr(b\models\alpha)\leq gr(b\models\beta)$, $\forall b\in B$. As
$gr(b\models\alpha)=gr(b\models\alpha)$, the order relation $\leq$ is
reflexive. It is easy to see that if $[\alpha_{1}]\leq[\beta_{1}]$ and
$[\beta_{1}]\leq[\beta_{3}]$ then $[\alpha_{1}]\leq[\beta_{3}]$. Thus the
order relation $\leq$ is transitive. Now, if $[\alpha]\leq[\beta]$ and
$[\beta]\leq[\alpha]$ then by the defined order relation we have
$gr(b\models\alpha)=gr(b\models\beta)$, $\forall b\in B$. Hence,
$\alpha\equiv\beta$. As a result, $[\alpha]=[\beta]$. So the order relation
$\leq$ is antisymmetric. Therefore, $\mathcal{E}$ is a poset with this order
relation. As
$gr(b\models\alpha\wedge\beta)=[[\alpha\wedge\beta]]_{\mathcal{B}}(b)=[[\alpha]]_{\mathcal{B}}(b)\wedge[[\beta]]_{\mathcal{B}}(b)=gr(b\models\alpha)\wedge
gr(b\models\beta)$, hence $[\alpha\wedge\beta]\in\mathcal{E}$. As a result,
$[\alpha]\wedge[\beta]\in\mathcal{E}$. Similarly, arbitrary join exists in
$\mathcal{E}$. We observe that, $[\alpha]\wedge\displaystyle\bigvee_{j\in
J}[\beta_{j}]=[\alpha]\wedge[\bigvee_{j\in
J}\beta_{j}]=[\alpha\wedge\bigvee_{j\in J}\beta_{j}]$. Now, we have
$gr(b\models\alpha\wedge\bigvee_{j\in J}\beta_{j})=gr(b\models\alpha)\wedge
gr(b\models\bigvee_{j\in
J}\beta_{j})=[[\alpha]]_{\mathcal{B}}(b)\wedge[[\displaystyle\bigvee_{j\in
J}\beta_{j}]]_{\mathcal{B}}(b)=[[\alpha]]_{\mathcal{B}}(b)\wedge Sup_{j\in
J}\\{[[\beta_{j}]]_{\mathcal{B}}(b)\\}=Sup_{j\in
J}\\{[[\alpha]]_{\mathcal{B}}(b)\wedge[[\beta_{j}]]_{\mathcal{B}}(b)\\}=[[\displaystyle\bigvee_{j\in
J}(\alpha\wedge\beta_{j})]](b)=gr(b\models\displaystyle\bigvee_{j\in
J}(\alpha\wedge\beta_{j}))$, $\forall b\in B$. Consequently,
$\displaystyle[\alpha\wedge\bigvee_{j\in J}\beta_{j}]=[\bigvee_{j\in
J}(\alpha\wedge\beta_{j})]$. Henceforth,
$\displaystyle[\alpha]\wedge\bigvee_{j\in J}[\beta_{j}]=[\bigvee_{j\in
J}(\alpha\wedge\beta_{j})]=\bigvee_{j\in
J}[\alpha\wedge\beta_{j}]=\bigvee_{j\in J}([\alpha]\wedge[\beta_{j}])$.
Therefore, $\mathcal{E}$ is a frame. ∎

###### Definition 34

Let $\mathcal{F}=\it{PT}(\mathcal{E})$. A map $\tilde{f}:B\to\mathcal{F}$ is
defined by $\tilde{f}(b)=h_{b}$, where $h_{b}$ is a frame homomorphism from
$\mathcal{E}$ to $[0,1]$ defined by $h_{b}([\alpha])=gr(b\models\alpha)$.

###### Note 5

The mapping $\tilde{f}:B\to\mathcal{F}$ is fuzzy continuous. Let
$[\alpha]\in\mathcal{E}$. Then we show that
$\tilde{f}^{-1}(\Psi([\alpha]))=[[\alpha]]_{\mathcal{B}}$ by the following:

$\tilde{f}^{-1}(\Psi([\alpha]))(b)=\Psi([\alpha])\tilde{f}(b)=\tilde{f}(b)([\alpha])=gr(b\models\alpha)=[[\alpha]]_{\mathcal{B}}(b).$

Hence, $\forall\alpha\in\textbf{FGML}(\Sigma)$,
$\tilde{f}^{-1}(\Psi([\alpha]))$ is a fuzzy open set on $B$. Therefore,
$\tilde{f}$ is a fuzzy continuous map.

Let $\mathcal{G}=\mathcal{P}\circ T\circ\it{PT}$. Then
$\mathcal{G}:\textbf{FRM}\to\textbf{FRM}$ is a functor. Since the category
S-FRM of spatial frames is equivalent to the opposite category of SFuzzy-Top,
the endofunctor defined on the category S-FRM is a restriction of
$\mathcal{G}$. As $\Sigma$ is characteristic, so the collection
$\\{\lambda_{B}(\widehat{[\alpha_{1}]},\cdots,\widehat{[\alpha_{n}]}):\lambda\in\Sigma,\alpha_{i}\in\textbf{FGML}(\Sigma),\widehat{[\alpha_{i}]}\in\mathcal{Q}(PT\mathcal{E}),i=1,\cdots,n\\}$
generates the frame $\mathcal{G}(\mathcal{E})$. So, an assignment can be
defined on the generators of $\mathcal{G}(\mathcal{E})$, and by Remark 2, it
can be extended to a frame homomorphism from $\mathcal{G}(\mathcal{E})$ to
$\mathcal{E}$.

