# Representations of Domains via CF-approximation Spaces

Guojun Wu  Luoshan Xu  College of Mathematical Science
Yangzhou University
Yangzhou 225002, P. R. China

###### Abstract

Representations of domains mean in a general way representing a domain as a
suitable family endowed with set-inclusion order of some mathematical
structures. In this paper, representations of domains via CF-approximation
spaces are considered. Concepts of CF-approximation spaces and CF-closed sets
are introduced. It is proved that the family of CF-closed sets in a CF-
approximation space endowed with set-inclusion order is a continuous domain
and that every continuous domain is isomorphic to the family of CF-closed sets
of some CF-approximation space endowed with set-inclusion order. The concept
of CF-approximable relations is introduced using a categorical approach, which
later facilitates the proof that the category of CF-approximation spaces and
CF-approximable relations is equivalent to that of continuous domains and
Scott continuous maps.

###### keywords:

CF-approximation space; CF-closed set; CF-approximable relation; continuous
domain; abstract base

††volume: NN††journal: Electronic Notes in Theoretical Informatics and
Computer Science††thanks: Supported by National Natural Science Foundation of
China (11671008).††thanks: Email<EMAIL_ADDRESS>Email:
<EMAIL_ADDRESS>
##  1 Introduction

Domain theory is one of the important research fields of theoretical computer
science [2]. In recent years of research in domain theory, there is a growing
body of scholarly work towards synthesizing various mathematical fields such
as ordered structures, topological spaces, formal contexts, rough sets, and
various kinds of logic. One of such syntheses is to create representation for
various kinds of domains using abstract bases [8, 12], formal topologies [13],
information systems [10, 12], formal contexts [4]-[11], and so on. Amongst
these, representation via abstract bases appears to be most natural due to its
simplicity.

By representation of domains, we mean any general way by which one can
characterize a domain using a suitable family of some mathematical structures
ordered by the set-theoretic inclusion. With this understanding, clearly,
every continuous domain can be represented by c-infs [10], abstract bases,
formal topologies [13], etc. Recently, Qingguo Li, et. al. in [11] introduced
attribute continuous formal contexts which are quadruples, and showed that
every continuous domain can be represented by attribute continuous formal
contexts.

While representation using abstract bases appears to be the most natural and
simple, its scope of study is unfortunately too narrow in that it is easy to
miss out on something deeper. Noticing from rough set theory [7] that abstract
bases are all special generalized approximation spaces (GA-spaces, for short)
[14], we consider generalize an abstract base to a CF-approximation space
which is a GA-space with some coordinating family of finite sets. Since the
lower approximation operator $\underline{R}$ and the upper approximation
operator $\overline{R}$ are mutually dual in a CF-approximation space, we
mainly use the upper approximation operator $\overline{R}$ and introduce CF-
closed sets which are generalizations of round ideals in abstract bases. With
these concepts, representations of domains via CF-approximation spaces are
obtained. We will see that this approach of representing domains is more
general than the approach of representing domains by abstract bases. We also
introduce the concept of CF-approximable relations using a categorical
approach and prove that the category of CF-approximation spaces and CF-
approximable relations is equivalent to that of continuous domains and Scott
continuous maps. This work makes links naturally between domains and rough
sets.

##  2 Preliminaries

We quickly recall some basic notions and results of domain theory. For a set
$U$ and $X\subseteq U$, we use $\mathcal{P}(U)$ to denote the power set of
$U$, $\mathcal{P}_{fin}(U)$ to denote the family of all nonempty finite
subsets of $U$ and $X^{c}$ to denote the complement of $X$ in $U$. The symbol
$F\subseteq_{fin}X$ means $F$ is a finite subset of $X$. For notions which we
do not explicitly define herein, the reader may refer to [2, 3].

Let ($L$, $\leqslant$) be a poset. A _principal ideal_ (resp., _principal
filter_) of $L$ is a set of the form ${\mathord{\downarrow}x}=\\{y\in L\mid
y\leqslant x\\}$ (resp., ${\uparrow\\!x}=\\{y\in L\mid x\leqslant y\\}$). For
$A\subseteq L$, we write $\mathord{\downarrow}A=\\{y\in L\mid\exists\ x\in
A,\,y\leqslant x\\}$ and $\uparrow\\!A=\\{y\in L\mid\exists\ x\in
A,\,x\leqslant y\\}$. A subset $A$ is a _lower set_ (resp., an _upper set_) if
$A=\mathord{\downarrow}A$ (resp., $A=\uparrow\\!A$). We say that $z$ is a
_lower bound_ (resp., an _upper bound_) of $A$ if $A\subseteq\uparrow\\!z$
(resp., $A\subseteq\downarrow\\!z$). The supremum of $A$ is the least upper
bound of $A$, denoted by $\bigvee A$ or $\sup A$. The infimum of $A$ is the
greatest lower bound of $A$, denoted by $\bigwedge A$ or $\inf A$. A nonempty
subset $D$ of $L$ is _directed_ if every finite subset of $D$ has an upper
bound in $D$. A subset $C$ of $L$ is _consistent_ if $C$ has an upper bound in
$L$. A poset $L$ is a _directed complete partially ordered set_ (dcpo, for
short) if every directed subset of $L$ has a supremum. A _semilattice_ (resp.,
sup-semilattice) is a poset in which every pair of elements has an infimum
(resp., a supremum). A _complete lattice_ is a poset in which every subset has
a supremum (equivlently, has an infimum). If any finite consistent subset $A$
of $L$ has a supremum, then $L$ is called a cusl. If any consistent subset $B$
of $L$ has a supremum, then $L$ is called a bc-poset.

Let $L$ be a semilattice and $K\subseteq L$. If for all $x,y\in K$ it holds
that $x\wedge y\in K$, then $K$ is called a subsemilattice of $L$.

Recall that in a poset $P$, we say that $x$ _way-below_ $y$, written $x\ll y$
if whenever $D$ is a directed set that has a supremum with $\sup D\geqslant
y$, then $x\leqslant d$ for some $d\in D$. If $x\ll x$, then $x$ is called a
compact element of $P$, the set $\\{x\in P\mid x\ll x\\}$ is denoted by
$K(P)$. The set $\\{y\in P\mid x\ll y\\}$ will be denoted by
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{1.72218pt}{$\uparrow$}}$\uparrow$}}x$
and $\\{y\in P\mid y\ll x\\}$ denoted by
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x$.
A poset $P$ is said to be _continuous_ (resp., algebraic) if for all $x\in P$,
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x$
is directed (resp., ${\downarrow}x\cap K(P)$ is directed) and
$x=\bigvee\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x$
(resp., $x=\bigvee({\downarrow}x\cap K(P))$). If a dcpo $P$ is continuous
(resp., algebraic), then $P$ is called a continuous domain (resp., an
algebraic domain). If a continuous domain $P$ is a semilattice (resp., sup-
semilattice, complete lattice), then $P$ is called a continuous semilattice
(resp., continuous sup-semilattice, continuous lattice). If a bc-poset $P$ is
also a continuous domain, then $P$ is called a bc-domain. If an algebraic
domain $L$ is a semilattice and $K(L)$ is a subsemilattice of $L$, then $L$ is
called arithmetic semmilattice.

Let $L$ and $P$ be dcpos, and $f:L\longrightarrow P$ a map. If for any
directed subset $D\subseteq L$, $f(\bigvee D)=\bigvee f(D)$, then $f$ is
called a Scott continuous map.

###### Lemma 2.1

([2]) Let $P$ be a poset. Then for all $x,y,u,z\in P$,

(1) $x\ll y\Rightarrow x\leqslant y$;(2) $u\leqslant x\ll y\leqslant
z\Rightarrow u\ll z$.

###### Lemma 2.2

If $P$ is a continuous poset, then the way-below relation $\ll$ has the
interpolation property:

$x\ll z\Rightarrow\exists y\in P$ such that $x\ll y\ll z$.

###### Definition 2.3

Let $P$ be a poset, $B\\!\subseteq\\!P$. The set $B$ is called a basis for $P$
if for all $a\in\\!P$, there is a directed set
$D_{a}\subseteq\\!B\cap\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}a$
such that $\sup_{P}D_{a}=\\!a$, where the subscripted $P$ indicates that the
operation (in this case, the supremum) is taken in poset $P$.

It is well known that a poset $P$ is continuous iff it has a basis and that
$P$ is algebraic iff $K(P)$ is a basis.

A binary relation $R\subseteq U\times U$ on a set $U$ is called _transitive_
if $xRy$ and $yRz$ implies $xRz$ for all $x,y,z\in U$. A binary relation $R$
is said a preorder if it is reflexive and transitive.

###### Definition 2.4

(see [2, 3]) Let $(U,\prec)$ be a set equipped with a binary relation. The
binary relation $\prec$ is called fully transitive if it is transitive and
satisfies the strong interpolation property:

$\forall|F|<\infty,F\prec z\Rightarrow\exists y\prec z$ such that $F\prec y$,

where $F\prec y$ means for all $t\in F$, $t\prec y$. If $(B,\prec)$ is a set
equipped with a binary relation which is fully transitive, then $(B,\prec)$ is
called an abstract basis.

###### Definition 2.5

([2, 3]) Let $(B,\prec)$ be an abstract basis. A non-empty subset $I$ of $B$
is a round ideal if

$(1)$ $\forall y\in I$, $x\prec y\Rightarrow x\in I$;

$(2)$ $\forall x$, $y\in I$, $\exists z\in I$ such that $x\prec z$ and $y\prec
z$.

All the round ideals of $B$ in set-inclusion order is called the round ideal
completion of $B$, denoted by $RI(B)$.

Observe that if $B$ is a basis for a continuous domain $P$, then $(B,\,\ll)$,
the restriction of the way-below relation to $B$, is an abstract basis. And it
is known (see in [10]) that $P$ in this case is isomorphic to $RI(B)$.

###### Proposition 2.6

If $P$ is a continuous domain, then $(P,\ll)$ is an abstract basis and
$RI(P,\ll)\cong(P,\leqslant)$.

Next, we introduce some terminologies imported from rough set theory. A set
$U$ with a binary relation $R$ is called a _generalized approximation space_
(_GA-space_ , for short). Let $(U,R)$ be a GA-space. Define
$R_{s},R_{p}:U\rightarrow\mathcal{P}(U)$ such that for all $x\in U$,
$R_{s}(x)=\\{y\in U\mid xRy\\},R_{p}(x)=\\{y\in U\mid yRx\\}.$

Lower and upper approximation operators are key notions in GA-spaces.

###### Definition 2.7

(cf. [17]) Let $(U,R)$ be a GA-space. For $A\subseteq U$, define

$\underline{R}(A)=\\{x\in U\mid\ R_{s}(x)\subseteq A\\}$,
$\overline{R}(A)=\\{x\in U\mid\ R_{s}(x)\cap A\neq\emptyset\\}.$

The operators
$\underline{R},\overline{R}:\mathcal{P}(U)\rightarrow\mathcal{P}(U)$ are
respectively called the lower and upper approximation operators in $(U,R)$,
where $\mathcal{P}(U)$ is the power set of $U$.

