First we recall some definitions and notations discussed in the following content.

If $(L, \leq)$ is a partially ordered set, for simplicity, it will always be denoted by $L$ for short. We write $\vee A$ and $\wedge A$ for the least upper bound and the greatest lower bound of $A$ in $L$ respectively if they exist. If $a\vee b$ and $a\wedge b$ exist for all $a, b\in L$, then we say $L$ is a lattice. If $L$ is a partially ordered set, a set $A\subseteq L$ is said to be directed, if for all $a, b\in A$, there is a $c\in A$ such that $a, b\leq c$. A subset $A$ of $L$ is called a lower set, if $A=\{x\in L: x\leq a \ \mbox{for some}\ a\in A\}$. If $A$ is a directed lower set, then it is called an ideal.

Let $a, b$ be two elements of a partially ordered set $L$. We call $a$ way-below $b$ [13], in symbols $a\ll b$, if and only if for all directed subsets $X \subseteq L$, if $\vee X$ exists and $b\leq \vee X$, then there exists an $x\in X$ such that $a\leq x$. If $a\ll a$, we call $a$ a compact or finite element.

Here we adopt the axiomatic definition of information algebras presented in [3], which was given by Kohlas. It should be noted that information algebras in this paper are often called domain-free information algebras.

Definition 1.1[3]   Let $(\Phi, D)$ be a tuple with two operations defined, where $D$ is a lattice:

1. Combination: $\Phi\times \Phi \rightarrow \Phi; (\phi, \psi)\mapsto \phi\otimes\psi$,

2. Focusing: $\Phi\times D \rightarrow \Phi; (\phi, x)\mapsto \phi^{\Rightarrow x}$.

The tuple $(\Phi, D)$ is called an information algebra, if it satisfies the following axioms:

(1) Semigroup: $\Phi$ is associative and commutative under combination. There is an element $e\in\Phi$ such that for all $\phi\in \Phi$ with $e\otimes \phi=\phi\otimes e=\phi$. We call $e$ a neutral element.

(2) Transitivity: For $\phi\in \Phi$ and $x, y\in D, (\phi^{\Rightarrow y})^{\Rightarrow x}=\phi^{\Rightarrow x\wedge y}$.

(3) Combination: For $\phi, \psi\in \Phi$ and $x\in D$, $(\phi^{\Rightarrow x}\otimes\psi)^{\Rightarrow x}=\phi^{\Rightarrow x}\otimes\psi^{\Rightarrow x}$.

(4) Neutrality: For $x\in D$, $e^{\Rightarrow x}=e$.

(5) Support: For $\phi\in \Phi$, there is an $x$ such that $\phi^{\Rightarrow x}=\phi$. $x$ is called a support set of $\phi$.

(6) Idempotency: For $\phi\in \Phi$ and $x\in D, \phi\otimes \phi^{\Rightarrow x}=\phi$.

If $(\Phi, D)$ is an information algebra and $\phi, \psi\in\Phi$, we write $\psi\leq \phi$ if $\psi\otimes \phi=\phi$. The order relation means that the information $\phi$ is more informative than another information $\psi$. This defines a partial order on $\Phi$, if $(\Phi, D)$ is an information algebra [3]. The following lemma gives some basic properties of this ordering relation.

Lemma 1.1   If $(\Phi, D)$ is an information algebra, then for $\phi, \psi\in \Phi$ and $x, y\in D$,

(1) $\phi^{\Rightarrow x}\leq \phi$;

(2) $\phi\otimes\psi=\sup\{\phi, \psi\}$;

(3) $\phi\leq \psi$ implies $\phi^{\Rightarrow x}\leq\psi^{\Rightarrow x}$;

(4) $x\leq y$ implies $\phi^{\Rightarrow x}\leq\phi^{\Rightarrow y}$;

(5) $\phi_1\leq\phi_2$ and $\psi_1\leq\psi_2$ imply $\phi_1\otimes\psi_1\leq \phi_2\otimes\psi_2$.