###### Definition 35

Define a morphism $\xi:\mathcal{G}(\mathcal{E})\to\mathcal{E}$ in
$\mathcal{FRM}$ by
$\xi(\lambda_{\mathcal{F}}(\widehat{[\alpha_{1}]},\widehat{[\alpha_{2}]},\cdots,\widehat{[\alpha_{n}]}))=[\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})]$.

The well-definedness of the morphism $\xi$ is shown by Lemma 1.

###### Lemma 1

Suppose $\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\lambda_{\mathcal{F}}^{i,j}(\widehat{[\alpha_{1}]}^{i,j},\widehat{[\alpha_{2}]}^{i,j},\cdots,\widehat{[\alpha_{n_{i,j}}]}^{i,j}))=\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\lambda_{\mathcal{F}}^{r,s}(\widehat{[\alpha_{1}^{\prime}]}^{r,s},\widehat{[\alpha_{2}^{\prime}]}^{r,s},\cdots,\widehat{[\alpha_{n_{r,s}}^{\prime}]}^{r,s}))$,
where $\Lambda$ and $\mathcal{I}$ are the arbitrary index sets, and $K_{i}$,
$\mathcal{J}_{r}$ are the finite index sets. Then, formulas
$\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\heartsuit^{\lambda^{i,j}}(\alpha_{1}^{i,j},\alpha_{2}^{i,j},\cdots,\alpha_{n_{i,j}}^{i,j}))$
and
$\displaystyle\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\heartsuit^{\lambda^{r,s}}(\alpha_{1}^{\prime
r,s},\alpha_{2}^{\prime r,s},\cdots,\alpha_{n_{r,s}}^{\prime r,s}))$ are
equivalent in $FMOD(T)$.

###### Proof.

We shall show that, for an object $\mathcal{B}=(B,\gamma,\mathcal{V}_{B})$ in
$FMOD(T)$, $gr(b\models\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\heartsuit^{\lambda^{i,j}}(\alpha_{1}^{i,j},\alpha_{2}^{i,j},\cdots,\alpha_{n_{i,j}}^{i,j})))=gr(b\models\displaystyle\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\heartsuit^{\lambda^{r,s}}(\alpha_{1}^{\prime
r,s},\alpha_{2}^{\prime r,s},\cdots,\alpha_{n_{r,s}}^{\prime r,s})))$,
$\forall b\in B$.
Now we observe that,

$\displaystyle\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\lambda_{B}^{i,j}([[\alpha_{1}^{i,j}]]_{\mathcal{B}},[[\alpha_{2}^{i,j}]]_{\mathcal{B}},\cdots,[[\alpha_{n_{i,j}}^{i,j}]]_{\mathcal{B}}))$
$\displaystyle=\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\lambda_{B}^{i,j}(\tilde{f}^{-1}(\Psi([\alpha_{1}^{i,j}])),\tilde{f}^{-1}(\Psi([\alpha_{2}^{i,j}])),\cdots,\tilde{f}^{-1}(\Psi([\alpha_{n_{i,j}}^{i,j}]))))\text{
[By Note \ref{nt-1}]}$
$\displaystyle=\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}(T\tilde{f})^{-1}(\lambda_{\mathcal{F}}^{i,j}(\Psi([\alpha_{1}^{i,j}]),\Psi([\alpha_{2}^{i,j}]),\cdots,\Psi([\alpha_{n_{i,j}}^{i,j}])))\text{
[ By naturality of $\lambda$ ]}$
$\displaystyle=(T\tilde{f})^{-1}(\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\lambda_{\mathcal{F}}^{i,j}(\Psi([\alpha_{1}^{i,j}]),\Psi([\alpha_{2}^{i,j}]),\cdots,\Psi([\alpha_{n_{i,j}}^{i,j}]))))\text{
[ By Definition \ref{IVP} ]}$
$\displaystyle=(T\tilde{f})^{-1}(\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\lambda_{\mathcal{F}}^{r,s}(\Psi([\alpha_{1}^{\prime
r,s}]),\Psi([\alpha_{2}^{\prime r,s}]),\cdots,\Psi([\alpha_{n_{r,s}}^{\prime
r,s}]))))\text{ [ By the given hypothesis]}$
$\displaystyle=\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}(T\tilde{f})^{-1}(\lambda_{\mathcal{F}}^{r,s}(\Psi([\alpha_{1}^{\prime
r,s}]),\Psi([\alpha_{2}^{\prime r,s}]),\cdots,\Psi([\alpha_{n_{r,s}}^{\prime
r,s}]))))\text{ [ By Definition \ref{IVP}]}$
$\displaystyle=\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\lambda_{B}^{r,s}(\tilde{f}^{-1}(\Psi([\alpha_{1}^{\prime
r,s}])),\tilde{f}^{-1}(\Psi([\alpha_{2}^{\prime
r,s}])),\cdots,\tilde{f}^{-1}(\Psi([\alpha_{n_{r,s}}^{\prime r,s}]))))\text{ [
By naturality of $\lambda$ ]}$
$\displaystyle=\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\lambda_{B}^{r,s}([[\alpha_{1}^{\prime
r,s}]]_{\mathcal{B}},[[\alpha_{2}^{\prime
r,s}]]_{\mathcal{B}},\cdots,[[\alpha_{n_{r,s}}^{\prime r,s}]]_{\mathcal{B}}))$