###### Lemma 2.8

(cf. [9]) Let $(U,R)$ be a GA-space. Then the lower and upper approximation
operators $\underline{R}$ and $\overline{R}$ have the following properties.

(1) $\underline{R}(A^{c})=(\overline{R}(A))^{c}$,
$\overline{R}(A^{c})=(\underline{R}(A))^{c}$, where $A^{c}$ is the complement
of $A\subseteq U$.

(2) $\underline{R}(U)=U$, $\overline{R}(\emptyset)=\emptyset$.

(3) Let $\\{A_{i}\mid i\in I\\}\subseteq\mathcal{P}(U)$. Then
$\underline{R}(\bigcap_{i\in I}A_{i})=\bigcap_{i\in I}\underline{R}(A_{i}),\
\overline{R}(\bigcup_{i\in I}A_{i})=\bigcup_{i\in I}\overline{R}(A_{i}).$

(4) If $A\subseteq B\subseteq U$, then
$\underline{R}(A)\subseteq\underline{R}(B),\overline{R}(A)\subseteq\overline{R}(B)$.

(5) For all $x\in U,\ \overline{R}(\\{x\\})=R_{p}(x)$.

###### Lemma 2.9

[16] Let $(U,R)$ be a GA-space. Then $R$ is reflexive iff for all $X\subseteq
U$, $X\subseteq\overline{R}(X)$.

###### Lemma 2.10

[16] Let $(U,R)$ be a GA-space. Then $R$ is transitive iff for all $X\subseteq
U$, $\overline{R}(\overline{R}(X))\subseteq\overline{R}(X)$.

###### Definition 2.11

([15]) Let $(U,R)$ be a GA-space and $A\subseteq U.$ The set $A$ is called
$R$-open if $A\subseteq\underline{R}(A)$ and $R$-closed if
$\overline{R}(B)\subseteq B$.

For a preorder $R$, the operator $\underline{R}$ is an interior operator of a
topology, so we have

###### Definition 2.12

([15]) If $R$ is a preorder, then GA-space $(U,R)$ is called a topological GA-
space.

The next proposition shows that all $R$-open sets of $(U,R)$ form a topology
on $U$.

###### Proposition 2.13

Let $(U,R)$ be a GA-space. Then $\tau_{R}=\\{A\subseteq U\mid
A\subseteq\underline{R}(A)\\}$ is an Alexandrov topology on $U$.

Proof. Obviously, $\emptyset,U\in\tau_{R}$. By Lemma 2.8(3), we have
$\tau_{R}$ is closed under arbitrary intersections. Let $A_{i}\in\tau_{R}$,
namely, $A_{i}\subseteq\underline{R}(A_{i})$ $(i\in\\!I)$. Then
$\bigcup_{i\in\\!I}A_{i}\subseteq\bigcup_{i\in\\!I}(\underline{R}(A_{i}))$. It
follows from $\underline{R}(A_{i})\subseteq\underline{R}(\bigcup_{i\in
I}A_{i})$ that
$\bigcup_{i\in\\!I}(\underline{R}(A_{i}))\subseteq\underline{R}(\bigcup_{i\in\\!I}A_{i})$.
Then $\bigcup_{i\in\\!I}A_{i}\subseteq\underline{R}(\bigcup_{i\in\\!I}A_{i})$.
So $\bigcup_{i\in\\!I}A_{i}\in\tau_{R}$, namely, $\tau_{R}$ is closed under
arbitrary unions. This shows that $\tau_{R}$ is an Alexandrov topology. $\Box$

The above topology $\tau_{R}$ is called a _topology induced by relation_ $R$.
Obviously, all the R-closed sets of $(U,R)$ are precisely all the closed sets
of $\tau_{R}$ .

##  3 CF-approximation Spaces and CF-closed Sets

For an abstract base $(B,\prec)$, we naturally have the triple
$(B,\prec,\\{\\{b\\}\mid b\in B\\})$ and that the family
$\\{{\downarrow^{\prec}b}\mid b\in B\\}$ is a base of the continuous domain
$RI(B)$, where $\downarrow^{\prec}b=\\{c\in B\mid c\prec b\\}$. We generalize
an abstract base to a GA-space with consistent family of finite subsets (CF-
approximation space, for short) by changing $(B,\prec)$ to a GA-space $(U,R)$
with $R$ being transitive and changing the family $\\{\\{b\\}\mid b\in B\\}$
to a suitable family $\mathcal{F}$ of some finite subsets of $U$. We hope that
the family $\mathcal{F}$ can also induce a base of a continuous domain.

###### Definition 3.1

Let $(U,R)$ be a GA-space, $R$ a transitive relation and
$\mathcal{F}\subseteq\mathcal{P}_{fin}(U)\cup\\{\emptyset\\}$. If for all
$F\in\mathcal{F}$, whenever $K\subseteq_{fin}\overline{R}(F)$, there always
exists $G\in\mathcal{F}$ such that $K\subseteq\overline{R}(G)$ and
$G\subseteq\overline{R}(F)$, then $(U,R,\mathcal{F})$ is called a generalized
approximation space with consistent family of finite subsets, or a CF-
approximation space, for short.

###### Lemma 3.2

Let $(U,R)$ be a GA-space, $A,B\subseteq U$. If $R$ is a transitive relation,
then $\overline{R}(B)\subseteq\overline{R}(A)$ when
$B\subseteq\overline{R}(A)$.

Proof. It follows from Lemma 2.8(4) and 2.10. $\Box$

###### Definition 3.3

Let $(U,R,\mathcal{F})$ be a CF-approximation space, $E\subseteq U$. If for
all $K\subseteq_{fin}E$, there always exists $F\in\mathcal{F}$ such that
$K\subseteq\overline{R}(F)\subseteq E$ and $F\subseteq E$, then $E$ is called
a CF-closed set of $(U,R,\mathcal{F})$. The collection of all CF-closed sets
of $(U,R,\mathcal{F})$ is denoted by $\mathfrak{C}(U,R,\mathcal{F})$.

###### Remark 3.4

$(1)$ If $\emptyset\in\mathfrak{C}(U,R,\mathcal{F})$, then
$\emptyset\in\mathcal{F}$ by $\overline{R}(\emptyset)=\emptyset$.

$(2)$ For CF-approximation space $(U,R,\mathcal{F})$, if
$\mathcal{F}=\\{\\{x\\}\mid x\in U\\}$, then $(U,R)$ is an abstract base by
Lemma 2.8(5), and all the CF-closed sets of $(U,R,\mathcal{F})$ are precisely
all the round ideals of $(U,R)$.

The following example shows that $(U,R)$ is not necessarily an abstract base
when $(U,R,\mathcal{F})$ is a CF-approximation space, showing that CF-
approximation spaces is a generalization of abstract bases.

###### Example 3.5

Let $U=\mathbb{N}$,
$R=\\{(1,1),(1,2),(1,3),(1,4),(2,3),(2,4),(4,3)\\}\cup\\{(i,i)\mid i\geqslant
5\\}$, $\mathcal{F}=\\{\\{1\\},\\{1,2\\},\emptyset\\}\cup\\{\\{i\\}\mid
i\geqslant 5\\}$. It is easy to check that $(U,R,\mathcal{F})$ is a CF-
approximation space and
$\mathfrak{C}(U,R,\mathcal{F})=\\{\emptyset,\\{1\\}\\}\cup\\{\\{i\\}\mid
i\geqslant 5\\}$ is a continuous domain. Notice that $(1,4),(2,4)\in R$, but
there is no $t\in U$ such that $(1,t),(2,t),(t,4)\in R$. So $(U,R)$ is not an
abstract base.

###### Proposition 3.6

Let $(U,R,\mathcal{F})$ be a CF-approximation space. If
$E\in\mathfrak{C}(U,R,\mathcal{F})$, then $E$ is an $R$-closed set.

Proof. If $x\in\overline{R}(E)$, then $R_{s}(x)\cap E\neq\emptyset$. So there
is $y\in U$ such that $xRy$ and $y\in E$. By Definition 3.3, there exists
$F\in\mathcal{F}$, such that $y\in\overline{R}(F)\subseteq E$ and $F\subseteq
E$. By the transitivity of $R$, we know that
$\overline{R}(\\{y\\})\subseteq\overline{R}(\overline{R}(F))\subseteq\overline{R}(F)\subseteq
E$. Thus $\overline{R}(\\{y\\})\subseteq E$. It is clear that
$x\in\overline{R}(\\{y\\})\subseteq E$ because of $xRy$. By the arbitrariness
of $x\in\overline{R}(E)$, we know that $\overline{R}(E)\subseteq E$. This
shows that $E$ is an $R$-closed set. $\Box$

###### Proposition 3.7

Let $(U,R,\mathcal{F})$ be a CF-approximation space, then the following
statements hold:

$(1)$ For any $F\in\mathcal{F}$,
$\overline{R}(F)\in\mathfrak{C}(U,R,\mathcal{F})$;

$(2)$ If $E\in\mathfrak{C}(U,R,\mathcal{F})$, $A\subseteq E$, then
$\overline{R}(A)\subseteq E$;

$(3)$ If $\\{E_{i}\\}_{i\in I}\subseteq\mathfrak{C}(U,R,\mathcal{F})$ is a
directed family, then $\bigcup_{i\in I}E_{i}\in\mathfrak{C}(U,R,\mathcal{F})$.

Proof. (1) Follows directly by Definition 3.1 and the transitivity of $R$.

(2) Follows from $A\subseteq E$ and Lemma 2.8(4) that
$\overline{R}(A)\subseteq\overline{R}(E)$. By Proposition 3.6, we know that
$\overline{R}(E)\subseteq E$. Thus $\overline{R}(A)\subseteq E$.

(3) Follows directly from Definition 3.3. $\Box$

The proposition above shows that $(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$
is a dcpo. The following proposition gives equivalent characterizations of CF-
closed sets.

###### Proposition 3.8

The following statements are equivalent for a CF-approximation space
$(U,R,\mathcal{F})$:

$(1)$ $E\in\mathfrak{C}(U,R,\mathcal{F})$;

$(2)$ The family $\mathcal{A}=\\{\overline{R}(F)\mid
F\in\mathcal{F},F\subseteq E\\}$ is directed and $E=\bigcup\mathcal{A}$;

$(3)$ There exists a family $\\{F_{i}\\}_{i\in I}\subseteq\mathcal{F}$ such
that $\\{\overline{R}(F_{i})\\}_{i\in I}$ is directed, and $E=\bigcup_{i\in
I}\overline{R}(F_{i})$;

$(4)$ There always exists $F\in\mathcal{F}$ such that
$K\subseteq\overline{R}(F)\subseteq E$ whenever $K\subseteq_{fin}E$.