Computers can treat only ``finite" information. Therefore, a new kind of structures named compact information algebras was introduced by Kohlas. The main characteristic of a compact information algebra is that the focusing of each piece of information $\phi$ on a set $x$ can be approximated by a family of finite information with the support set $x$, which is coarser than $\phi$.

Definition 1.2[3,7]   A system $(\Phi, \Phi_f, D)$, where $(\Phi, D)$ is an information algebra, the lattice $D$ has top element, $\Phi_f$ is closed under combination and contains the neutral element $e$, satisfying the three conditions of convergence, density and compactness formulated following, is called a compact information algebra:

1. Convergency: If $X\subseteq \Phi_f$ is a directed set, then the supremum $\vee X$ over $X$ exists and belongs to $\Phi$.

2. Density: For $\phi\in \Phi$ and $x\in D$, $\phi^{\Rightarrow x}=\vee\{\psi\in \Phi_f:\psi\leq \phi, \psi^{\Rightarrow x}=\psi\}.$

3. Compactness: If $X\subseteq \Phi_f$ is a directed set, and $\phi\in \Phi_f$ such that $\phi\leq \vee X$, then there exists a $\psi\in X$ such that $\phi\leq \psi$.

In [3], it was shown that, for a compact information algebra $(\Phi, \Phi_f, D)$, an information $\phi\in \Phi$ belongs to $\Phi_f$ if and only if $\phi\ll\phi$. More important properties on compact information algebras can be found in [3, 4, 7, 8]

In the following we provide two simple examples of compact information algebras.

Example 1.1   Let $L=[0, 1]$ and $D=\{0, 1\}$. $\Phi=Id\ L$ is the set of all nonempty ideals. The operations on $(\Phi, D)$ are defined as follows: $\forall I_1, I_2, I\in \Phi$,

$ \begin{array}{l} {\rm{combination}}: I_1\otimes I_2=\{x\in[0, 1]: x\leq a \ \mbox{for some}\ a\in I_1\cup I_2\};\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{focusing}}: I^{\Rightarrow 1}=I, I^{\Rightarrow 0}=\{0\}. \end{array} $

We can progressively show that $(\Phi, D)$ is an information algebra with these two operations. Let $I_x=\{y\in[1,0]: y\leq x\}$ and $\Phi_f=\{I_x: x\in [1,0]\}$. Then the set $\Phi_f$, which is closed under combination, satisfies the axioms of compactness. We can obtain that $(\Phi, \Phi_f, D)$ is a compact information algebra. Here we can refer to some conclusions shown in [13].

Example 1.2   For any set $X$, let the operations combination and focusing be defined as follows:

$ \begin{array}{l} {\rm{combination}}: \forall A, B\subseteq X, A\otimes B=A\cup B;\\ \;\;\;\;\;{\rm{focusing}}: \forall A, B\subseteq X, A^{\Rightarrow B}=A\cap B. \end{array} $

Then $({\cal P}(X), {\cal P}_f(X), {\cal P}(X))$ is a compact information algebra according to Definition 1.1, where ${\cal P}_f(X)$ is the set of all finite subsets of $X$.

In this section we give the method named compact extension for obtaining compact information algebras from general information algebras. Here we say a compact information algebra $(\Psi, D)$ is a compact extension of an information algebra $(\Phi, D)$, if there exists an injective $f: (\Phi, D)\rightarrow (\Psi, D)$ such that $f(\phi\otimes \psi)=f(\phi)\otimes f(\psi)$, $f(\phi^{\Rightarrow x})=[f(\phi)]^{\Rightarrow x}$, where $\phi, \psi\in\Phi$ and $x\in D$. We often call $f$ a embedding map.

Let ${\cal D}_\Phi$ denote the set of all the directed subsets of an information algebra $(\Phi, D)$. We define the operations combination and focusing on ${\cal D}_\Phi$ as follows: For all $A, B\in {\cal D}_\Phi$,

$ \begin{array}{l} {\rm{combination}}: A\otimes B=\{\phi\otimes \psi:\phi\in A, \psi\in B\};\\ \;\;\;\;\;{\rm{focusing}}: A^{\Rightarrow x} =\{\phi ^{\Rightarrow x}:\phi \in A\}. \end{array} $

Lemma 2.1  If $A, B\in {\cal D}_\Phi$, then $A\otimes B, A^{\Rightarrow x}\in {\cal D}_\Phi$.

