Suppose now that the axiom is true in $A_{2}$. Since it is an universal
formula, then it is trivially true in $A_{1}$.

Consider now that Ax is the axiom $(C^{t}E)$
$a\overline{C}^{t}b\Rightarrow(\exists c)(a\overline{C}^{t}c$ and
$c^{*}\overline{C}^{t}b)$. Suppose first that $(C^{t}E)$ is true in $A_{1}$.
In order to show that it is true in $A_{2}$ suppose
$a_{2}\overline{C}^{t}b_{2}$. Then by the condition ($C^{t}$-separation) there
exist $a_{1},b_{1}$ such that $a_{1}\overline{C}^{t}b_{1}$, $a_{2}\leq a_{1}$
and $b_{2}\leq b_{1}$. By the assumption that $(C^{t}E)$ is true in $A_{1}$,
$a_{1}\overline{C}^{t}b_{1}$ implies that $(\exists
c_{1})(a_{1}\overline{C}^{t}c_{1}$ and $c_{1}^{*}\overline{C}^{t}b_{1}$). From
here we obtain $a_{2}\overline{C}^{t}c_{1}$ and
$c_{1}^{*}\overline{C}^{t}b_{2}$. Obviously $c_{1}$ and $c_{1}^{*}$ are in
$B_{A_{2}}$ which shows that $(C^{t}E)$ is true in $A_{2}$.

Suppose now that $(C^{t}E)$ is true in $A_{2}$ and in order to prove it in
$A_{1}$ suppose $a_{1}\overline{C}^{t}b_{1}$. Since $a_{1},b_{1}$ are also in
$B_{A_{2}}$, then by the assumption there is $c_{2}$ such that
$a_{1}\overline{C}^{t}c_{2}$ and $c_{2}^{*}\overline{C}^{t}b_{1}$. Then by the
condition ($C^{t}$-separation) applied to $a_{1}\overline{C}^{t}c_{2}$ there
exist $a_{1}^{\prime},c_{1}^{\prime}$ such that $a_{1}\leq a_{1}^{\prime}$,
$c_{2}\leq c_{2}\leq c_{1}^{\prime}$ and
$a_{1}^{\prime}\overline{C}^{t}c_{1}^{\prime}$. Analogously from
$c_{2}^{*}\overline{C}^{t}b_{1}$ we infer that there exist
$c_{1}^{\prime\prime},b_{1}^{\prime}$ such that $b_{1}\leq b_{1}^{\prime}$,
$c_{2}^{*}\leq c_{1}^{\prime\prime}$, $b_{1}\leq b_{1}^{\prime}$ and
$c_{1}^{\prime\prime}\overline{C}^{t}b_{1}^{\prime}$. Manipulating with
inequalities and monotonicity conditions for $C^{t}$ we finally obtain
$a_{1}\overline{C}^{t}c_{1}^{\prime}$ and
$c_{1}^{\prime*}\overline{C}^{t}b_{1}$ which shows that $(C^{t}E)$ holds in
$A_{1}$.

In a similar way one can treat the case for the axioms $(C^{t}\mathcal{B})$
and $(\mathcal{B}C^{t})$.

As an example we will treat one case for time axioms just to show that the
tings go in a similar way. We consider the axiom (lin) $a\not=0$ and
$b\not=0\Rightarrow a\mathcal{B}b$ or $b\mathcal{B}a$. Suppose first that
(lin) is true in $A_{1}$ and in order to show that it is true in $A_{2}$
suppose $a_{2}\not=0$ and $b_{2}\not=0$. Then by the condition (dence) there
exists $a_{1}\not=0$ such that $a_{1}\leq a_{2}$ and there exists
$b_{1}\not=0$ such that $b_{1}\leq b_{2}$. By the assumption $a_{1}\not=0$ and
$b_{1}\not=0$ imply $a_{1}\mathcal{B}b_{1}$ or $b_{1}\mathcal{B}a_{1}$. By
monotonicity conditions for $\mathcal{B}$ we get $a_{2}\mathcal{B}b_{2}$ or
$b_{2}\mathcal{B}a_{2}$ which finishes the proof for this direction. For the
converse direction suppose that (lin) is true in $A_{2}$. Since (lin) is an
universal sentence it trivially holds in the subalgebra $A_{1}$.∎