Proof. If $E=\emptyset\in\mathfrak{C}(U,R,\mathcal{F})$, the proposition holds
obviously. Let $E\neq\emptyset$.

$(1)\Rightarrow(2)$ By Definition 3.3, we know that $\mathcal{A}$ is not
empty. Let $X_{1},X_{2}\in\mathcal{A}$, then there exist
$F_{1},F_{2}\in\mathcal{F}$ and $F_{1},F_{2}\subseteq E$, such that
$X_{1}=\overline{R}({F_{1}})$, $X_{2}=\overline{R}({F_{2}})$. By $F_{1}\cup
F_{2}\subseteq_{fin}E$ and Definition 3.3 we know that there exists
$F_{3}\in\mathcal{F}$, such that $F_{1}\cup F_{2}\subseteq\overline{R}(F_{3})$
and $F_{3}\subseteq E$. By the transitivity of $R$ and Lemma 3.2 we know that
$\overline{R}(F_{1})\subseteq\overline{R}(F_{3})$,
$\overline{R}(F_{2})\subseteq\overline{R}(F_{3})$. This shows that
$\mathcal{A}$ is directed. Next we prove $E=\bigcup\mathcal{A}$. By
Proposition 3.7(2) we know that $\bigcup\mathcal{A}\subseteq E$ holds.
Conversely, if $x\in E$, then by Definition 3.3, there is $F\in\mathcal{F}$
such that $x\in\overline{R}(F)\subseteq E$ and $F\subseteq E$. So
$x\in\bigcup\mathcal{A}$. By the arbitrariness of $x\in E$ we know that
$E\subseteq\bigcup\mathcal{A}$. Thus $E=\bigcup\mathcal{A}$.

$(2)\Rightarrow(3)$ Trivial.

$(3)\Rightarrow(4)$ Follows directly from the finiteness of $K$ and the
directedness of $\\{\overline{R}(F_{i})\\}_{i\in I}$.

$(4)\Rightarrow(1)$ If $K\subseteq_{fin}E$, then there exists
$F\in\mathcal{F}$ such that $K\subseteq\overline{R}(F)\subseteq E$. By
Definition 3.1, there exists $G\in\mathcal{F}$ such that
$K\subseteq\overline{R}(G)$ and $G\subseteq\overline{R}(F)$. By Lemma 3.2 we
know that $\overline{R}(G)\subseteq\overline{R}(F)\subseteq E$. Thus
$K\subseteq\overline{R}(G)\subseteq E$. Noticing that
$G\subseteq\overline{R}(F)\subseteq E$, by Definition 3.3 we obtain that
$E\in\mathfrak{C}(U,R,\mathcal{F})$. $\Box$

The following theorem characterizes the way-below relation $\ll$ in dcpo
$(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$.

###### Theorem 3.9

Let $(U,R,\mathcal{F})$ be a CF-approximation space,
$E_{1},E_{2}\in\mathfrak{C}(U,R,\mathcal{F})$. Then $E_{1}\ll E_{2}$ if and
only if there exists $F\in\mathcal{F}$ such that
$E_{1}\subseteq\overline{R}(F)$ and $F\subseteq E_{2}$.

Proof. $\Rightarrow$: It follows from $E_{2}\in\mathfrak{C}(U,R,\mathcal{F})$
and Proposition 3.8(2) that $E_{2}=\bigcup\\{\overline{R}(F)\mid
F\in\mathcal{F},F\subseteq E_{2}\\}$ and that $\\{\overline{R}(F)\mid
F\in\mathcal{F},F\subseteq E_{2}\\}$ is directed. If $E_{1}\ll E_{2}$, then
there exists $F\in\mathcal{F}$ such that $F\subseteq E_{2}$,
$E_{1}\subseteq\overline{R}(F)$.

$\Leftarrow$: For any directed family $\\{C_{i}\\}_{i\in
I}\subseteq\mathfrak{C}(U,R,\mathcal{F})$, if $E_{2}\subseteq\bigvee_{i\in
I}C_{i}=\bigcup_{i\in I}C_{i}$, then by $F\subseteq E_{2}$ and the finiteness
of $F$ we know that there exists $i_{0}\in I$ such that $F\subseteq
C_{i_{0}}$. By Proposition 3.7(2) and $E_{1}\subseteq\overline{R}(F)$, we know
that $E_{1}\subseteq\overline{R}(F)\subseteq C_{i_{0}}$, showing that
$E_{1}\ll E_{2}$. $\Box$

###### Corollary 3.10

Let $(U,R,\mathcal{F})$ be a CF-approximation space,
$E\in\mathfrak{C}(U,R,\mathcal{F})$, $F\in\mathcal{F}$. The following
statements hold:

$(1)$ If $F\subseteq E$, then $\overline{R}(F)\ll E$;

$(2)$ $\overline{R}(F)\ll\overline{R}(F)$ if and only if there exists
$G\in\mathcal{F}$, such that $G\subseteq\overline{R}(G)=\overline{R}(F)$.

Proof. Follows directly from Theorem 3.9. $\Box$

###### Theorem 3.11

Let $(U,R,\mathcal{F})$ be a CF-approximation space. Then
$(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ is a continuous domain.

Proof. By Proposition 3.7 we see that
$(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ is a dcpo. Set
$\mathcal{B}=\\{\overline{R}(F)\mid F\in\mathcal{F}\\}$. Then $\mathcal{B}$ is
a base of $(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ by Proposition 3.8(2)
and Corollary 3.10(1). Thus $(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ is a
continuous domain. $\Box$

The following theorem shows that any continuous domain $(L,\leqslant)$ can
induce a CF-approximation space.

###### Theorem 3.12

Let $(L,\leqslant)$ be a continuous domain, $R_{L}$ the way-below relation
“$\ll$” of $(L,\leqslant)$; $\mathcal{F}_{L}=\\{F\subseteq_{fin}L\mid F\
\mbox{has a top element}\\}$. For any $F\in\mathcal{F}_{L}$, let $c_{F}$ be
the top element of $F$. Then $(L,R_{L},\mathcal{F}_{L})$ is a CF-approximation
space.

Proof. By Lemma 2.1, we know that $R_{L}=\ll$ is transitive. For any
$F\in\mathcal{F}_{L}$, by Lemma 2.8(5), we have that
$\overline{R}_{L}(F)=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}c_{F}$.
For
$K\subseteq_{fin}\overline{R}_{L}(F)=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}c_{F}$,
by that $L$ is a continuous domain, we know that
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}c_{F}$
is directed. Then there exists
$x\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}c_{F}$
such that $K\subseteq{\downarrow}x$. It follows from $x\ll c_{F}$ and Lemma
2.2 that there is $y\in L$ such that $x\ll y\ll c_{F}$. Thus
$K\subseteq\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y$.
Set $G=\\{y\\}\in\mathcal{F}_{L}$. By that
$K\subseteq\overline{R}_{L}(G)=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y$
and
$G\subseteq\overline{R}_{L}(F)=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}c_{F}$,
we have that $(L,R_{L},\mathcal{F}_{L})$ is a CF-approximation space. $\Box$

###### Theorem 3.13

Let $(L,\leqslant)$ be a continuous domain, $(L,R_{L},\mathcal{F}_{L})$ the
one constructed in Theorem 3.12. Then
$\mathfrak{C}(L,R_{L},\mathcal{F}_{L})=\\{\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\mid
x\in L\\}$.

Proof. By the proof of Theorem 3.12 and Proposition 3.7(1), we know that
$\\{\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\mid
x\in L\\}\subseteq\mathfrak{C}(L,R_{L},\mathcal{F}_{L})$. Conversely, let
$E\in\mathfrak{C}(L,R_{L},\mathcal{F}_{L})$. Then by Proposition 3.8, there is
a directed set $D\subseteq L$ such that
$E=\bigcup\\{\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}d\mid
d\in D\\}$. Next we prove
$E=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}\bigvee
D$. Obviously,
$E\subseteq\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}\bigvee
D$. Conversely, if
$x\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}\bigvee
D$, then by Lemma 2.2, there is $y\in L$ such that $x\ll y\ll\bigvee D$. So
there is $d\in D$ such that $x\ll y\leqslant d$. Thus
$x\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}d\subseteq
E$, and
$E=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}\bigvee
D$. This shows that
$\mathfrak{C}(L,R_{L},\mathcal{F}_{L})\subseteq\\{\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\mid
x\in L\\}$ and
$\mathfrak{C}(L,R_{L},\mathcal{F}_{L})=\\{\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\mid
x\in L\\}$. $\Box$

###### Theorem 3.14

(Representation Theorem) Let $(L,\leqslant)$ be a poset. Then $L$ is a
continuous domain iff there exists a CF-approximation space
$(U,R,\mathcal{F})$ such that
$(L,\leqslant)\cong(\mathfrak{C}(U,R,\mathcal{F}),\subseteq))$.

Proof. $\Leftarrow$: Follows directly by Theorem 3.11.

$\Rightarrow$: If $L$ is a continuous domain, then by Theorem 3.12 we know
that $(L,R_{L},\mathcal{F}_{L})$ is a CF-approximation space. Define a map
$f:L\to\mathfrak{C}(L,R_{L},\mathcal{F}_{L})$ such that for all $x\in L$,
$f(x)=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x$.
Then it follows from Theorem 3.13 and the continuity of $L$ that $f$ is an
order isomorphism. $\Box$

##  4 Representations of some special domains

In this section, we add some conditions to CF-approximation spaces, and then
discuss representations of some special types of continuous domains.

###### Lemma 4.1

Let $(L,\leqslant)$ be a continuous domain and $B\subseteq L$ a base. If
$(B,\leqslant)$ is a semilattice (resp., sup-semilattice, poset with bottom
element, poset with top element, sup-semilattice with bottom element, cusl),
then $(L,\leqslant)$ is a continuous semilattice (resp., continuous sup-
semilattic, continuous domain with bottom element, continuous domain with top
element, continuous lattice, bc-domain).

Proof. (1) Let $(B,\leqslant)$ be a semilattice. For any $x,y\in L$, set
$D=\\{a\wedge_{B}b\mid
a\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B,b\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B\\}$. It is easy to show that
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B$ and
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B$ are directed. Therefore $D$ is directed and $\bigvee D$ exists. It is clear
that $\bigvee
D\leqslant\bigvee(\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B)=x$, $\bigvee
D\leqslant\bigvee(\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B)=y$. If $z\leqslant x,y$, then for any
$t\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B$, we have $t\ll
x=\bigvee(\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B)$, $t\ll
y=\bigvee(\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B)$. Therefore there exist
$t_{1}\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B$,
$t_{2}\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B$ such that $t\leqslant t_{1},t_{2}$. Thus $t\leqslant t_{1}\wedge_{B}t_{2}$.
Noticing that $t_{1}\wedge_{B}t_{2}\in D$ and the arbitrariness of
$t\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B$, we have that
$z=\bigvee(\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B)\leqslant\bigvee D$. This shows that $\bigvee D$ is a greatest lower bound
of $x,y$, namely, $x\wedge y=\bigvee D$. Thus $L$ is a continuous semilattice.