Therefore, for a fuzzy geometric model $\mathcal{B}$, we have
$gr(b\models\displaystyle\bigvee_{i\in\Lambda}(\bigwedge_{j\in
K_{i}}\heartsuit^{\lambda^{i,j}}(\alpha_{1}^{i,j},\alpha_{2}^{i,j},\cdots,\alpha_{n_{i,j}}^{i,j})))=gr(b\models\displaystyle\bigvee_{r\in\mathcal{I}}(\bigwedge_{s\in\mathcal{J}_{r}}\heartsuit^{\lambda^{r,s}}(\alpha_{1}^{\prime
r,s},\alpha_{2}^{\prime r,s},\cdots,\alpha_{n_{r,s}}^{\prime r,s})))$,
$\forall b\in B$. ∎

So, $(\mathcal{E},\xi)$ is a $\mathcal{G}$-algebra. Now we construct a
$T$-coalgebra structure on $\mathcal{F}=\textit{PT}\mathcal{E}$.

###### Definition 36

Consider a morphism
$\phi=\eta^{-1}_{T\mathcal{F}}\circ\it{PT}(\xi):\mathcal{F}\to T\mathcal{F}$,
where the morphism
$\it{PT}(\xi):\it{PT}(\mathcal{E})\to\it{PT}(\mathcal{G}(\mathcal{E}))$ is
defined by
$\it{PT}(\xi)(h)(\lambda_{\mathcal{F}}(\widehat{[\alpha_{1}]},\widehat{[\alpha_{2}]},\cdots,\widehat{[\alpha_{n}]}))=gr([\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})]\in
h)$, where $h\in\textit{PT}(\mathcal{E})=\textit{PT}\mathcal{E}$, and the
morphism
$\eta_{T\mathcal{F}}:T\mathcal{F}\to\it{PT}(\mathcal{G}(\mathcal{E}))$ is
defined by
$\eta_{T\mathcal{F}}(h^{*})(\lambda_{\mathcal{F}}(\widehat{[\alpha_{1}]},\widehat{[\alpha_{2}]},\cdots,\widehat{[\alpha_{n}]}))=\lambda_{\mathcal{F}}(\widehat{[\alpha_{1}]},\widehat{[\alpha_{2}]},\cdots,\widehat{[\alpha_{n}]})(h^{*})$,
where $h^{*}\in T\mathcal{F}$.

###### Note 6

Since $T\mathcal{F}$ is a sober fuzzy topological space, so by Theorem 1,
$\eta_{T\mathcal{F}}$ is an isomorphism. Consequently, the morphism $\phi$ is
well-defined.

###### Definition 37

The triple $(\mathcal{F},\phi,\mathcal{V}_{\mathcal{F}})$ is an object in
$FMOD(T)$, where $(\mathcal{F},\phi)$ is a $T$-coalgebra and the valuation
function $\mathcal{V}_{\mathcal{F}}:\Phi\to\mathcal{Q}(\mathcal{F})$ is
defined by
$\mathcal{V}_{\mathcal{F}}(p)(\tilde{g})=gr(p\in\tilde{g})=\tilde{g}(p)$,
where $p\in\Phi$ and $\tilde{g}\in\mathcal{Q}(\mathcal{F})$.

###### Proposition 4

The mapping
$\tilde{f}:\mathcal{B}\to(\mathcal{F},\phi,\mathcal{V}_{\mathcal{F}})$ is a
morphism in $FMOD(T)$.

###### Proof.

We are to show that $\tilde{f}$ is a coalgebra morphism from $\mathcal{B}$ to
$\mathcal{F}$, and
$\tilde{f}^{-1}\circ\mathcal{V}_{\mathcal{F}}=\mathcal{V}_{B}$. It is observed
that for every propositional variable $p$,

$\begin{split}\tilde{f}^{-1}\circ\mathcal{V}_{\mathcal{F}}(p)(b)&=\tilde{f}^{-1}(\mathcal{V}_{\mathcal{F}}(p))(b)\\\
&=\mathcal{V}_{\mathcal{F}}(p)(\tilde{f}(b))\\\ &=gr(p\in\tilde{f}(b))\\\
&=\tilde{f}(b)(p)\\\ &=gr(b\models p)\text{ [ By Definition \ref{p14} ] }\\\
&=\mathcal{V}_{B}(p)(b).\end{split}$

Henceforth, $\tilde{f}^{-1}\circ\mathcal{V}_{\mathcal{F}}=\mathcal{V}_{B}$. To
prove $\tilde{f}$ is a $T$-coalgebra morphism, we show that
$T\tilde{f}\circ\gamma=\phi\circ\tilde{f}$ i.e., the diagram shown in Fig.6
commutes.