Proof   Assume that $\phi_1\otimes \psi_1, \phi_2\otimes \psi_2\in A\otimes B$, where $\phi_i\in A, \psi_i\in B(i=1, 2)$. There exist $\phi\in A, \psi\in B$ such that $\phi_i\leq \phi, \psi_i\leq \psi$, since $A, B\in {\cal D}_\Phi$. Then $\phi_i\otimes \psi_i\leq \phi\otimes\psi \in A\otimes B$ by Lemma 1.1(5). Hence $A\otimes B\in {\cal D}_\Phi$.

By the directness of $A$ and Lemma 1.1(3), the directness of $A^{\Rightarrow x}$ also can be obtained easily.

An equivalence relation $\theta$ on ${\cal D}_\Phi$ is defined as follows: For $A, B\in {\cal D}_\Phi$,

$ \begin{array}{l} A \equiv B({\rm{mod}}\theta )\;{\rm{iff}}\;{\rm{for}}\;{\rm{any}}\;\phi \in A, {\rm{there}}\;{\rm{exists}}\;{\rm{a}}\;\psi \in B\;{\rm{such}}\;{\rm{that}}\;\phi \le \psi, \\ {\rm{and}}\;{\rm{meanwhile, for}}\;{\rm{any}}\;\beta \in B, {\rm{there}}\;{\rm{exists}}\;{\rm{a}}\;\alpha \in A\;{\rm{such}}\;{\rm{that}}\;\beta \le \alpha . \end{array} $

The characteristic of $\theta$ demonstrates that, in a sense, information content of $A$ and $B$ are equivalent. Next we define the operations combination and focusing on the quotient object ${\cal D}_\Phi/\theta$: For $A, B\in {\cal D}_\Phi$,

$ \begin{array}{l} {\rm{combination}}: [A]_{\theta}\otimes[B]_{\theta}=[A\otimes B]_{\theta};\\ \;\;\;\;\;{\rm{focusing}}: [A]_{\theta}^{\Rightarrow x} =[A^{\Rightarrow x}]_{\theta}. \end{array} $

The two operations defined as above are well-defined on ${\cal D}_\Phi/\theta$.

Lemma 2.2   The system $({\cal D}_\Phi/\theta, D)$ is an information algebra, where $(\Phi, D)$ is an information algebra and $D$ has a top element $\top$.

Proof  (1) Semigroup: It is easy to see that ${\cal D}_\Phi/\theta$ is associative and commutative under combination, since $(\Phi, D)$ is an information algebra. $[\{e\}]_\theta$ is the neutral element such that $[\{e\}]_\theta \otimes [A]_\theta =[A]_\theta$, where $ e\in\Phi $ is the neutral element of $(\Phi, D)$. Meanwhile, the axiom of neutrality also can be obtained easily.

(2) Combination: For $A, B\in {\cal D}_\Phi$ and $x\in D$, we have

$ \begin{eqnarray*} \begin{array}{lll} (A^{\Rightarrow x} \otimes B)^{\Rightarrow x} &=&\{(\phi^{\Rightarrow x}\otimes\psi)^{\Rightarrow x}:\phi\in A, \psi\in B\}\\ &=&\{\phi^{\Rightarrow x}\otimes \psi^{\Rightarrow x}:\phi\in A, \psi\in B\}\\ &=&\{\phi^{\Rightarrow x}:\phi\in A\}\otimes \{\psi^{\Rightarrow x}:\psi\in B\}\\ &=&A^{\Rightarrow x} \otimes B^{\Rightarrow x}. \end{array} \end{eqnarray*} $

Then $([A]_{\theta}^{\Rightarrow x} \otimes [B]_{\theta})^{\Rightarrow x}=([A^{\Rightarrow x} \otimes B]_{\theta})^{\Rightarrow x}$ $=[(A^{\Rightarrow x} \otimes B)^{\Rightarrow x}]_{\theta} =[A^{\Rightarrow x} \otimes B^{\Rightarrow x}]_{\theta}$ $=[A]_{\theta}^{\Rightarrow x} \otimes [B]_{\theta}^{\Rightarrow x}$.