(2) Let $(B,\leqslant)$ be a sup-semilattice. For any $x,y\in L$, set
$D=\\{a\vee_{B}b\mid
a\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B,b\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B\\}$. Clearly, $D$ is directed and $\bigvee D$ exists. It is obvious that
$x,y\leqslant\bigvee D$. Let $x,y\leqslant z$. Then for all
$a\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B$ and
$b\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B$, we have that $a\ll z$, $b\ll z$. By the directedness of
$\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B$, there exists
$t\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B$ such that $a,b\leqslant t$. Therefore $a\vee_{B}b\leqslant t$. Noticing
that $a\vee_{B}b\in D$, we have $\bigvee
D\leqslant\bigvee(\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B)=z$. This shows that $\bigvee D$ is a least upper bound of $x,y$, namely,
$x\vee y=\bigvee D$. Thus $L$ is a continuous sup-semilattice.

(3)/(4) If $\perp$/$\top$ is a bottom/top element of $(B,\leqslant)$, then
$\perp$/$\top$ is also a bottom/top element of $L$.

(5) Let $(B,\leqslant)$ be a sup-semilattice with bottom element $\perp$. Then
by (2) and (3), we know that $L$ is a sup-semilattice with bottom element.
Since $L$ is a dcpo, $L$ is a complete lattice. Thus $L$ is a continuous
lattice.

(6) Let $(B,\leqslant)$ be a cusl. For any $x,y,z\in L$ which satisfy
$x,y\leqslant z$, we show that for all
$a\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x\cap
B$ and
$b\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}y\cap
B$, $a\vee_{B}b$ exists. By that $x,y\leqslant z$, we have $a\ll z,b\ll z$.
Since $B$ is a base, there exists
$c\in\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}z\cap
B$ such that $a,b\leqslant c$. That $a\vee_{B}b$ exists by that
$(B,\leqslant)$ is a cusl. Similar to the proof of (2), we have that $L$ is a
cusl. As $L$ is a continuous domain, we see that $L$ is a bc-domain. $\Box$

###### Theorem 4.2

Let $(U,R,\mathcal{F})$ be a CF-approximation space. If
$(\\{\overline{R}(F)\mid F\in\mathcal{F}\\},\subseteq)$ is a semilattice
(resp., sup-semilattice, poset with bottom element, poset with top element,
sup-semilattice with bottom element, cusl), then
$\mathfrak{C}(U,R,\mathcal{F})$ is a continuous semilattice (resp., continuous
sup-semilattic, continuous domain with bottom element, continuous domain with
top element, continuous lattice, bc-domain). Conversely, any continuous
semilattice (resp., continuous sup-semilattic, continuous domain with bottom
element, continuous domain with top element, continuous lattice, bc-domain)
$L$ is isomorphic to $(\mathfrak{C}(L,R_{L},\mathcal{F}_{L}),\subseteq)$ of
corresponding CF-approximation spaces, respectively.

Proof. The first half of the theorem follows directly from Theorem 3.11 and
Lemma 4.1.

For the second half, let $(L,R_{L},\mathcal{F}_{L})$ be the one in Theorem
3.12. Define a map $f:L\longrightarrow\\{\overline{R}(F)\mid
F\in\mathcal{F}_{L}\\}$ such that for all $x\in L$,
$f(x)=\mathord{\mbox{\makebox[0.0pt][l]{\raisebox{-1.72218pt}{$\downarrow$}}$\downarrow$}}x$.
Since $L$ is continuous, $f$ is an order isomorphism. Thus
$\\{\overline{R}(F)\mid F\in\mathcal{F}_{L}\\}$ is a semilattice (resp., sup-
semilattice, poset with bottom element, poset with top element, sup-
semmilattice with bottom element, cusl) whenever $L$ is a continuous
semilattice (resp., continuous sup-semilattic, continuous domain with bottom
element, continuous domain with top element, continuous lattice, bc-domain).
By Theorem 3.13, $L$ is isomorphic to
$(\mathfrak{C}(L,R_{L},\mathcal{F}_{L}),\subseteq)$. $\Box$

Next, we consider algebraic cases.

###### Definition 4.3

Let $(U,R)$ be a GA-space,
$\mathcal{F}\subseteq\mathcal{P}_{fin}(U)\cup\\{\emptyset\\}$. If $R$ is a
preorder, then $(U,R,\mathcal{F})$ is called a topological CF-approximation
space.

###### Remark 4.4

A topological CF-approximation space must be a CF-approximation space. In
fact, for all $F\in\mathcal{F}$ and $K\subseteq_{fin}\overline{R}(F)$, taking
$G=F$, then by Lemma 2.9 we have $K\subseteq\overline{R}(G)$ and
$G\subseteq\overline{R}(F)$. By Definition 3.1, $(U,R,\mathcal{F})$ is a CF-
approximation space.

###### Proposition 4.5

Let $(U,R,\mathcal{F})$ be a topological CF-approximation space,
$E_{1},E_{2}\in\mathfrak{C}(U,R,\mathcal{F})$. Then $E_{1}\ll E_{2}$ iff there
exists $F\in\mathcal{F}$ such that $E_{1}\subseteq\overline{R}(F)\subseteq
E_{2}$. Thus
$K((\mathfrak{C}(U,R,\mathcal{F}),\subseteq))=(\\{\overline{R}(F)\mid
F\in\mathcal{F}\\},\subseteq)$.

Proof. It follows directly from Lemma 2.9 and Theorem 3.9. $\Box$

###### Theorem 4.6

Let $(U,R,\mathcal{F})$ be a topological CF-approximation space. Then
$(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ is an algebraic domain.
Conversely, any algebraic domain can be represented by some topological CF-
approximation space.

Proof. The first half of the theorem follows from Proposition 4.5 and Theorem
3.8 that $(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ is an algebraic domain.

For the second half, let $(L,\leqslant)$ be an algebraic domain. Set
$(K(L),R_{K(L)},\mathcal{F}_{K(L)})$, where
$\mathcal{F}_{K(L)}=\\{F\subseteq_{fin}K(L)\mid F\ \mbox{has top element}\\}$,
$R_{K(L)}=\leqslant$ is a partial order. Thus
$(K(L),R_{K(L)},\mathcal{F}_{K(L)})$ is a topological CF-approximation space.
For any $F\in\mathcal{F}_{K(L)}$, let $c_{F}$ be the top element of $F$. By
Lemma 2.8(5), we know that for all $F\in\mathcal{F}_{K(L)}$,
$\overline{R_{K(L)}}(F)={\downarrow}c_{F}\cap K(L)$. Similar to the proof of
Theorem 3.13, we have that
$\mathfrak{C}(K(L),R_{K(L)},\mathcal{F}_{K(L)})=\\{{\downarrow}x\cap K(L)\mid
x\in L\\}.$ Since $L$ is an algebraic domain, we know that
$(\\{{\downarrow}x\cap K(L)\mid x\in L\\},\subseteq)\cong(L,\leqslant)$. The
proof is finished. $\Box$

###### Lemma 4.7

Let $(L,\leqslant)$ be an algebraic domain. If $(K(L),\leqslant)$ is a
semilattice, then $L$ is an arithmetic semilattice.

Proof. For any $x,y\in L$, let $D=\\{a\wedge_{K(L)}b\mid a\in\downarrow x\cap
K(L),b\in\downarrow y\cap K(L)\\}$. By Lemma 4.1, we see that $x\wedge
y=\bigvee D$ and $L$ is a semilattice. Next we prove $(K(L),\leqslant)$ is a
subsemilattice of $L$. If $x,y\in K(L)$, then $x\wedge_{K(L)}y\in D$ and
$x\wedge_{K(L)}y$ is the top element of $D$. So $x\wedge_{K(L)}y=\bigvee
D=x\wedge y$. Hence $x\wedge_{K(L)}y=x\wedge y$. This shows that
$(K(L),\leqslant)$ is a subsemilattice of $L$, and $L$ is an arithmetic
semilattice. $\Box$

###### Theorem 4.8

Let $(U,R,\mathcal{F})$ be a topological CF-approximation space. If poset
$\\{\overline{R}(F)\mid F\in\mathcal{F}\\},\subseteq)$ is a semilattice, then
$(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)$ is an arithmetic semilattice.
Conversely, any arithmetic semilattice can be represented in this way.

Proof. The first half of the theorem follows directly from Theorem 4.6 and
Lemma 4.7.

For the second half, let $L$ be an arithmetic semilattice and
$(K(L),R_{K(L)},\mathcal{F}_{K(L)})$ be the one in Theorem 4.6. Therefore
$(K(L),R_{K(L)},\mathcal{F}_{K(L)})$ is a topological CF-approximation space.
For $F_{1},F_{2}\in\mathcal{F}_{K(L)}$, then
$\overline{R_{K(L)}}(F_{1})={\downarrow}c_{F_{1}}\cap K(L)$,
$\overline{R_{K(L)}}(F_{2})={\downarrow}c_{F_{2}}\cap K(L)$. Since $L$ is an
arithmetic semilattice, we know that $c_{F_{1}}\wedge c_{F_{2}}=c\in K(L)$. So
$\overline{R_{K(L)}}(F_{1})\cap\overline{R_{K(L)}}(F_{2})={\downarrow}c\cap
K(L)$. This shows that there exists $\\{c\\}\in\mathcal{F}_{L}$ such that
$\overline{R_{K(L)}}(F_{1})\wedge\overline{R_{K(L)}}(F_{2})=\overline{R_{K(L)}}(\\{c\\})$.
This shows that $\\{\overline{R_{K(L)}}(F)\mid F\in\mathcal{F}\\}$ is a
semilttice. By Theorem 4.6, we see that the second half of the theorem holds.
$\Box$

##  5 CF-approximable Relations and Equivalence of Categories

In this section, we define and study CF-approximable relations between CF-
approximation spaces and prove that the category of CF-approximation spaces
and CF-approximable relations is equivalent to the category of continuous
domains and Scott continuous maps.