###  5.3 Dynamic Mereotopological Spaces (DMS)

######  Definition 5.3.

Dynamic Mereotopological Space. A system $S=(X^{t}_{S},X^{s}_{S},\newline
T_{S},\prec_{S},M_{S})$ is called Dynamic Mereotopological Space (DMS, DM-
space) if the next axioms are satisfied.

The axioms of DMS:

$\bullet$ (S1) $X^{t}_{S}$ is a nonempty topological space, the elements of
$X^{t}_{S}$ are called partial time points of $S$.

$\bullet$ (S2) $M_{S}$ is a subalgebra of the algebra $RC(X^{t}_{S})$ of
regular closed sets of $X^{t}_{S}$ and $M_{S}$ is a closed base of the
topology of $X^{t}_{S}$.

$\bullet$ (S3) The sets $X^{t}_{S}$, $X^{s}_{S}$ and $T_{S}$ are non-empty
sets satisfying the following inclusions:

$X^{s}_{S}\subseteq X^{t}_{S}$, $T_{S}\subseteq X^{t}_{S}$.

The elements of $X^{s}_{S}$ are called space points of $S$, hence every space
point is a partial time point. The elemnts of $T_{S}$ are called time points
of $S$.

$\bullet$ (S4) For $a\in RC(X^{t}_{S})$: if $a\not=\varnothing$, then $a\cap
X^{s}_{S}\not=\varnothing$ and

$\bullet$ (S5) $\prec_{S}$ is a binary relation in $X^{t}_{S}$ called before-
after relation. The subsystem $(T_{S},\prec_{S})$ is called the time structure
of $S$.

Definitions: For $a,b\in RC(X^{t}_{S})$ define:

> $aC^{t}_{S}b$ iff $a\cap b\not=\varnothing$, time contact,
>
> $aC^{s}_{S}b$ iff $a\cap b\cap X^{s}_{S}\not=\varnothing$, space contact,
>
> $a\mathcal{B}_{S}b$ iff there exist $x,y\in X_{S}^{t}$ such that
> $x\prec_{S}y$, $x\in a$ and $y\in b$, precedence,
>
> $RC(S)=_{def}(RC(X^{t}_{S}),C^{t}_{S},C^{s}_{S},\mathcal{B}_{S})$, regular-
> sets algebra of $S$,
>
> For $x\in X^{t}_{S}$ set $\rho_{S}(x)=_{def}\\{a\in M_{S}:x\in a\\}$.
>
> $S^{+}=_{def}(M_{S},C^{t}_{S},C^{s}_{S},\mathcal{B}_{S})$ with the above
> defined relations restricted to $M_{S}$.

It can easily be seen that $C^{s}_{S}$ and $C^{t}_{S}$ are contact relations
in $RC(X^{t}_{S})$ and that $\mathcal{B}$ is a precontact relation (for
$C^{s}_{S}$ use axiom (S4)).

$\bullet$ (S6) The system $S^{+}$ is a DCA. $S^{+}$ is called the canonical
DCA of $S$ or the dual of $S$.

$\bullet$ (S7) For $x,y\in X^{t}_{S}$, $x\prec_{S}y$ iff $(\forall a,b\in
M_{S})(x\in a,y\in b\Rightarrow a\mathcal{B}_{S}b)$.

$\bullet$ (S8) If $x\in T_{S}$ then $\rho_{S}(x)$ is a cluster in $S^{+}$,

We say that $S$ is a $T0$ space if $X^{t}_{S}$ is a $T0$ space.

Let $\widehat{Ax}$ be a subset of the time conditions from the list (RS),
(LS), (Up Dir), (Down Dir), (Circ), (Dens), (Ref), (Lin), (Tri), (Tr). We say
that $S$ satisfies the axioms from the list $\widehat{Ax}$ if the time
structure $(T_{S},\prec_{S})$ satisfies these conditions.