${B}$${\mathcal{F}}$${TB}$${T\mathcal{F}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\gamma}$$\scriptstyle{\phi}$$\scriptstyle{T\tilde{f}}$
Figure 6: Illustration of $T$-coalgebra morphism

Now we observe that,

$\displaystyle
gr(T\tilde{f}\circ\gamma(b)\in\lambda_{\mathcal{F}}(\Psi([\alpha_{1}]),\Psi([\alpha_{2}]),\cdots,\Psi([\alpha_{n}])))$
$\displaystyle=gr(\gamma(b)\in(T\tilde{f})^{-1}\circ\lambda_{\mathcal{F}}(\Psi([\alpha_{1}]),\Psi([\alpha_{2}]),\cdots,\Psi([\alpha_{n}])))$
$\displaystyle=gr(\gamma(b)\in\lambda_{B}([[\alpha_{1}]]_{\mathcal{B}},[[\alpha_{2}]]_{\mathcal{B}},\cdots,[[\alpha_{n}]]_{\mathcal{B}}))\text{
[Since $\lambda$ is the natural transformation ]}$
$\displaystyle=\lambda_{B}([[\alpha_{1}]]_{\mathcal{B}},[[\alpha_{2}]]_{\mathcal{B}},\cdots,[[\alpha_{n}]]_{\mathcal{B}})\circ\gamma(b)$
$\displaystyle=[[\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})]]_{\mathcal{B}}(b)$
$\displaystyle=gr(b\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))$
$\displaystyle=\tilde{f}(b)([\heartsuit^{\lambda}([\alpha_{1}],[\alpha_{2}],\cdots,[\alpha_{n}])])\text{
[By Definition \ref{p14} ] }$
$\displaystyle=gr(\phi\circ\tilde{f}(b)\in\lambda_{\mathcal{F}}(\Psi([\alpha_{1}]),\Psi([\alpha_{2}]),\cdots,\Psi([\alpha_{n}]))\text{
[ By Definition \ref{16} ] }$

As $T\mathcal{F}$ is a sober fuzzy topological space, hence it is a
$T_{0}$-space (Kolmogorov space). Therefore, we have
$T\tilde{f}\circ\gamma=\phi\circ\tilde{f}$. ∎

###### Theorem 2

The fuzzy geometric model
$\mathfrak{F}=(\mathcal{F},\phi,\mathcal{V}_{\mathcal{F}})$ in $FMOD(T)$ is a
final model in $FMOD(T)$.

###### Proof.

We prove it by showing that for an object
$\mathcal{B}=(B,\gamma,\mathcal{V}_{B})$ in $FMOD(T)$, a unique $T$-coalgebra
morphism exists from $\mathcal{B}$ to $\mathfrak{F}$. Following the
Proposition 4, a $T$-coalgebra morphism $\tilde{f}:\mathcal{B}\to\mathfrak{F}$
exists. The only part that remains to be proven here is that $\tilde{f}$ is
unique. Consider a morphism $f^{*}:\mathcal{B}\to\mathfrak{F}$ in $FMOD(T)$.
By Proposition 1, we have $gr(b\models\alpha)=gr(f^{*}(b)\models[\alpha])$.
Now
$gr(\tilde{f}(b)\models[\alpha])=\tilde{f}(b)[\alpha]=gr(b\models\alpha)=gr(f^{*}(b)\models[\alpha])$.
Consequently, $\tilde{f}(b)=f^{*}(b)$. Therefore, $\mathfrak{F}$ is final in
$FMOD(T)$. ∎

By Theorem 2, we derive the following result.

###### Theorem 3

Modal equivalence implies behavioural equivalence.

###### Proof.

Let $\mathcal{B}=(B,\gamma,\mathcal{V}_{B})$ and
$\mathcal{B}_{1}=(B_{1},\gamma_{1},\mathcal{V}_{B_{1}})$ be fuzzy geometric
models for $T$. Let $b\in B$ and $b_{1}\in B_{1}$ be states. If $b$ and
$b_{1}$ are modally equivalent then we have
$gr(b\models\alpha)=gr(b_{1}\models\alpha)$, for all formulas $\alpha$. By
Proposition 4, there exist morphisms $\tilde{f}:\mathcal{B}\to\mathfrak{F}$
and $\tilde{f}_{1}:\mathcal{B}_{1}\to\mathfrak{F}$ in $FMOD(T)$. Using
Proposition 1, we have
$gr(\tilde{f}(b)\models[\alpha])=gr(b\models\alpha)=gr(b_{1}\models\alpha)=gr(\tilde{f}_{1}(b_{1})\models[\alpha])$,
for all formulas $\alpha$. Therefore, $\tilde{f}(b)=\tilde{f}_{1}(b_{1})$.
Hence, $b$ and $b_{1}$ are behaviourally equivalent. ∎

###### Remark 3

The converse of the statement mentioned in Theorem 3 is true by Proposition 2.
Thus, modal equivalence and behavioural equivalence coincide.

##  5 Bisimulations for Fuzzy geometric models

The aim of this section is to develop bisimulations for fuzzy geometric models
for an endofunctor $T$, where $T$ is defined on Fuzzy-Top.

######  Definition 38 ([33])

Consider that $F$ and $F^{\prime}$ are any two sets, and $R$ is a relation
between $F$ and $F^{\prime}$. Then, for a subset $E$ of $F$,
$R[E]=\\{d^{\prime}\in F^{\prime}:\exists\text{ e}\in E,eRd^{\prime}\\}$ and
for a subset $E^{\prime}$ of $F^{\prime}$ $R^{-1}[E^{\prime}]=\\{d\in
F:\exists\text{ e}^{\prime}\in E^{\prime},dRe^{\prime}\\}$.