###### Definition 5.1

Let $(U_{1},R_{1},\mathcal{F}_{1})$, $(U_{2},R_{2},\mathcal{F}_{2})$ be CF-
approximation spaces, and
$\mathrel{\Theta}\subseteq\mathcal{F}_{1}\times\mathcal{F}_{2}$ a binary
relation. If

$(1)$ for all $F\in\mathcal{F}_{1}$, there is $G\in\mathcal{F}_{2}$ such that
$F\mathrel{\Theta}G$;

$(2)$ for all $F,F^{\prime}\in\mathcal{F}_{1}$, $G\in\mathcal{F}_{2}$, if
$F\subseteq\overline{R_{1}}(F^{\prime})$, $F\mathrel{\Theta}G$, then
$F^{\prime}\mathrel{\Theta}G$;

$(3)$ for all $F\in\mathcal{F}_{1}$, $G,G^{\prime}\in\mathcal{F}_{2}$, if
$F\mathrel{\Theta}G$, $G^{\prime}\subseteq\overline{R_{2}}(G)$, then
$F\mathrel{\Theta}G^{\prime}$;

$(4)$ for all $F\in\mathcal{F}_{1}$, $G\in\mathcal{F}_{2}$, if
$F\mathrel{\Theta}G$, then there are $F^{\prime}\in\mathcal{F}_{1}$,
$G^{\prime}\in\mathcal{F}_{2}$ such that
$F^{\prime}\subseteq\overline{R_{1}}(F)$,
$G\subseteq\overline{R_{2}}(G^{\prime})$ and
$F^{\prime}\mathrel{\Theta}G^{\prime}$; and

$(5)$ for all $F\in\mathcal{F}_{1}$, $G_{1},G_{2}\in\mathcal{F}_{2}$, if
$F\mathrel{\Theta}G_{1}$ and $F\mathrel{\Theta}G_{2}$, then there is
$G_{3}\in\mathcal{F}_{2}$ such that $G_{1}\cup
G_{2}\subseteq\overline{R_{2}}(G_{3})$ and $F\mathrel{\Theta}G_{3}$,

then $\mathrel{\Theta}$ is called a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$.

###### Proposition 5.2

Let $\mathrel{\Theta}$ be a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$. Then for
all $F\in\mathcal{F}_{1},G\in\mathcal{F}_{2}$, the following statements are
equivalent:

$(1)$ $F\mathrel{\Theta}G$;

$(2)$ There exists $F^{\prime}\in\mathcal{F}_{1}$ such that
$F^{\prime}\subseteq\overline{R_{1}}(F)$ and $F^{\prime}\mathrel{\Theta}G$;

$(3)$ There exists $G^{\prime}\in\mathcal{F}_{2}$ such that
$F\mathrel{\Theta}G^{\prime}$ and $G\subseteq\overline{R_{2}}(G^{\prime})$;

$(4)$ There exist $F^{\prime}\in\mathcal{F}_{1}$ and
$G^{\prime}\in\mathcal{F}_{2}$ such that
$F^{\prime}\subseteq\overline{R_{1}}(F)$,
$G\subseteq\overline{R_{2}}(G^{\prime})$ and
$F^{\prime}\mathrel{\Theta}G^{\prime}$.

Proof. Follows directly from Definition 5.1(2)-(4). $\Box$

Let $\mathrel{\Theta}$ be a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$. For all
$F\in\mathcal{F}_{1}$, set
$\widetilde{\mathrel{\Theta}}(F)=\bigcup\\{\overline{R_{2}}(G)\mid
F\mathrel{\Theta}G\mbox{\ and\ }G\in\mathcal{F}_{2}\\}$. Define a map
$f_{\mathrel{\Theta}}:\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})\longrightarrow\mathcal{P}(U_{2})$
such that for all $E\in\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})$,
$f_{\mathrel{\Theta}}(E)=\bigcup\\{\widetilde{\mathrel{\Theta}}(F)\mid
F\subseteq E\mbox{\ and\ }F\in\mathcal{F}_{1}\\}$.

###### Proposition 5.3

Let $\mathrel{\Theta}$ be a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$,
$F\in\mathcal{F}_{1}$, $E\in\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})$. Then
the following hold:

$(1)$ $\\{\overline{R_{2}}(G)\mid F\mathrel{\Theta}G\mbox{\ and\
}G\in\mathcal{F}_{2}\\}$ is directed;

$(2)$
$\widetilde{\mathrel{\Theta}}(F)\in\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$;

$(3)$ For any $F\in\mathcal{F}_{1}$,
$f_{\mathrel{\Theta}}(\overline{R_{1}}(F))=\widetilde{\mathrel{\Theta}}(F)$;

$(4)$ $\\{\widetilde{\mathrel{\Theta}}(F)\mid F\subseteq
E,F\in\mathcal{F}_{1}\\}$ is directed and
$f_{\mathrel{\Theta}}(E)\in\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$.

Proof. (1) By Definition 5.1(1), we know that $\\{\overline{R_{2}}(G)\mid
F\mathrel{\Theta}G,G\in\mathcal{F}_{2}\\}$ is not empty. By Definition 5.1(5)
and Lemma 3.2, we know that $\\{\overline{R_{2}}(G)\mid
F\mathrel{\Theta}G,G\in\mathcal{F}_{2}\\}$ is directed.

(2) Follows directly from (1) and Proposition 3.7(1).

(3) It is easy to check that

$\begin{array}[]{lll}f_{\mathrel{\Theta}}(\overline{R_{1}}(F))&=&\bigcup\\{\widetilde{\mathrel{\Theta}}(F^{\prime})\mid
F^{\prime}\subseteq\overline{R_{1}}(F),F^{\prime}\in\mathcal{F}_{1}\\}\\\
&=&\bigcup\\{\overline{R_{2}}(G)\mid
F^{\prime}\in\mathcal{F}_{1},G\in\mathcal{F}_{2},F^{\prime}\subseteq\overline{R_{1}}(F)\mbox{~{}and~{}}F^{\prime}\mathrel{\Theta}G\\}~{}~{}(\mbox{by
the definition of~{}}\widetilde{\mathrel{\Theta}}(F^{\prime}))\\\
&=&\bigcup\\{\overline{R_{2}}(G)\mid
G\in\mathcal{F}_{2},F\mathrel{\Theta}G\\}~{}~{}(\mbox{by
Proposition~{}}\ref{pn1-CF-apprel})\\\
&=&\widetilde{\mathrel{\Theta}}(F)~{}~{}(\mbox{by the definition
of~{}}\widetilde{\mathrel{\Theta}}(F)).\end{array}$

(4) Firstly, we show that $\\{\widetilde{\mathrel{\Theta}}(F)\mid F\subseteq
E,F\in\mathcal{F}_{1}\\}$ is directed. Let $F_{1},F_{2}\in\mathcal{F}_{1}$. If
$F_{1},F_{2}\subseteq E$, then by Proposition 3.8(4), there exists
$F_{3}\in\mathcal{F}_{1}$ such that $F_{1}\cup
F_{2}\subseteq\overline{R_{1}}(F_{3})\subseteq E$. And by Definition 5.1(2),
it is easy to deduce that $\widetilde{\mathrel{\Theta}}(F_{1})$,
$\widetilde{\mathrel{\Theta}}(F_{2})\subseteq\widetilde{\mathrel{\Theta}}(F_{3})$.
This shows that the family $\\{\widetilde{\mathrel{\Theta}}(F)\mid F\subseteq
E,F\in\mathcal{F}_{1}\\}$ is directed. Noticing that
$f_{\mathrel{\Theta}}(E)=\bigcup\\{\widetilde{\mathrel{\Theta}}(F)\mid
F\subseteq E,F\in\mathcal{F}_{1}\\}$, by (2) and Proposition 3.7(3), we have
that $f_{\mathrel{\Theta}}(E)\in\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$.
$\Box$

Proposition 5.3 shows that $f_{\mathrel{\Theta}}$ can be seen as a map from
$\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})$ to
$\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$.

###### Proposition 5.4

Let $\mathrel{\Theta}$ be a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$ ,
$F\in\mathcal{F}_{1}$, $G\in\mathcal{F}_{2}$. Then
$G\subseteq\widetilde{\mathrel{\Theta}}(F)\Leftrightarrow F\mathrel{\Theta}G$.

Proof. It is easy to check that

$\begin{array}[]{lll}G\subseteq\widetilde{\mathrel{\Theta}}(F)&\Leftrightarrow&G\subseteq\bigcup\\{\overline{R_{2}}(G^{\prime})\mid
F\mathrel{\Theta}G^{\prime},G^{\prime}\in\mathcal{F}_{2}\\}\\\
&\Leftrightarrow&\exists
G^{\prime}\in\mathcal{F}_{2}\mbox{~{}s.t.~{}}F\mathrel{\Theta}G^{\prime},G\subseteq\overline{R_{2}}(G^{\prime})~{}~{}(\mbox{by
Proposition~{}}\ref{pn2-CF-apprel}(1)\mbox{~{}and the finiteness of~{}}G)\\\
&\Leftrightarrow&F\mathrel{\Theta}G~{}~{}(\mbox{by Proposition~{}}\ref{pn1-CF-
apprel}).\end{array}$

$\Box$

###### Theorem 5.5

Let $\mathrel{\Theta}$ be a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$. Then
$f_{\mathrel{\Theta}}$ is a Scott continuous map from
$\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})$ to
$\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$.

Proof. It follows from the definition of $f_{\mathrel{\Theta}}$ that
$f_{\mathrel{\Theta}}$ is order preserving. In order to prove
$f_{\mathrel{\Theta}}$ is Scott continuous, by Proposition 3.7(3), it suffices
to show that for any directed family $\\{E_{i}\\}_{i\in
I}\subseteq\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})$, we have
$f_{\mathrel{\Theta}}(\bigcup_{i\in I}E_{i})=\bigcup_{i\in
I}f_{\mathrel{\Theta}}(E_{i})$. In fact,

$\begin{array}[]{lll}f_{\mathrel{\Theta}}(\bigcup_{i\in
I}E_{i})&=&\bigcup\\{\widetilde{\mathrel{\Theta}}(F)\mid
F\subseteq\bigcup_{i\in I}E_{i},F\in\mathcal{F}_{1}\\}\\\ &=&\bigcup_{i\in
I}(\bigcup\\{\widetilde{\mathrel{\Theta}}(F)\mid F\subseteq
E_{i},F\in\mathcal{F}_{1}\\}\\\ &&(\mbox{by the finitness of~{}}F\mbox{ ~{}and
the directedness of ~{}}\\{E_{i}\\}_{i\in I})\\\ &=&\bigcup_{i\in
I}f_{\mathrel{\Theta}}(E_{i})~{}~{}(\mbox{by the definition
of~{}}f_{\mathrel{\Theta}}(E_{i})).\end{array}$

$\Box$

Theorem 5.5 shows that a CF-approximable relation between CF-approximation
spaces can induce a Scott continuous map between continuous domains.
Conversely, a Scott continuous map between relative continuous domains can
also induce a CF-approximable relation between CF-approximation spaces as
follows.

###### Theorem 5.6

Let
$f:\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})\longrightarrow\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$
be a Scott continuous map between CF-approximation spaces
$(U_{1},R_{1},\mathcal{F}_{1})$ and $(U_{2},R_{2},\mathcal{F}_{2})$. Define
$\mathrel{\Theta}_{f}\subseteq\mathcal{F}_{1}\times\mathcal{F}_{2}$ such that

$\forall
F\in\mathcal{F}_{1},G\in\mathcal{F}_{2},F\mathrel{\Theta}_{f}G\Leftrightarrow
G\subseteq f(\overline{R_{1}}(F)).$

Then $\mathrel{\Theta}_{f}$ is a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$.