(3) Transitivity: For $A\in {\cal D}_\Phi$ and $x, y \in D$,

$ \begin{eqnarray*} \begin{array}{lll} (A^{\Rightarrow x})^{\Rightarrow y} &=&\{\phi ^{\Rightarrow x}:\phi\in A\}^{\Rightarrow y}\\ &=&\{(\phi ^{\Rightarrow x})^{\Rightarrow y}:\phi\in A\}\\ &=&\{\phi^{\Rightarrow {x\wedge y}}:\phi\in A\}\\ &=& A^{\Rightarrow {x\wedge y}}. \end{array} \end{eqnarray*} $

Then $([A]_{\theta}^{\Rightarrow x})^{\Rightarrow y} =[(A^{\Rightarrow x})^{\Rightarrow y}]_{\theta} =[A^{\Rightarrow x \wedge y}]_{\theta} =[A]_{\theta}^{\Rightarrow x \wedge y}$.

(4) Support: For $ A\in {\cal D}_\Phi$, $A^{\Rightarrow \top}=A$. Then $[A]_{\theta} ^{\Rightarrow \top}=[A]_{\theta}$.

(5) Idempotency: For $ A\in {\cal D}_\Phi$ and $x\in D$, we need to verify that $A\otimes A^{\Rightarrow x}\equiv A({\mbox{mod}} \ \theta).$ In fact, for $\phi, \psi \in A$, since $A$ is directed, then there exists a $\varphi \in A$ such that $\phi, \psi\leq \varphi$. So $\phi\otimes\psi ^{\Rightarrow x}\leq \phi\otimes\psi\leq \varphi$. Meanwhile, for $\phi \in A$, we have $\phi=\phi\otimes\phi ^{\Rightarrow x}\in A\otimes A^{\Rightarrow x}$. Then $A\otimes A^{\Rightarrow x}\equiv A$ (mod $\theta)$ by the definition of $\theta$. Thus

$ [A]_{\theta}\otimes [A]_{\theta}^{\Rightarrow x} =[A\otimes A^{\Rightarrow x}]_{\theta}=[A]_{\theta}. $

Lemma 2.3   Let $A, B\in {\cal D}_\Phi$. Then, $[A]_{\theta}\leq [B]_{\theta}$ if and only if for any $\phi \in A$, there exists a $\psi \in B$ such that $\phi\leq \psi$.

Proof   If $[A]_{\theta}\leq [B]_{\theta}$, then $A\otimes B\equiv B$ (mod $\theta$). For any $\phi\in A$ and $\eta\in B$, by the definition of $\theta$, there exists a $\psi\in B$ such that $\phi\otimes\eta\leq \psi$. Then $\phi\leq\psi$.

Conversely, we need to show $A\otimes B\equiv B$(mod $\theta$). On one hand, let $\phi\in A$ and $\psi\in B$. By the assumption here, there exists a $\eta\in B$ such that $\phi\leq\eta$. Then $\phi\otimes\psi\leq\eta\otimes\psi$. Since $B$ is directed, there exists a $\gamma\in B$ such that $\eta, \psi\leq \gamma$. Thus $\phi\otimes\psi\leq \gamma$. On the other hand, for any $\psi\in B$ and $\phi\in A$, we have $\psi\leq\phi\otimes\psi\in A\otimes B$. Thus $A\otimes B\equiv B$(mod $\theta$). It implies that $[A]_{\theta}\leq [B]_{\theta}$.

Let $\Gamma=\{[\{\phi\}]_{\theta}:\phi\in \Phi\}$. To facilitate the writing, $[\{\phi\}]_{\theta}$ is simply written as $[\phi]_{\theta}$.