Let $\mu$ be a fuzzy set on $F$. Then a fuzzy set $R[\mu]$ on $F^{\prime}$ can
be defined by $R[\mu](d^{\prime})=\displaystyle\bigvee_{d\in
F}\\{\mu(d):dRd^{\prime}\\}$. For a fuzzy set $\eta$ on $F^{\prime}$, we
define an inverse image of $\eta$ under the relation $R$ by
$R^{-1}[\eta](d)=\displaystyle\bigvee_{d^{\prime}\in
F^{\prime}}\\{\eta(d^{\prime}):dRd^{\prime}\\}$. It is clear that
$R^{-1}[\eta]$ is a fuzzy set on $F$.
We define the Aczel-Mendler bisimulation between fuzzy geometric models for
$T$.

###### Definition 39

Let $\mathcal{B}_{1}=(B_{1},\gamma_{1},\mathcal{V}_{B_{1}})$ and
$\mathcal{B}_{2}=(B_{2},\gamma_{2},\mathcal{V}_{B_{2}})$ be two fuzzy
geometric models for $T$. Then, a relation $\mathcal{R}\subseteq B_{1}\times
B_{2}$ is said to be an Aczel-Mendler bisimulation between $\mathcal{B}_{1}$
and $\mathcal{B}_{2}$ if for each $(b_{1},b_{2})\in\mathcal{R}$ and
$p\in\Phi$, $\mathcal{V}_{B_{1}}(p)(b_{1})=\mathcal{V}_{B_{2}}(p)(b_{2})$,
i.e., $gr(b_{1}\models p)=gr(b_{2}\models p)$ and there exists a coalgebra
morphism $\gamma^{*}:\mathcal{R}\to T\mathcal{R}$ for which the projection
maps $\pi_{1}:\mathcal{R}\to B_{1}$ and $\pi_{2}:\mathcal{R}\to B_{2}$ are
coalgebra morphisms and satisfy the relations
$\gamma_{1}\circ\pi_{1}=T(\pi_{1})\circ\gamma^{*}$,
$\gamma_{2}\circ\pi_{2}=T(\pi_{2})\circ\gamma^{*}$, i.e. the diagram shown in
Fig. 7 commutes:

${B_{1}}$${\mathcal{R}}$${B_{2}}$${TB_{1}}$${T\mathcal{R}}$${TB_{2}}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{T\pi_{1}}$$\scriptstyle{T\pi_{2}}$
Figure 7: Illustration of Aczel-Mendler bisimulation between fuzzy geometric
models

We now introduce a notion of $\Sigma$-bisimulation between fuzzy geometric
models for $T$, adapting the “$\Lambda$-bisimulation” concepts discussed in
[7, 8]. First, we introduce the notion of coherent pairs.

###### Definition 40

Assume that $\mathcal{R}$ is a relation between $B$ and $B^{\prime}$. Let
$\pi_{1}:\mathcal{R}\to B$ and $\pi_{2}:\mathcal{R}\to B^{\prime}$ be
projection maps. Then, a pair $(\tilde{r_{1}},\tilde{r_{2}})$, where
$\tilde{r_{1}}$ and $\tilde{r_{2}}$ are respectively the fuzzy sets on $B$ and
$B^{\prime}$, is called $\mathcal{R}$-coherent if
$\mathcal{R}[\tilde{r_{1}}]\leq\tilde{r_{2}}$ and
$\mathcal{R}^{-1}[\tilde{r_{2}}]\leq\tilde{r_{1}}$.

###### Definition 41

Let $\mathcal{B}_{1}=(B_{1},\gamma_{1},\mathcal{V}_{B_{1}})$ and
$\mathcal{B}_{2}=(B_{2},\gamma_{2},\mathcal{V}_{B_{2}})$ be two fuzzy
geometric models for $T$. A relation $\mathcal{R}\subseteq B_{1}\times B_{2}$
is said to be a $\Sigma$-bisimulation between $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ if for all $(b_{1},b_{2})\in\mathcal{R}$, $p\in\Phi$ and
each pair of fuzzy open sets
$(\mu_{i},\eta_{i})\in\mathcal{Q}(B_{1})\times\mathcal{Q}(B_{2})$ such that
$\mathcal{R}[\mu_{i}]\leq\eta_{i}$ and
$\mathcal{R}^{-1}[\eta_{i}]\leq\mu_{i}$, we have :

  1. (i)

$gr(b_{1}\models p)=gr(b_{2}\models p)$,  and

  2. (ii)

$gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n}))=gr(\gamma_{2}(b_{2})\in\lambda_{B_{2}}(\eta_{1},\eta_{2},\cdots,\eta_{n}))$.

Two states $b_{1}\in B_{1}$ and $b_{2}\in B_{2}$ are said to be
$\Sigma$-bisimilar if there exists a $\Sigma$-bisimulation $\mathcal{R}$ such
that $(b_{1},b_{2})\in\mathcal{R}$.

###### Lemma 2

Consider $\mathcal{R}\subseteq B_{1}\times B_{2}$ is a relation that is
equipped with the fuzzy subspace topology. Let $\pi_{1}:\mathcal{R}\to B_{1}$
and $\pi_{2}:\mathcal{R}\to B_{2}$ be projection maps. Then, a pair of fuzzy
open sets $(\mu,\eta)\in\mathcal{Q}(B_{1})\times\mathcal{Q}(B_{2})$ is
$\mathcal{R}$-coherent $\iff$ $\pi_{1}^{-1}(\mu)=\pi_{2}^{-1}(\eta)$.