Intuitively DMS is abstracted from the clan-structure of DCA by introducing in
it a topology.

######  Lemma 5.4.

Let $S=(X^{t}_{S},X^{s}_{S},T_{S},\prec_{S},M_{S})$ be a DMS. Then:

(i) If $x\in X^{t}_{S}$, then $\rho_{S}(x)$ is a t-clan in $S^{+}$.

(ii) If $x\in X^{s}_{S}$, then $\rho_{S}(x)$ is an s-clan in $S^{+}$.

(iii) If $x\in T_{S}$, then $\rho_{S}(x)$ is a cluster in $S^{+}$.

(iv) Let $\prec_{S^{+}}$ be the canonical relation of $\mathcal{B}$ between
t-clans of $S^{+}$ (see (9) for the definition). Then Axiom (S7) of DMS is
equivalent to the following statement: for all $x,y\in X^{t}_{S}$,
$x\prec_{S}y$ iff $\rho_{S}(x)\prec_{S^{+}}\rho_{S}(y)$.

(v) $S$ is T0 space iff $(\forall x,y\in
X^{t}_{S})(\rho_{S}(x)=\rho_{S}({y})\Rightarrow x=y)$. (or, equivalently, $S$
is T0 iff $\rho_{S}$ is an injective mapping from $X^{t}_{S}$ into the t-clans
of $S^{+}$).

###### Proof.

For (i) and (ii) - by an easy verification of the corresponding definitions.
For (iii) this is just the axiom (S8) for DMS. (iv) is trivial on the base of
the definition of the relation $\prec_{M^{+}}$. (v) is easy if we take in
consideration the definition $T0$ property, the definition of $\rho_{s}$ and
the fact that $M_{S}$ is a closed base of the topology of $X^{t}_{S}$. ∎

######  Definition 5.5.

(1) A t-clan (s-clan, t-cluster) $\Gamma$ of $S^{+}$ is called a point t-clan
(s-clan, t-cluster) if there is a point $x\in X^{t}_{S}$ ($x\in X^{s}_{S}$,
$x\in T_{S}$) such that $\Gamma=\rho_{S}(x)$.

(2) $S$ is a DM-compact (dynamic mereoompact) space if every t-clan, s-clan
and t-cluster of $S^{+}$ is respectively a point t-clan, s-clan and a
t-cluster.

The following Lemma is obvious.

######  Lemma 5.6.

Let $S$ be a DMS. Then the following two conditions are equivalent:

(i) $S$ is DM-compact,

(ii) $\rho_{S}$ is a surjective mapping from $X^{t}_{S}$ onto the set of all
t-clans of $S^{+}$. More over $\rho_{S}$ maps $X^{s}_{S}$ onto the set of all
s-clans of $S^{+}$ and it maps $T_{S}$ onto the set of all clusters of
$S^{+}$.

######  Corollary 5.7.

Let $S$ be a T0 and DM-compact DMS. Then $\rho_{S}$ is a one-one mapping from
$X^{t}_{S}$ onto the set of all t-clans of $S^{+}$ which preserves the sets
$X^{s}_{S}$ and $T_{S}$.