Proof. It follows from
$f(\overline{R_{1}}(F))\in\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$ and
Definition 3.3 that $\mathrel{\Theta}_{f}$ satisfies Definition 5.1(1).

To check that $\mathrel{\Theta}_{f}$ satisfies Definition 5.1(2), let
$F,F^{\prime}\in\mathcal{F}_{1}$, $G\in\mathcal{F}_{2}$. Then

$\begin{array}[]{lll}&&F\subseteq\overline{R_{1}}(F^{\prime}),F\mathrel{\Theta}_{f}G\\\
&\Rightarrow&F\subseteq\overline{R_{1}}(F^{\prime}),G\subseteq
f(\overline{R_{1}}(F))~{}~{}(\mbox{by the definiton
of~{}}\mathrel{\Theta}_{f})\\\ &\Rightarrow&G\subseteq
f(\overline{R_{1}}(F))\subseteq f(\overline{R_{1}}(F^{\prime}))~{}~{}(\mbox{by
Lemma\ }\ref{lm2-tr-up}\mbox{~{}and the order preservation of~{}}f)\\\
&\Rightarrow&G\subseteq f(\overline{R_{1}}(F^{\prime}))\Leftrightarrow
F^{\prime}\mathrel{\Theta}_{f}G.\end{array}$

To check that $\mathrel{\Theta}_{f}$ satisfies Definition 5.1(3), let
$F\in\mathcal{F}_{1}$, $G,G^{\prime}\in\mathcal{F}_{2}$. Then

$\begin{array}[]{lll}&&F\mathrel{\Theta}_{f}G,G^{\prime}\subseteq\overline{R_{2}}(G)\\\
&\Rightarrow&G\subseteq
f(\overline{R_{1}}(F)),G^{\prime}\subseteq\overline{R_{2}}(G)\\\
&\Rightarrow&G^{\prime}\subseteq\overline{R_{2}}(G)\subseteq
f(\overline{R_{1}}(F))~{}~{}(\mbox{by Proposition\ }\ref{pn2-cf-
clo}(2)\mbox{~{}and~{}}f(\overline{R_{1}}(F))\in\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2}))\\\
&\Rightarrow&G^{\prime}\subseteq f(\overline{R_{1}}(F))\Leftrightarrow
F\mathrel{\Theta}_{f}G^{\prime}.\end{array}$

To check that $\mathrel{\Theta}_{f}$ satisfies Definition 5.1(4), let
$F\in\mathcal{F}_{1}$, $G\in\mathcal{F}_{2}$. If $F\mathrel{\Theta}_{f}G$,
then $G\subseteq f(\overline{R_{1}}(F))$. By Proposition 3.7, 3.8(2) and the
Scott continuity of $f$, we know that

$(\ast)$ $f(\overline{R_{1}}(F))=\bigcup\\{f(\overline{R_{1}}(F^{\prime}))\mid
F^{\prime}\subseteq\overline{R_{1}}(F),F^{\prime}\in\mathcal{F}_{1}\\}$.

Therefore we have that

$\begin{array}[]{lll}G\subseteq f(\overline{R_{1}}(F))&\Rightarrow&\exists
G^{\prime}\in\mathcal{F}_{2},\mbox{~{}s.t.~{}}G\subseteq\overline{R_{2}}(G^{\prime})\mbox{~{}and~{}}G^{\prime}\subseteq
f(\overline{R_{1}}(F))~{}~{}(\mbox{by Definition~{}}\ref{dn-cf-cl})\\\
&\Rightarrow&\exists
F^{\prime}\in\mathcal{F}_{1},G^{\prime}\in\mathcal{F}_{2},\mbox{~{}s.t.~{}}G\subseteq\overline{R_{2}}(G^{\prime}),F^{\prime}\subseteq\overline{R_{1}}(F)\mbox{~{}and~{}}G^{\prime}\subseteq
f(\overline{R_{1}}(F^{\prime}))\\\ &&(\mbox{by equation~{}}(\ast)\mbox{~{}and
the finiteness of~{}}G^{\prime})\\\ &\Rightarrow&\exists
F^{\prime}\in\mathcal{F}_{1},G^{\prime}\in\mathcal{F}_{2},\mbox{~{}s.t.~{}}G\subseteq\overline{R_{2}}(G^{\prime}),F^{\prime}\subseteq\overline{R_{1}}(F)\mbox{~{}and~{}}F^{\prime}\mathrel{\Theta}_{f}G^{\prime}\\\
&&(\mbox{by the definition of~{}}\mathrel{\Theta}_{f}).\end{array}$

To check that $\mathrel{\Theta}_{f}$ satisfies Definition 5.1(5), let
$F\in\mathcal{F}_{1}$, $G_{1},G_{2}\in\mathcal{F}_{2}$. If
$F\mathrel{\Theta}_{f}G_{1}$ and $F\mathrel{\Theta}_{f}G_{2}$, then $G_{1}\cup
G_{2}\subseteq f(\overline{R_{1}}({F}))$. By Definition 3.3 and
$f(\overline{R_{1}}({F}))\in\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$, there
exists $G_{3}\in\mathcal{F}_{2}$ such that $G_{1}\cup
G_{2}\subseteq\overline{R_{2}}(G_{3})\subseteq f(\overline{R_{1}}({F}))$ and
$G_{3}\subseteq f(\overline{R_{1}}({F}))$. So $F\mathrel{\Theta}_{f}G_{3}$,
showing that $\mathrel{\Theta}_{f}$ satisfies Definition 5.1(5). $\Box$

###### Theorem 5.7

Let
$f:\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})\longrightarrow\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$
be a Scott continuous map between CF-approximation spaces
$(U_{1},R_{1},\mathcal{F}_{1})$ and $(U_{2},R_{2},\mathcal{F}_{2})$,
$\mathrel{\Theta}$ a CF-approximable relation from
$(U_{1},R_{1},\mathcal{F}_{1})$ to $(U_{2},R_{2},\mathcal{F}_{2})$. Then
$\mathrel{\Theta}_{f_{\mathrel{\Theta}}}=\mathrel{\Theta}$ and
$f_{\mathrel{\Theta}_{f}}=f$.

Proof. Let $F\in\mathcal{F}_{1},G\in\mathcal{F}_{2}$. Then by Propositions
5.3(3) and 5.4, we have

$(F,G)\in\mathrel{\Theta}_{f_{\mathrel{\Theta}}}\Leftrightarrow G\subseteq
f_{\mathrel{\Theta}}(\overline{R_{1}}({F}))=\widetilde{\mathrel{\Theta}}(F)\Leftrightarrow(F,G)\in\mathrel{\Theta},$

showing that $\mathrel{\Theta}_{f_{\mathrel{\Theta}}}=\mathrel{\Theta}$.

For any $E\in\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})$, we have that

$\begin{array}[]{lll}f_{\mathrel{\Theta}_{f}}(E)&=&\bigcup\\{\widetilde{\mathrel{\Theta}_{f}}(F)\mid
F\subseteq E\mbox{~{}and~{}}F\in\mathcal{F}_{1}\\}\\\
&=&\bigcup\\{\overline{R_{2}}(G)\mid
F\in\mathcal{F}_{1},G\in\mathcal{F}_{2},F\subseteq
E\mbox{~{}and~{}}F\mathrel{\Theta}_{f}G\\}\\\
&=&\bigcup\\{\overline{R_{2}}(G)\mid
F\in\mathcal{F}_{1},G\in\mathcal{F}_{2},F\subseteq E\mbox{~{}and~{}}G\subseteq
f(\overline{R_{1}}({F}))\\}\\\ &=&\bigcup\\{f(\overline{R_{1}}({F}))\mid
F\in\mathcal{F}_{1},F\subseteq E\\}~{}~{}(\mbox{by Proposition~{}}\ref{pn3-cf-
cl}(2))\\\ &=&f(\bigcup\\{\overline{R_{1}}({F})\mid
F\in\mathcal{F}_{1},F\subseteq E\\})~{}~{}(\mbox{by Scott continuity
of~{}}f)\\\ &=&f(E).\end{array}$

This shows that $f_{\mathrel{\Theta}_{f}}=f$. $\Box$

Given a CF-approximation space $(U,R,\mathcal{F})$, define the identity on
$(U,R,\mathcal{F})$ to be a binary relation
$\operatorname{Id}_{(U,R,\mathcal{F})}\subseteq\mathcal{F}\times\mathcal{F}$
such that for all
$F,G\in\mathcal{F},(F,G)\in\operatorname{Id}_{(U,R,\mathcal{F})}\Leftrightarrow
G\subseteq\overline{R}(F).$

Let $(U_{1},R_{1},\mathcal{F}_{1})$, $(U_{2},R_{2},\mathcal{F}_{2})$,
$(U_{3},R_{3},\mathcal{F}_{3})$ be CF-approximation spaces,
$\mathrel{\Theta}\subseteq\mathcal{F}_{1}\times\mathcal{F}_{2}$,
$\mathrel{\Upsilon}\subseteq\mathcal{F}_{2}\times\mathcal{F}_{3}$ be CF-
approximable relations. Define
$\mathrel{\Upsilon}\circ\mathrel{\Theta}\subseteq\mathcal{F}_{1}\times\mathcal{F}_{3}$,
the composition of $\mathrel{\Upsilon}$ and $\mathrel{\Theta}$ by that for any
$F_{1}\in\mathcal{F}_{1},F_{3}\in\mathcal{F}_{3}$,
$(F_{1},F_{3})\in\mathrel{\Upsilon}\circ\mathrel{\Theta}$ iff there exists
$F_{2}\in\mathcal{F}_{2}$ satisfying $(F_{1},F_{2})\in\mathrel{\Theta}$ and
$(F_{2},F_{3})\in\mathrel{\Upsilon}$.

It is a routine work to check that $\operatorname{Id}_{(U,R,\mathcal{F})}$ is
a CF-approximable relation from $(U,R,\mathcal{F})$ to itself. Thus, CF-
approximation spaces as objects and CF-approximable relations as morphisms
with the identities and compositions defined above, form a category, and
denoted by CF-GA.

Let CDOM be the category of continuous domains and Scott continuous maps. We
next show that categories CF-GA and CDOM are equivalent.