###### Proof.

Suppose the pair of fuzzy open sets $(\mu,\eta)$ is $\mathcal{R}$-coherent. We
shall show that $\pi_{1}^{-1}(\mu)=\pi_{2}^{-1}(\eta)$. First, we show that
$\pi_{1}^{-1}(\mu)$ is a fuzzy subset of $\pi_{2}^{-1}(\eta)$, i.e.
$\pi_{1}^{-1}(\mu)\leq\pi_{2}^{-1}(\eta)$. We notice that
$\pi_{2}(\pi_{1}^{-1}(\mu))$ and $\mathcal{R}[\mu]$ are both fuzzy sets on
$B_{2}$. It is simple to demonstrate that
$\pi_{2}(\pi_{1}^{-1}(\mu))=\mathcal{R}[\mu]$.
Now,

$\begin{split}\pi^{-1}(\mu)&\leq\pi_{2}^{-1}(\pi_{2}(\pi_{1}^{-1}(\mu)))\\\
&=\pi_{2}^{-1}(\mathcal{R}[\mu])\text{ [As
$\pi_{2}(\pi_{1}^{-1}(\mu))=\mathcal{R}[\mu]$ ]}\\\
&\leq\pi_{2}^{-1}(\eta)\text{ [ As $\mathcal{R}[\mu]\leq\eta$ ]}\end{split}$

Similarly, we can show that $\pi_{2}^{-1}(\eta)\leq\pi_{1}^{-1}(\mu)$. It can
be easily proved that if $\pi_{1}^{-1}(\mu)=\pi_{2}^{-1}(\eta)$ then the pair
$(\mu,\eta)$ is $\mathcal{R}$-coherent. ∎

Now, we show that $\Sigma$-bisimilar states are modally equivalent.

###### Corollary 2

Assume that $T$ is an endofunctor on Fuzzy-Top. Then $\Sigma$-bisimilarity
implies modal equivalence.

###### Proof.

Let $\mathcal{R}$ be a $\Sigma$-bisimulation between fuzzy geometric models
$\mathcal{B}_{1}=(B_{1},\gamma_{1},\mathcal{V}_{B_{1}})$ and
$\mathcal{B}_{2}=(B_{2},\gamma_{2},\mathcal{V}_{B_{2}})$. Let $b_{1}\in
B_{1}\text{ and }b_{2}\in B_{2}$ be two states. Suppose
$b_{1}\mathcal{R}b_{2}$. We shall show that
$gr(b_{1}\models\alpha)=gr(b_{2}\models\alpha)$,
$\forall\alpha\in\textbf{FGML}(\Sigma)$. If $p$ is a propositional variable,
then it follows from the definition of $\Sigma$-bisimulation that
$gr(b_{1}\models p)=gr(b_{2}\models p)$. It can be easily shown that
$gr(b_{1}\models\alpha_{1}\wedge\alpha_{2})=gr(b_{2}\models\alpha_{1}\wedge\alpha_{2})$
and $gr(b_{1}\models\displaystyle\bigvee_{j\in
J}\alpha_{j})=gr(b_{2}\models\displaystyle\bigvee_{j\in J}\alpha_{j})$, $J$ is
an index set. Now,
$gr(b_{1}\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))=gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}([[\alpha_{1}]]_{\mathcal{B}_{1}},[[\alpha_{2}]]_{\mathcal{B}_{1}},\cdots,[[\alpha_{n}]]_{\mathcal{B}_{1}}))$.
By induction principle, we can show that, for each $i=1,2,\cdots,n$,
$\mathcal{R}[[[\alpha_{i}]]_{\mathcal{B}_{1}}]\leq[[\alpha_{i}]]_{\mathcal{B}_{2}}$
and
$\mathcal{R}^{-1}[[[\alpha_{i}]]_{\mathcal{B}_{2}}]\leq[[\alpha_{i}]]_{\mathcal{B}_{1}}$.
As $\mathcal{R}$ is a $\Sigma$-bisimulation, we have
$gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}([[\alpha_{1}]]_{\mathcal{B}_{1}},[[\alpha_{2}]]_{\mathcal{B}_{1}},\cdots,[[\alpha_{n}]]_{\mathcal{B}_{1}}))=gr(\gamma_{2}(b_{2})\in\lambda_{B_{2}}([[\alpha_{1}]]_{\mathcal{B}_{2}},[[\alpha_{2}]]_{\mathcal{B}_{2}},\cdots,[[\alpha_{n}]]_{\mathcal{B}_{2}})$.
Consequently,
$gr(b_{1}\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))=gr(b_{2}\models\heartsuit^{\lambda}(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}))$.
Therefore, $b_{1}$ and $b_{2}$ are modally equivalent. ∎

###### Corollary 3

Let $T$ be an endofunctor on Fuzzy-Top; let $\Sigma$ be a monotone fuzzy
geometric modal signature for $T$. Then Aczel-Mendler bisimulation is a
$\Sigma$-bisimulation.