###### Proof.

By Lemma 5.4 and Lemma 5.6 ∎

######  Remark 5.8.

The notions of DM-space and DM-compactness can be considered as dynamic
versions of the notions of mereotopological space and mereocompactness
introduced by Goldblatt and Grice in [34]. Their definitions are the
following. A mereotopological space is a pair $S=(X_{S},M_{S})$ where $X$ is a
topological space and $M_{S}$ is a subalgebra of the Boolean algebra
$RC(X_{S})$ of regular closed sets of $X_{S}$ considered as closed base of the
topology of $X$. Let $S^{+}$ be the contact algebra $(M_{S},C_{S})$ where
$C_{S}$ is the standard topological contact between regular closed sets. $S$
is mereocompact if every clan of the contact algebra $S^{+}$ is a point clan
in the sense of Definition 5.5 (in fact the definition of mereocompactness in
[34] is slightly different but equivalent to the given here). So, if
$S=(X^{t}_{S},X^{s}_{S},T_{S},\prec_{S},M_{S})$ is a DM-space then the pair
$(X^{t},M_{S})$ is a mereotopological space and if $S$ is DM-compact then
$(X^{t},M_{S})$ is mereocompact. Mereotopological spaces have been introduced
by Goldblatt and Grice in order to develop a topological duality theory for
contact algebras. Similarly, we introduce the notion of DM-space to be used in
the topological representation theory and duality theory of DCAs.

Because our exposition is quite similar to that of Goldblatt and Grice and in
some sense is an adaptation of their method to the case of DCAs, we recommend
the paper [34] to the reader of the present text. For convenience we even use
similar and compatible notations with [34].

######  Lemma 5.9.

Let $S$ be a DM-compact space. Then the topological space $X^{t}_{S}$ is
compact.

###### Proof.

According to Remark 5.8 DM-compactness of $S$ implies that the pair
$(X^{t}_{S},M_{S})$ is a mereocompact space and then the statement follows
from Theorem 4.2.(3) of [34]. We present below the proof illustrating our
definition of DM-compactness.

In order to prove the compactness of $X^{t}_{S}$, it suffices to prove the
following. Let $I\subseteq M_{S}$ be a nonempty set and let $A=\bigcap\\{a\in
M_{S}:a\in I\\}$. If for every finite $I_{0}\subseteq I$ the set
$\bigcap\\{a\in M_{S}:a\in I_{0}\\}\not=\varnothing$, then
$A\not=\varnothing$. The fact that $\bigcap\\{a\in M_{S}:a\in
I_{0}\\}\not=\varnothing$ for every finite subset $I_{0}$ of $I$ guarantees
the existence of an ultrafilter $U$ in the subset of all subsets of
$X^{t}_{S}$ such that $\\{a\in M_{S}:a\in I\\}\subseteq U$. Let
$\Gamma=\\{a\in M_{S}:a\in U\\}$. Then it is easy to see that $\Gamma$ is a
t-clan. Then by DM-compactness there exists $x\in X^{t}_{S}$ such that
$\Gamma=\rho_{S}(x)$. Hence for every $a\in I$ we have the following:

$a\in I$ $\Longrightarrow$ $a\in U$ $\Longrightarrow$ $a\in\Gamma$
$\Longrightarrow$ $a\in\rho_{S}(x)$ $\Longrightarrow$ $x\in a$
$\Longrightarrow$ $x\in A$ $\Longrightarrow$ $A\not=\varnothing$ ∎

######  Lemma 5.10.

Let $S=(X^{t}_{S},X^{s}_{S},T_{S},\gamma_{S},\prec_{S},M_{S})$ be a DM-compact
DMS. Then the set $X^{s}_{S}$ of space points of $S$ with a subset topology is
a $T0$ dense subset of $X^{t}_{S}$.

###### Proof.

Let $Cl$ denote the closure operation of $X^{t}_{S}$. We have to show that
$ClX^{s}_{S}=X^{t}_{S}$. Suppose that this is not true, i.e. there exists
$x\in X^{t}_{S}$ such that $x\not\in ClX^{s}_{S}$. Since $M_{S}$ is a closed
base of the topology of $X^{t}_{S}$ then there exits $a\in M_{S}$ such that
$X^{s}\subseteq a$ and $x\not\in a$. Then $a\not\in\rho_{S}(x)$, which is a
t-clan in $S^{+}$. Then for all ultrafilters $U\subseteq\rho_{S}(x)$ we have
that $a\not\in U$, and let $U$ be such one. But $U$ is an s-clan, so by DM-
compactness there is a point $y\in X^{s}_{S}$ such that $U=\rho_{S}(y)$.
Because $U\subseteq\rho_{S}(x)$ we obtain $\rho_{S}(y)\subseteq\rho_{S}(x)$.
From here we obtain that $a\not\in\rho_{S}(y)$ and consequently $y\not\in a$.
But $y\in X^{s}\subseteq a$, so $y\in a$ \- a contradiction.∎