Theorem 2.1   The system $({\cal D}_\Phi/\theta, \Gamma, D)$ is a compact information algebra, where $(\Phi, D)$ is an information algebra and $D$ has a top element $\top$.

Proof   Clearly $[e]_\theta\in \Gamma$ is the empty information and $\Gamma$ is closed under combination. Following we show the axioms of convergence, density and compactness of $({\cal D}_\Phi/\theta, \Gamma, D)$. Then the conclusion is proved.

(1) Convergence: Assume that $\{[\phi_{i}]_{\theta}:i\in I\}\subseteq\Gamma$ is a directed subset, we claim that $\bigvee\limits_{i\in I}[\phi_{i}]_{\theta}=[E]_{\theta}$, where $E=\{\phi_{i_1}\otimes \phi_{i_2}\otimes \cdots \otimes \phi_{i_n}: i_1, i_2, \cdots, i_n\in I, n\in {\bf N} \}$. Obviously, $E$ is a directed subset of $\Phi$ and $[\phi_i]_{\theta}\leq [E]_{\theta}$ for all $i\in I$. Next suppose that $[A]_{\theta}$ is another upper bound of $\{[\phi_{i}]_{\theta}:i\in I\}$. For each $i\in I $, there exists a $\psi_i\in A$ such that $\phi_i\leq \psi_i$. Then

$ \phi_{i_1}\otimes \phi_{i_2}\otimes\cdots \otimes \phi_{i_n} \leq \psi_{i_1}\otimes \psi_{i_2}\otimes\cdots \otimes \psi_{i_n}. $

Since $A$ is directed, there is a $\psi\in A$ such that $\psi_{i_j}\leq \psi(j=1, 2, \cdots, n)$. Thus $\phi_{i_1}\otimes \phi_{i_2}\otimes\cdots \otimes \phi_{i_n}\leq \psi$. So $[E]_{\theta}\leq [A]_{\theta}$. We obtain that $\bigvee\limits_{i\in I}[\phi_{i}]_{\theta}=[E]_{\theta}$.

(2) Density: For $A\in {\cal D}_\Phi$ and $ x\in D$, we show that

$ [A]_{\theta}^{\Rightarrow x} =\vee\{[\phi]_{\theta}:[\phi]_{\theta}\leq [A]_{\theta}, [\phi]_{\theta}=[\phi]_{\theta}^{\Rightarrow x}\}. $

Let $\Theta=\{[\phi]_{\theta}:[\phi]_{\theta}\leq [A]_{\theta}, [\phi]_{\theta}=[\phi]_{\theta}^{\Rightarrow x}\}.$ For any $[\phi]_{\theta}\in\Theta$, there exists a $\psi\in A$ such that $\phi\leq \psi$. Meanwhile, by Lemma 2.3 we have $\phi=\phi^{\Rightarrow x}$ for $[\phi]_{\theta}\in \Theta$. Thus $\phi=\phi^{\Rightarrow x}\leq \psi^{\Rightarrow x}$. By Lemma 2.3 again, $[\phi]_{\theta}\leq [A]_{\theta} ^{\Rightarrow x}$. This implies that $[A]_{\theta}^{\Rightarrow x}$ is an upper bound of $\Theta$.

Assume that $[B]_{\theta}$ is another upper bound of $\Theta$. For $\psi\in A$, we have $[\psi^{\Rightarrow x}]_{\theta}\in \Theta$, because $\psi^{\Rightarrow x}=(\psi^{\Rightarrow x})^{\Rightarrow x}$ and $\psi^{\Rightarrow x}\leq \psi$. Then $[\psi^{\Rightarrow x}]_{\theta}\leq [B]_{\theta}$. By Lemma 2.3 there is a $\eta\in B$ such that $\psi^{\Rightarrow x}\leq \eta$. Thus we obtain that $[A]_{\theta}^{\Rightarrow x} \leq [B]_{\theta}$. So $[A]_{\theta}^{\Rightarrow x}=\vee \Theta$.