###### Proof.

Consider $\mathcal{B}_{1}=(B_{1},\gamma_{1},\mathcal{V}_{B_{1}})$ and
$\mathcal{B}_{2}=(B_{2},\gamma_{2},\mathcal{V}_{B_{2}})$ are fuzzy geometric
models for $T$. Let $\mathcal{R}$ be an Aczel-Mendler bisimulation between
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$. Then the diagram shown in Fig.8
commutes.

${B_{1}}$${\mathcal{R}}$${B_{2}}$${TB_{1}}$${T\mathcal{R}}$${TB_{2}}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\gamma^{*}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{T\pi_{1}}$$\scriptstyle{T\pi_{2}}$
Figure 8: Illustration of Aczel-Mendler bisimulation between fuzzy geometric
models

We shall show that, $\mathcal{R}$ is a $\Sigma$-bisimulation. Suppose
$b_{1}\mathcal{R}b_{2}$, where $b_{1}\in B_{1}$ and $b_{2}\in B_{2}$ are the
states. Since $\mathcal{R}$ is an Aczel-Mendler bisimulation, we have
$gr(b_{1}\models p)=gr(b_{2}\models p)$, for any propositional variable
$p\in\Phi$. Assume that for each pair of fuzzy open sets
$(\mu_{i},\eta_{i})\in\mathcal{Q}(B_{1})\times\mathcal{Q}(B_{2})$,
$\mathcal{R}[\mu_{i}]\leq\eta_{i}$ and
$\mathcal{R}^{-1}[\eta_{i}]\leq\mu_{i}$.
Now,

$\displaystyle
gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n}))$
$\displaystyle=\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n})(\gamma_{1}(b_{1}))$
$\displaystyle=\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n})(T\pi_{1})(\gamma^{*}(b_{1},b_{2}))\text{
[ As $\gamma_{1}\circ\pi_{1}=T\pi_{1}\circ\gamma^{*}]$ }$
$\displaystyle=gr(\gamma^{*}(b_{1},b_{2})\in(T\pi_{1})^{-1}(\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n})))$
$\displaystyle=gr(\gamma^{*}(b_{1},b_{2})\in\lambda_{\mathcal{R}}(\mu_{1}\circ\pi_{1},\mu_{2}\circ\pi_{1},\cdots,\mu_{n}\circ\pi_{1}))\text{
[As $\lambda$ is a natural transformation]}$ $\displaystyle\leq
gr(\gamma^{*}(b_{1},b_{2})\in\lambda_{\mathcal{R}}(\pi_{2}^{-1}(\pi_{2}(\pi_{1}^{-1}(\mu_{1})))),\cdots,\pi_{2}^{-1}(\pi_{2}(\pi_{1}^{-1}(\mu_{n})))\text{
[As $\lambda$ is monotone] }$
$\displaystyle=gr(\gamma^{*}(b_{1},b_{2})\in\lambda_{\mathcal{R}}(\pi_{2}^{-1}(\mathcal{R}[\mu_{1}]),\pi_{2}^{-1}(\mathcal{R}[\mu_{2}]),\cdots,\pi_{2}^{-1}(\mathcal{R}[\mu_{n}])))$
$\displaystyle\leq
gr(\gamma^{*}(b_{1},b_{2})\in\lambda_{\mathcal{R}}(\pi_{2}^{-1}(\eta_{1}),\pi_{2}^{-1}(\eta_{2}),\cdots,\pi_{2}^{-1}(\eta_{n})))\text{[As
$\lambda$ is monotone ]}$
$\displaystyle=gr(\gamma^{*}(b_{1},b_{2})\in\lambda_{\mathcal{R}}(\eta_{1}\circ\pi_{2},\cdots,\eta_{n}\circ\pi_{2}))$
$\displaystyle=gr(\gamma^{*}(b_{1},b_{2})\in(T\pi_{2})^{-1}(\lambda_{B_{2}}(\eta_{1},\eta_{2},\cdots,\eta_{n})))\text{
[As $\lambda$ is natural] }$
$\displaystyle=gr(\gamma_{2}(b_{2})\in\lambda_{B_{2}}(\eta_{1},\eta_{2},\cdots,\eta_{n}))$

Therefore,
$gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n}))\leq
gr(\gamma_{2}(b_{2})\in\lambda_{B_{2}}(\eta_{1},\eta_{2},\cdots,\eta_{n}))$.
Similarly, it can be shown that
$gr(\gamma_{2}(b_{2})\in\lambda_{B_{2}}(\eta_{1},\eta_{2},\cdots,\eta_{n}))\leq
gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n}))$.
Finally, we have
$gr(\gamma_{1}(b_{1})\in\lambda_{B_{1}}(\mu_{1},\mu_{2},\cdots,\mu_{n}))=gr(\gamma_{2}(b_{2})\in\lambda_{B_{2}}(\eta_{1},\eta_{2},\cdots,\eta_{n}))$.
Hence, the result follows. ∎

###### Theorem 4

Let $\mathcal{B}=(B,\gamma,\mathcal{V}_{B})$ be a fuzyy geometric model for
$T$. Then $\mathcal{R}^{\prime}=\bigcup\\{\mathcal{R}:\mathcal{R}\text{ is a
$\Sigma$-bisimulation from $\mathcal{B}$ to $\mathcal{B}$ }\\}$ is a
$\Sigma$-bisimulation from $\mathcal{B}$ to itself.