######  Lemma 5.11.

([12], page 271) Let $X$ be a dense subspace of a topological space $Y$ and
let $RC(X)$ and $RC(Y)$ be the corresponding Boolean algebras of regular
closed sets of $X$ and $Y$. Let for $a\in RC(X)$, $h(a)=Cl_{Y}(a)$. Then
$h:RC(X)\rightarrow RC(Y)$ is an isomorphism from $RC(X)$ onto $RC(Y)$. For
$b\in RC(Y)$ converse mapping $h^{-1}$ acts as follows: $h^{-1}(b)=b\cap X$.

######  Corollary 5.12.

The Boolean algebra $RC(X^{s}_{S})$ of regular closed subsets of $X^{s}_{S}$
is isomorphic to the Boolean algebra $RC(X^{t}_{S})$.

###### Proof.

The lemma is a corollary of Lemma 5.10 and Lemma 5.11. ∎

In the next section we study some other consequences of DM-compactness.

###  5.4 Canonical filters in DM-compact spaces

We assume in this section that $S$ is a DM-compact space. The aim of the
section is to introduce a technical notion - _canonical filter_ , generalizing
a similar notion from [69]. By means of canonical filters and the assumption
of DM-compactness of a given $S$ we will establish that the algebra $S^{+}$ is
a stable subalgebra of $RC(S)$ in the sense of Definition 5.1 which fact
implies several important consequences.

######  Definition 5.13.

Let $A\in RC(X^{t}_{S})$. Then the set $F_{A}=_{def}\\{a\in M_{S}:A\subseteq
a\\}$ is called canonical filter of $S^{+}$.

######  Lemma 5.14.

Let $A,B\in RC(X^{t}_{S})$. Then:

(i) $F_{A}$ is a filter in $S^{+}$.

(ii) $\forall x\in X^{t}_{S}$: $x\in A$ iff $F_{A}\subseteq\rho_{S}(x)$.

(iii) $A\not=X^{t}_{S}$ iff there exists $a\in M_{S}$ such that $A\subseteq a$
and $a\not=X^{t}_{S}$.

Let $R^{t},R^{s},\prec$ be the canonical relations between filters
corresponding to the relations $C^{t}_{S},C^{s}_{S},\mathcal{B}_{S}$ from the
DCA algebra $S^{+}$.

(iv) The following conditions are equivalent:

(1.1) $AC^{t}_{S}B$. (1.2) $F_{A}R^{t}F_{B}$. (1.3) $A\cap B\cap
T_{S}\not=\varnothing$.

(v) The following conditions are equivalent:

(2.1) $AC^{s}_{S}B$. (2.2) $F_{A}R^{s}F_{B}$.

(vi) The following conditions are equivalent

(3.1) $A\mathcal{B}_{S}B$. (3.2) $F_{A}\prec F_{B}$. (3.3) There exist $x\in
A\cap T_{S}$ and $y\in B\cap T_{S}$ such that $x\prec_{S}y$.

###### Proof.

(i) The proof is by a direct checking the corresponding definitions.

(ii) The implication from left to right is by straightforward checking. For
the converse direction we will reason by contraposition. Suppose $x\not\in A$.
Now we will apply the fact that $M_{S}$ is a closed base of the topology of
$X$. Because $A$ is a regular closed set then $A$ is a closed set and then
there exists $a\in M_{S}$ such that $A\subseteq a$ and $x\not\in a$. Then
$a\in F_{A}$ and $a\not\in\rho_{S}(x)$, so $F_{A}\not\subseteq\rho_{S}(x)$.

(iii) can be derived by direct application of (ii).