###### Lemma 5.8

([1]) Let $\mathcal{C},\mathcal{D}$ be two categories. If there is a functor
$\Phi:\mathcal{C}\longrightarrow\mathcal{D}$ such that

$(1)$ $\Phi$ is full, namely, for all $A,B\in ob(\mathcal{C})$, $g\in
Mor_{\mathcal{D}}(\Phi(A),\Phi(B))$, there is $f\in Mor_{\mathcal{C}}(A,B)$
such that $\Phi(f)=g$;

$(2)$ $\Phi$ is faithful, namely, for all $A,B\in ob(\mathcal{C})$, $f,g\in
Mor_{\mathcal{C}}(A,B)$, if $f\neq g$, then $\Phi(f)\neq\Phi(g)$;

$(3)$ for all $B\in ob(\mathcal{D})$, there is $A\in ob(\mathcal{C})$ such
that $\Phi(A)\cong B$,

then $\mathcal{C}$ and $\mathcal{D}$ are equivalent.

###### Theorem 5.9

The categories CF-GA and CDOM are equivalent.

Proof. Define $\Psi:$ CF-GA$\to$ CDOM such that for all $(U,R,\mathcal{F})\in
ob$(CF-GA),
$\Psi((U,R,\mathcal{F}))=(\mathfrak{C}(U,R,\mathcal{F}),\subseteq)\in ob({\bf
CDOM})$; for all $\mathrel{\Theta}\in Mor$(CF-GA),
$\Psi(\mathrel{\Theta})=f_{\mathrel{\Theta}}\in Mor({\bf CDOM})$.

Give a CF-approximation space $(U,R,\mathcal{F})$, for any
$E\in\mathfrak{C}(U,R,\mathcal{F})$, we have

$\begin{array}[]{lll}\Psi(\operatorname{Id}_{(U,R,\mathcal{F})})(E)&=&f_{\operatorname{Id}_{(U,R,\mathcal{F})}}(E)\\\
&=&\bigcup\\{\overline{R}({G})\mid F,G\in\mathcal{F},F\subseteq
E\mbox{~{}and~{}}(F,G)\in\operatorname{Id}_{(U,R,\mathcal{F})}\\}\\\
&=&\bigcup\\{\overline{R}({G})\mid F,G\in\mathcal{F},F\subseteq
E\mbox{~{}and~{}}G\subseteq\overline{R}({F})\\}\\\
&=&\bigcup\\{\overline{R}({F})\mid F\in\mathcal{F},F\subseteq
E\\}~{}~{}(\mbox{by Proposition~{}}\ref{pn2-cf-
clo}(1)\mbox{~{}and~{}}\ref{pn3-cf-cl}(2))\\\ &=&E~{}~{}(\mbox{by
Proposition~{}}\ref{pn3-cf-cl}(2))\\\
&=&id_{\mathfrak{C}(U,R,\mathcal{F})}(E).\end{array}$

This shows that
$\Psi(\operatorname{Id}_{(U,R,\mathcal{F})})=id_{\mathfrak{C}(U,R,\mathcal{F})}$.

Let $\mathrel{\Theta}\subseteq\mathcal{F}_{1}\times\mathcal{F}_{2}$,
$\mathrel{\Upsilon}\subseteq\mathcal{F}_{2}\times\mathcal{F}_{3}$ be CF-
approximable relations. Then for any $E\in\mathfrak{C}(U,R,\mathcal{F})$, we
have

$\begin{array}[]{lll}&&\Psi(\mathrel{\Upsilon})\circ\Psi(\mathrel{\Theta})(E)=f_{\mathrel{\Upsilon}}(f_{\mathrel{\Theta}}(E))\\\
&=&\bigcup\\{\widetilde{\mathrel{\Upsilon}}(F)\mid F\subseteq
f_{\mathrel{\Theta}}(E),F\in\mathcal{F}_{2}\\}\\\
&=&\bigcup\\{\overline{R_{3}}({G})\mid F\subseteq
f_{\mathrel{\Theta}}(E),F\in\mathcal{F}_{2},G\in\mathcal{F}_{3}\mbox{~{}and~{}}F\mathrel{\Upsilon}G\\}~{}~{}(\mbox{by
the definiton of ~{}}\widetilde{\mathrel{\Upsilon}})\\\
&=&\bigcup\\{\overline{R_{3}}({G})\mid F_{1}\in\mathcal{F}_{1},F_{1}\subseteq
E,G_{1}\in\mathcal{F}_{2},F_{1}\mathrel{\Theta}G_{1},F\subseteq\overline{R_{2}}(G_{1}),F\in\mathcal{F}_{2},G\in\mathcal{F}_{3}\mbox{~{}and~{}}F\mathrel{\Upsilon}G\\}\\\
&&(\mbox{by the definiton of\ }f_{\mathrel{\Theta}}(E),\mbox{
Proposition~{}}\ref{pn2-CF-apprel}(1),\mbox{ Theorem~{}}\ref{tm-cfap-
sco}\mbox{~{}and finiteness of members in~{}}\mathcal{F}_{2})\\\
&=&\bigcup\\{\overline{R_{3}}({G})\mid F_{1}\in\mathcal{F}_{1},F_{1}\subseteq
E,G_{1}\in\mathcal{F}_{2},F_{1}\mathrel{\Theta}G_{1},G\in\mathcal{F}_{3}\mbox{~{}and~{}}G_{1}\mathrel{\Upsilon}G\\}\\\
&&(\mbox{by~{}}F\subseteq\overline{R_{2}}(G_{1}),F\mathrel{\Upsilon}G,\mbox{~{}and
Definition~{}}\ref{dn-CF-app}(2))\\\ &=&\bigcup\\{\overline{R_{3}}({G})\mid
F_{1}\in\mathcal{F}_{1},F_{1}\subseteq
E,G\in\mathcal{F}_{3}\mbox{~{}and~{}}(F_{1},G)\in\mathrel{\Upsilon}\circ\mathrel{\Theta}\\}~{}~{}(\mbox{by~{}}F_{1}\mathrel{\Theta}G_{1}\mbox{~{}and~{}}G_{1}\mathrel{\Upsilon}G)\\\
&=&\bigcup\\{\widetilde{\mathrel{\Upsilon}\circ\mathrel{\Theta}}(F_{1})\mid
F_{1}\in\mathcal{F}_{1},F_{1}\subseteq E\\}~{}~{}(\mbox{by the definition
of~{}}\widetilde{\mathrel{\Upsilon}\circ\mathrel{\Theta}}(F_{1}))\\\
&=&f_{\mathrel{\Upsilon}\circ\mathrel{\Theta}}(E)=\Psi(\mathrel{\Upsilon}\circ\mathrel{\Theta})(E).\end{array}$

This shows that
$\Psi(\mathrel{\Upsilon})\circ\Psi(\mathrel{\Theta})=\Psi(\mathrel{\Upsilon}\circ\mathrel{\Theta})$,
and thus $\Psi$ is a functor.

To show that CF-GA is equivalent to CDOM, it suffices to check that $\Psi$
satisfies the three conditions in Lemma 5.8.

Let $(U_{1},R_{1},\mathcal{F}_{1})$, $(U_{2},R_{2},\mathcal{F}_{2})$ be CF-
approximation spaces, $\mathrel{\Theta}_{1},\mathrel{\Theta}_{2}$ be CF-
approximable relations from $(U_{1},R_{1},\mathcal{F}_{1})$ to
$(U_{2},R_{2},\mathcal{F}_{2})$. If
$\mathrel{\Theta}_{1}\neq\mathrel{\Theta}_{2}$, then by Theorem 3.14 we know
that
$\mathrel{\Theta}_{1}=\mathrel{\Theta}_{f_{\mathrel{\Theta}_{1}}}\neq\mathrel{\Theta}_{f_{\mathrel{\Theta}_{2}}}=\mathrel{\Theta}_{2}.$
Thus $f_{\mathrel{\Theta}_{1}}\neq f_{\mathrel{\Theta}_{2}}$, showing that
$\Psi$ is faithful.

Let
$f:\mathfrak{C}(U_{1},R_{1},\mathcal{F}_{1})\longrightarrow\mathfrak{C}(U_{2},R_{2},\mathcal{F}_{2})$
be a Scott continuous map, by Theorem 3.14, there is $\mathrel{\Theta}_{f}\in
Mor$(CF-GA) such that $\Psi(\mathrel{\Theta}_{f})=f_{\mathrel{\Theta}_{f}}=f$,
showing that $\Psi$ is full.

It is clear by Theorem 3.14 that $\Psi$ satisfies the condition (3) in Lemma
5.8. $\Box$

Similarly, we can also establish categorical equivalences between the category
of algebraic domains with Scott continuous maps as morphisms and the category
of topological CF-approxiamtion spaces with CF-approximable relations as
morphisms. For the details, we leave them to interested readers.

##  6 Conclusions

This paper generalizes abstract bases to CF-approximation spaces and
generalizes the family of round ideals of an abstract basis to the family of
CF-closed sets of a CF-approximation space. Thus a representation method of
various continuous domains including continuous semilattices, continuous sup-
semilattices, continuous domains with bottom, continuous domains with top,
continuous lattices, bc-domains, algebraic domains and arithmetic semilattices
in the framework of rough set theory is obtained. CF-approximable relations
between CF-approximation spaces are defined, and categorical equivalence
between categories CF-GA of CF-approximation spaces and continuous domains and
CDOM of continuous domains and Scott continuous maps is established. This work
strengthens the links among rough set theory, domain theory and topology,
widens the scope of application of rough set theory and domain theory.

Acknowledgment We would like to thank the referees for their valuable
suggestions and comments.

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# The earthquake network: the best time scale for network construction

Nastaran Lotfi  Instituto de Ciências Matemáticas e de Computação,
Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil.

###### Abstract

Scientists mapped the seismic time series into networks by considering the
geographical location of events as nodes and establishing links between the
nodes with different rules. Applying the successive defined laws to construct
the networks of seismic data, a variety of features of earthquake networks are
detected (scale-free and small-world structures). Network construction models
had changed in detail to optimize the performance of the verification of the
minimum geographical size defined for the node. In all the studies, people try
to use large data sets like years of data to ensure their results are good
enough. In this work, by proposing the temporal network construction and
employing the small-worldness property for data from Iran and California, we
could achieve the minimum time scale needed for the best results. We verified
the importance of this scale by analyzing two significant centrality measures
(degree centrality and PageRank) introduced in the concept of earthquake
network.

###### keywords:

Complex Networks, Temporal networks, earthquake networks

††journal: Journal of Acta Geophysica Templates

##  1 introduction

An earthquake is a sudden motion of a fault that releases an enormous amount
of energy and is considered a complex spatiotemporal phenomenon occurring in
the earth’s crust [19]. Transferring the stress of the movement of one fault
to the others results in triggering subsequent events [20, 10, 15]. Omori law
[27] and Gutenberg-Richter law [17] are empirical laws to characterize the
Temporal pattern of aftershocks, frequency, and magnitude, respectively.
Besides the visible properties, complex interaction exists in the internal of
the seismic system [9, 8, 16].