###### Proof.

Consider for each pair of fuzzy open sets $(\mu_{i},\mu_{i}^{\prime})$ in $B$,
$\mathcal{R}^{\prime}[\mu_{i}]\leq\mu_{i}^{\prime}$ and
$\mathcal{R^{\prime}}^{-1}[\mu_{i}^{\prime}]\leq\mu_{i}$. Let
$(b_{1},b_{2})\in\mathcal{R}^{\prime}$. Then there exists
$\mathcal{R}\in\mathcal{R}^{\prime}$ such that $(b_{1},b_{2})\in\mathcal{R}$.
Since $\mathcal{R}$ is $\Sigma$-bisimulation, we have
$\mathcal{V}_{B}(p)(b_{1})=\mathcal{V}_{B}(p)(b_{2})$ i.e., $gr(b_{1}\models
p)=gr(b_{2}\models p)$. We observe that
$\mathcal{R}[\mu_{i}]\leq\mathcal{R}^{\prime}[\mu_{i}]\leq\mu^{\prime}_{i}$
and
$\mathcal{R}^{-1}[\mu^{\prime}_{i}]\leq\mathcal{R}^{\prime-1}[\mu^{\prime}_{i}]\leq\mu_{i}$.
Now, we get
$gr(\gamma(b_{1})\in\lambda_{B}(\mu_{1},\mu_{2},\cdots,\mu_{n}))=gr(\gamma(b_{2})\in\lambda_{B}(\mu^{\prime}_{1},\mu^{\prime}_{2},\cdots,\mu^{\prime}_{n}))$.
It follows that $\mathcal{R}^{\prime}$ is $\Sigma$-bisimulation from
$\mathcal{B}$ to itself. ∎

We emphasize that the above result is important for further progress of this
work. In particular, the above result makes it possible for new theoretical
developments, such as coinductive proof principles [25].

##  6 Conclusion

This paper uses coalgebraic methods to define modal operators in fuzzy
geometric logic. In order to study modalities in fuzzy geometric logic, we
have introduced the idea of fuzzy-open predicate lifting for an endofunctor
$T$ on Fuzzy-Top. The structures referred to as the fuzzy geometric models for
$T$, provide the semantics for our coalgebraic logics. We have shown that a
final model exists in the category $FMOD(T)$ of fuzzy geometric models, where
$T$ is an endofunctor on SFuzzy-Top.
Finally, we have studied bisimulations for fuzzy geometric models. In this
study, we have developed the Aczel-Mendler bisimulation concept for fuzzy
geometric models, and we have looked at how it relates to the proposed
$\Sigma$-bisimulation concept for fuzzy geometric models. In addition, we have
demonstrated that $\Sigma$-bisimilarity implies modal equivalence. For an
endofunctor $T$ on SFuzzy-Top, behavioural equivalence coincides with modal
equivalence. As a result, $\Sigma$-bisimilarity implies behavioural
equivalence.
We conclude the study by mentioning future research directions.
In this study, we have not considered whether the behavioural equivalence
concept implies the $\Sigma$-bisimilarity concept. Following the topological
predicate lifting notion for an endofunctor on the category of Stone spaces
defined in [22], we can define a stronger fuzzy-open predicate lifting notion
for endofunctors on Fuzzy-Top. Then, it is possible to show that behavioural
equivalence implies $\Sigma$-bisimilarity. In future work, we will do it.
Considering an endofunctor $T$ on the category of compact fuzzy Hausdorff
spaces, we can show the bi-implication between modal equivalence and
$\Sigma$-bisimilarity. In this case, it will be fruitful to adopt a stronger
notion of fuzzy geometric modal signature $\Sigma$. We will make an effort to
do it in future work.
We have already observed that if $T$ is an endofunctor on SFuzzy-Top then
behavioural equivalence coincides with modal equivalence. In our future work,
we will verify whether behavioural equivalence and modal equivalence coincide
if $T$ is an endofunctor on the category of compact fuzzy Hausdorff spaces.

## Declarations

  * •

Conflict of interest
The authors declare that there is no conflict of interest.

  * •

Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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  *[GAN]: generative adversarial network
  *[CGAN]: conditional generative adversarial network
  *[GANs]: generative adversarial network
  *[WGAN]: Wasserstein generative adversarial network
  *[EEG]: electroencephalogram
  *[GRU]: gated recurrent unit
  *[BERT]: bidirectional encoder representations from transformers
  *[MLP]: multilayer perceptron
  *[EHR]: electronic health record
  *[STFT]: short-time Fourier transform
  *[RSSI]: radio signal strength indication
  *[IMU]: inertial measurement unit
  *[SVM]: support vector machine
  *[CNN]: convolutional neural network
  *[LSTM]: long short-term memory
  *[GNN]: graph neural network
  *[ECG]: electrocardiogram
  *[LBP]: linear binary pattern
  *[EMG]: electromyography
  *[BCI]: brain computer interface
  *[SSVEP]: steady state visually evoked potential
  *[SVR]: support vector regression
  *[SoC]: system on a chip
  *[PPG]: photoplethysmography
  *[GCN]: graph convolutional network
  *[SVMs]: support vector machine
  *[EHRs]: electronic health record
  *[MAML]: Model-Agnostic Meta-Learning
  *[FOMAML]: first-order model-agnostic meta-learning
  *[VOG]: video-oculography
  *[MSE]: mean squared error
  *[LLMs]: large language model
  *[NLP]: natural language processing