(iv) (1.1)$\Rightarrow$(1.2) Suppose $AC^{t}_{S}B$. Then there is a point
$x\in X^{t}_{S}$ such that $x\in A$ and $x\in B$. By (ii) this implies

(1) $F_{A}\subseteq\rho_{S}(x)$ and

(2) $F_{B}\subseteq\rho_{S}(x)$.

In order to show $F_{A}\prec F_{B}$ suppose $a\in F_{A}$ and $b\in F_{B}$ and
proceed to show $F_{A}R^{t}F_{B}$. Then by (1) and (2) we get
$a\in\rho_{S}(x)$ and hence $x\in a$, and $b\in\rho_{S}(x)$ and hence $x\in
b$, which shows $a\cap b\not=\varnothing$. So, $aC^{t}_{S^{+}}b$ which proves
that $F_{A}R^{t}F_{B}$.

(1.2)$\Rightarrow$(1.3) Suppose $F_{A}R^{t}F^{b}$. By Lemma 2.9 there exist
ultrafilters $U,V$ such that $F_{A}\subseteq U$, $F_{B}\subseteq V$ and
$UR^{t}V$. Let $\Gamma=U\cup V$. Obviously $F_{A}\subseteq\Gamma$ and
$F_{B}\subseteq\Gamma$. By Lemma 2.15 $\Gamma$ as a union of $R^{t}$-related
ultrafilters is a t-clan in $S^{+}$ and then it can be extended into a cluster
$\Delta$. By DM-compactness there is $x\in T_{S}$ such that
$\Delta=\rho_{s}(x)$. Hence $F_{A}\subseteq\rho_{s}(x)$ and
$F_{B}\subseteq\rho_{s}(x)$. By (ii) $x\in A$ and $x\in B$ hence $A\cap B\cap
T_{S}\not=\varnothing$.

(1.3)$\Rightarrow$(1.1) Suppose $A\cap B\cap T_{S}\not=\varnothing$. Then
$A\cap B\not=\varnothing$, so $AC^{t}_{S}B$.

(v) the proof is similar to (iv)- it is used that if $\Gamma$ is an s-clan in
$S^{+}$ then by the DM-compactness there is point $x\in X^{s}_{S}$ such that
$\Gamma=\rho_{S}(x)$.

(vi) (3.1)$(\Rightarrow)$ (3.2) Suppose $A\mathcal{B}_{S}B$. Then there exist
$x\in A\cap X^{t}_{S}$ and $y\in B\cap X^{t}_{S}$ such that $x\prec_{S}y$.
Then by (ii) we obtain $F_{A}\subseteq\rho_{S}(x)$,
$F_{B}\subseteq\rho_{S}(y)$ and by Lemma 5.4 we have
$\rho_{S}(x)\prec\rho_{S}(y)$ and $\rho_{S}(x)$ and $\rho_{S}(y)$ are t-clans.
Then by the definition of $\prec$ in the set of t-clans we get $F_{A}\prec
F_{B}$.

(3.2)$(\Rightarrow)$ (3.3) Suppose $F_{A}\prec F_{B}$. Then by Lemma 2.9 there
are ultrafilters $U,V$ such that $F_{A}\subseteq U$, $F_{B}\subseteq V$ and
$U\prec V$. Ultrafilters are t-clans and we can extend them into clusters
preserving the relation $\prec$, namely: there exist clusters $\Gamma,\Delta$
such that $U\subseteq\Gamma$, $V\subseteq\Delta$ and $\Gamma\prec\Delta$. By
DM-compactness there are $x^{\prime},y^{\prime}\in T_{S}$ such that
$\Gamma=\rho_{S}(x^{\prime})$ and $\Delta=\rho_{S}(y^{\prime})$, so
$\rho_{S}(x^{\prime})\prec\rho_{S}(y^{\prime})$ and hence
$x^{\prime}\prec_{S}y^{\prime}$. We can obtain also
$F_{A}\subseteq\rho_{S}(x^{\prime})$ and hence $x^{\prime}\in A$, and
$F_{B}\subseteq\rho_{S}(y^{\prime})$ and hence $y^{\prime}\in B$. All this
says: $\exists x^{\prime}\in A\cap T_{S}$, $\exists y^{\prime}\in B\cap T_{S}$
such that $x^{\prime}\prec_{S}y^{\prime}$.