While seismicity is assumed to be a complex phenomenon, the network approach
offers a powerful tool for analyzing the dynamic structures of it [4, 5, 6,
7]. Over the last decade, different models proposed to construct the
earthquake network [5, 21, 28, 26]. In the simple but basic model introduced
by Abe-Suzuki [5], the geographical region is divided into small square
(cubic) cells, and seismic events with time sequences get connected. Later,
Lacasa et al. [21] proposed a model to construct the network with a visibility
graph. They converted the time series into a graph by inheriting the
properties of the series in its structure. They explored periodicity,
fractality, chaoticity, and non-linearity of the seismic time series [22, 23,
14]. Rezaie et al. [28] introduced the hybrid model, which inherits the bases
of the Abe-Suzuki model mixed with a visibility graph. To better capture the
evolution of the earthquake network through time, a multiplex network was
employed [26]. Analyzing the seismic data with a network approach through
different models helped reveal many features of the seismic activity just by
knowing the basic information of magnitude, time of occurrence, and the
location of seismic events [8, 4, 24, 3, 25]. It had been verified that the
the earthquake networks that constructed from the seismic data taken from
California and Japan [4, 6, 5], Iran [24, 25], Chile [2] , Greece [12], and
Italy [28] are scale-free and small-world.

Most recent works focused on improving the proposed models to capture the best
minimum resolution of the cell size needed for network construction. It means
the cell size should be smaller than the specified limit to be trustable. The
main question is how we ensure that the time window, in the scale of dates,
months, or years, is large enough for constructing the network. In all the
studies done till now, scientists considered the time on such a big scale of
years. And the concept of the minimum necessary time window for achieving the
best results are missed. In this work, we employ the definition of temporal
network construction and capture the lowest time window essential for network
construction. We found that depending on the region of consideration, the
value of the time window threshold would change. We verified the trustiness of
this time window size by analyzing two important centrality parameters, degree
centrality, and PageRank. If the time window is small, we miss the information
in centrality, and if it is bigger than the threshold, we do not gain extra
knowledge than in the threshold time region.

The rest of the paper is organized as follows. In section 2, we provide
information about the data sets we employ, and Section 3 is devoted to our
results.

##  2 Database

We applied our model for the latest four years of data, 01 Jan 2018 to 31 Dec
2021, for Iran in the range of $24N-44N$ latitude and $40E–62E$ longitude with
$14062$ total events obtained from Iranian Seismological
Center111http://irsc.ut.ac.ir, and California in the range of $32N–42N$
latitude and $114W–124W$ longitude with $7575$ total events gained from the
Northern California Earthquake Catalog222http://www.usgs.gov/. In both of the
considered data sets, we examined only events with a magnitude larger than
2.5.

Figure 1: (a)-(c) The schematic representation of temporal earthquake networks
of Iran for three different time steps with time windows of (a) one month, (b)
10 months, and (c) 46 months with filtered data magnitudes $>4.0$, (d) to (f)
are the degree distribution of the networks for the three defined networks
respectively for data with magnitudes $>2.5$).

Figure 2: (a)-(c) The schematic representation of temporal earthquake networks
of California for three different time steps with time windows of (a) 10
months, (b) 19 months, and (c) 46 months with filtered data magnitudes $>3.0$,
(d) to (f) are the degree distribution of the networks for the three defined
networks respectively for data with magnitudes $>2.5$).  Figure 3: The
variation of small-worldness in the scale of time windows computed for
earthquake networks of Iran and California.

Figure 4: Degree centrality and PageRank for earthquake network of Iran for
time windows of (a), (d) one month, (b), (e) 10 months, and (c), (f) 46 months
respectively.

Figure 5: Degree centrality and PageRank for earthquake network of California
for time windows of (a), (d) 10 month, (b), (e) 19 months, and (c), (f) 46
months respectively.

##  3 Results

Through different models introduced for earthquake network construction, we
used the simple model introduced by Abe-Suzuki [5]. Dividing the geographical
region into small square cells and having seismic events data ordered by the
occurrence time, each square is regarded as one node if an earthquake with any
magnitude occurred, and two nodes with consecutive events are connected.

We also divided the seismic data of four years length into small time windows
in the following way; In the first step, we construct the Abe-Suzuki network
for the data for the length of one month and study the characteristics of
interest. Then, we added the data from the second month to the previous one
and reconstructed the network. The process of adding data by time windows of
the size of one month continues until the whole 48 months of data are covered.
The schematic representation of the temporal network construction is plotted
in Fig. 1 (a)-(c) for Iran and Fig. 2 (a)-(c) for California. Fig. 1 and 2
belong to the data of the length of one month for Iran and ten months for
California. As the number of events is low, having a sparse network is
predictable. The second Fig. 1 and 2 is in the middle time when the network is
not sparse as the first month and is not too connected as the last, and the
Fig. 1 and 2 represent the networks for the whole four years data which have a
very dense connection.

The second step would be building the adjacency matrix $A$ for facilitating
the analysis; $a_{ij}=1$ if nodes $i$ and $j$ are connected, and 0 otherwise.
In the network definitions, the degree of the node is the number of
connections a node could have and is calculated from the adjacency matrix
$k_{i}=\sum_{j}a_{ij}$. The degree distribution of the earthquake network of
different regions is power law [4, 6, 24]. To check the validity of this
characteristic, for each of the above mentioned networks (Fig. 1 and 2), we
plot the degree distribution Fig. 1 and 2. One could see that no matter the
time length, we would have approximately the power-law distribution.

The other famous characteristic of earthquakes is being small-world [5, 24].
In a small-world network with $N$ nodes and $M$ links, the value of the
shortest path is similar to the random network with the same number of nodes
and links, while the clustering coefficient has a higher value. The clustering
coefficient of a node $i$ is the fraction of connection existing among its
nearest neighbor nodes to the maximum number of possible links among them. The
clustering coefficient of the network would be the average clustering of all
nodes:

$C_{i}=\frac{1}{k_{i}(k_{i}-1)}\sum_{j,k}a_{ij}a_{jk}a_{ki}\;\;,\;\;C=\frac{1}{N}\sum_{i=1}^{N}C_{i}$
(1)

where $N$ is the total number of nodes in the network. In other words, the
clustering coefficient is the probability of the tendency of the nodes in the
graph to cluster together and has a value $0\leq C\leq 1$. On the other hand,
the shortest path is the minimum path length needed to traverse to get from
one node to the other. The average over all nodes would result in the shortest
path of the network:

$L=\frac{1}{N(N-1)}\sum_{i,j=1,N;i\neq j}d_{ij}$ (2)

in which $d_{ij}$ is the minimum length of the path between two nodes of $i$
and $j$.

By having the clustering coefficient and shortest path of the network,
Humphries et al. [18] introduced a small-worldness metric defined with the
averaged clustering coefficient and path length relative to these metrics for
random networks. This metric helps to provide an overview of connectivity in
the entire network:

$S_{w}=\frac{C/C_{rand}}{L/L_{rand}}$ (3)

$C_{rand}$ and $L_{rand}$ are the values obtained for random networks by
randomizing the connections of each earthquake network by keeping the same
number of nodes and links.

The variation of $S_{w}$ by time (in the scale of the length of the month) is
shown in Fig. 3. One could see that this value is small for the first months
of consideration. It starts to increase until a threshold and gets stationery
later. This behavior could emphasize that until a specific time window, the
variation of the parameters is high. The fluctuations disappear while a person
considers a large enough time window, and the system gets stationary. The
geographical region under consideration and frequency of the seismic event
could result in observing different values. This value for Iran’s data is
approximately ten months, while for California it is around 19 months.

To clarify the importance of having the minimum time window, we calculate two
of the most important centralities in the concept of earthquake networks and
compare them in three different time windows. Looking through the literature,
one could find different parameters to calculate the centrality of nodes in
the seismic networks. The simplest and most common centrality that uses the
local structure around the nodes is the degree centrality. In Fig. 4 and 5
(a)-(c), we plotted the degree centrality for three different time scales as
the following: 4 and 5 are for the time window of length 1 month for Iran and
10 months for California. Near the time window of the threshold, we selected
the network with the length of 10 months of data for Iran (Fig. 4) and 19
months for California (Fig.5). And Fig. 4 and 5 belong to the largest time
window (48 months).

The second famous centrality in the concept of earthquake network is PageRank
[13, 29]. PageRank is an algorithm used to assess the ranks of nodes in a
network based on their connections’ levels used in the Google search engine
for ranking web pages for the first time [11]. PageRank explained through the
random walk. The random walker starts from one node and selects the next one
randomly. In this definition, PageRank of node $i$ is the asymptotic
probability that the walker meets the node. One could infer that the
possibility of reaching one important node is higher than the unimportant
ones. This centrality is an iterative procedure in which the PageRank of nodes
depends on all its neighbors’ PageRanks. The following equation describes such
a random walking procedure:

$PR_{i}=\frac{d}{N}+(1-d)\sum_{j\in B_{i}}\frac{PR_{j}}{K_{j}^{out}}$ (4)

in which $PR_{i}$ is the PageRank of node $i$, $B_{i}$ is the set of nearest
neighbors of node $i$, and $k^{out}$ is the out-degree of each node. $d$ is a
fixed value (0.15) defined as the probability of jumping to any vertex. Fig. 4
and 5 (d)-(f) are representing the PageRank of the networks for the three
different time windows. Taking into account both above-introduced
centralities, one could see that in the small-time windows, we could not find
enough information about the central locations of the regions as it should. If
we increase the length of the time window up to the threshold, the results
capture the same central regions as the largest time window.

##  4 Conclusion

Recently, different models proposed to study the earthquake phenomena to
explore the features of this harmful disaster. Although this phenomenon is
very complex from a fault and inside earth interactions point of view, it is
possible to study it with the complex network with the minimum information:
geographical location, time, and magnitude. Among the most famous models
proposed, Abe-Suzuki and visibility models, scientists were trying to improve
the model’s performances. The main idea that got most of the attention from
those studying was how they could introduce the best minimum geographical cell
size.

Here, we proposed the temporal earthquake network construction for capturing
another essential factor of network analysis, the best time window size. We
start with constructing an earthquake network in windows of the month length
and adding data with a length of one month in each step. We used the most
straightforward model introduced by Abe-Suzuki to build our networks. For each
constructed network, the small-worldness is evaluated. Studying how this
parameter changes by increasing the time window, we could verify the minimum
length of time window needed. This value is smaller than its value in the
threshold time window and gets stationary by enlarging the time lengths. The
time threshold differs for disparate geographical regions as the construction
of the earth is different. One point of these differences appears in the
frequency of the events on the same time scale. Then, it is a delicate factor
to study the minimum and efficient time window size for different geographical
regions before the rest of the analysis to ensure obtaining the best results.
By considering two famous centralities measures in the concept of earthquake
networks (degree centrality, and PageRank), we show that if this size is
smaller than the threshold, we will miss the information we should have. If
the time window is too large, it doesn’t provide extra information.

##  5 Acknowledgments

N. Lotfi is thankful to the FAPESP (grant with number 2020/08359-1) for the
support given to this research.

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See pages 1 of interest.pdf