(3) Compactness: Suppose that $\{[\phi_{i}]_{\theta}:i\in I\}\subseteq\Gamma$ is a directed subset. Let $[\phi]_{\theta}\in \Gamma$ and $[\phi]_{\theta}\leq \vee\{[\phi_{i}]_{\theta}:i\in I\}$. By what we have proven in the convergency, there exists a set $\{i_1, i_2, \cdots, i_n\}\subseteq I$ such that $\phi\leq \phi_{i_1}\otimes \phi_{i_2}\otimes\cdots \otimes \phi_{i_n}$. Since $\{\phi_i: i\in I\}$ is directed, there exists a $j\in I$ such that $\phi\leq \phi_j$. So $[\phi]_{\theta}\leq [\phi_{j}]_{\theta}$. The compactness is true.

Theorem 2.2   The system $({\cal D}_\Phi/\theta, D)$ is a compact extension of $(\Phi, D)$, where the embedding map $f: (\Phi, D)\rightarrow ({\cal D}_\Phi/\theta, D)$ is defined as $f(\phi)=[\phi]_\theta$.

Proof   Let $\phi, \psi\in\Phi$. If $f(\phi)=f(\psi)$, then

$ [\phi]_\theta=[\psi]_\theta. $

By Lemma 2.3, we have $\phi\leq\psi$ and $\psi\leq\phi$. Thus $\phi=\psi$, that is, $f$ is injective. Also, we can show that $f$ preserves these two operations combination and focusing, and preserves the neutral element $e\in\Phi$ directly.

At last, let us see an example of compact extension of an information algebra.

Example 2.1    Let $\Phi=[0, 1]$, $D=\{0, 1\}$. The operations on $(\Phi, D)$ are defined as follows:

$ \begin{array}{l} {\rm{Combination}}: \forall \phi, \psi\in \Phi, \phi\otimes\psi=\max\{\phi, \psi\};\\ \;\;\;\;\;\;\;\;\;\;\;{\rm{Focusing}}: \forall \phi\in \Phi, \phi^{\Rightarrow 1}=\phi; \\ \;\;\;\;\;\;\;\;\;\;\; \phi^{\Rightarrow 0}=\left\{ \begin {array}{ll} \phi, &{\mbox{if}} \ \ \phi\in[0, \frac{1}{2}];\\ \frac{1}{2}, &{\mbox{if}} \ \ \phi\in[\frac{1}{2}, 1]. \end{array} \right. \end{array} $

We have $\phi\ll \psi$ if and only if $\phi<\psi$ or $\phi=\psi=0$. Then the tuple $(\Phi, D)$ is an information algebra but not compact [7].

Let $\theta$ be the equivalence relation defined as above. For any directed subset $\Upsilon$ of $[1,0]$ with a maximum element $\phi\in \Upsilon$, we have $\Upsilon\equiv \{\phi\}$(mod $\theta$). While, for any two directed subsets $\Upsilon_1$ and $\Upsilon_2$ with a same supremum $d$ and $d\not\in \Upsilon_1, \Upsilon_2$, we have $\Upsilon_1\equiv \Upsilon_2$(mod $\theta$). Then the quotient object

$ {\cal D}_\Phi/\theta=\{[\phi]_\theta: \phi\in [0, 1]\}\cup \{[\{\phi-\frac{1}{n}: n>[\frac{1}{\phi}], n\in {\bf Z} \}]_\theta: \phi\in[0, 1]\}, $

where $[x]$ denotes the largest integer not greater than $x$. The set ${\cal D}_\Phi/\theta$ has more finite elements than $\Phi$. In fact, according to the proof in Theorem 2.1, we have $[\phi]_\theta\ll [\phi]_\theta$ for all $\phi\in [0, 1]$. Hence $({\cal D}_\Phi/\theta, \Gamma, \{0, 1\})$ is compact clearly, where $\Gamma=\{[\phi]_\theta: \phi\in[0, 1]\}$.