(3.3)$(\Rightarrow)$ (3.1). This implication is obvious because
$T_{S}\subseteq X^{t}_{S}$. ∎

Note that conditions (i), (ii) and (iii) of the above lemma does not depend on
the assumption of DM-compactness.

######  Lemma 5.15.

The following conditions are true for $S$:

(i) The algebra $S^{+}$ is a stable Boolean sub-algebra of $RC(S)$.

(ii) $RC(S)$ is a DCA.

###### Proof.

(i) We first show that $S^{+}$ satisfies the lifting conditions (see
Definition 5.1) and then (i) is a corollary of Lemma 5.2. First we verify the
lifting condition (co-dense). Suppose $A\in RC(X^{t}_{S})$ and
$A\not=X^{t}_{S}$. Then by Lemma 5.14 (iii) there exists $a\not=M_{S}$ such
that $a\not=X^{t}_{S}$ and $A\subseteq a$. We do not treat (dense) because it
is equivalent to (co-dense).

To verify the condition (C-separation) for
$C\in\\{C^{t}_{S},C^{s}_{S},\mathcal{B}_{S}\\}$ we proceed as follows. Looking
at the conditions (iv), (v), (vi) of Lemma 5.14 we see that they have the
following common form. Let $R$ be the canonical relation between filters
corresponding to the relation $C$. Then for any $A,B\in RC(X^{t}_{S})$: $ACB$
iff $F_{A}RF_{B}$. Taking the negation in both sides we obtain:
$A\overline{C}B$ iff $F_{A}\overline{R}F_{B}$ iff there exists $a,b\in M_{S}$
such that $a\in F_{A}$, $b\in F_{B}$ and $a\overline{C}b$ iff there exists
$a,b\in M_{S}$ such that $A\subseteq a$, $B\subseteq b$ and $a\overline{C}b$.
Thus: $F_{A}\overline{R}F_{B}$ implies that for some $a,b\in M_{S}$,
$A\subseteq a$, $B\subseteq b$ and $a\overline{C}b$ which is the
(C-separation) condition. Note that just this implication needed DM-
compactness in Lemma 5.14.

(ii) is a corollary of (i) and the fact that $S^{+}$ is a DCA, so by Lemma 5.2
the axioms $(C^{s}\subseteq C^{t})$, $(C^{t}E)$, $(C^{t}\mathcal{B}$ and
$(\mathcal{B}C^{t})$ are lifted from $S^{+}$ to $RC(S)$.∎

######  Lemma 5.16.

Let $(\varphi)$ be any of the time axioms: (rs), (ls), (up dir), (down dir),
(circ), (dens), (ref), (lin), (tri), (tr). Then the following conditions are
equivalent:

(i) $(\varphi)$ is true in the algebra $S^{+}$.

(ii) $(\varphi)$ is true in the algebra $RC(S)$.

###### Proof.

The proof follows from Lemma 5.15 (i) and Lemma 5.2.∎

######  Lemma 5.17.

Let $S$ be DM-compact DMS, $RC(S)$ be its regular-sets algebra,
$(T_{S},\prec_{S})$ be its time structure and let $(T_{S^{+}},\prec_{S^{+}})$
be the canonical time structure of $S^{+}$ (see Definition 4.8). Let $(\Phi)$
be the time condition from the list (RS), (LS), (Up Dir), (Down Dir), (Circ),
(Dens), (Ref), (Irr), (Lin), (Tr) ( condition (Tri) is excluded). Then the
following conditions are true:

(i) $(\Phi)$ is true in $(T_{S},\prec_{S})$ iff $(\Phi)$ is true in
$(T_{S^{+}},\prec_{S^{+}})$.

(ii) If $S$ is $T0$ DMS, then: (Tri) is true in $(T_{S},\prec_{S})$ iff (Tri)
is true in $(T_{S^{+}},\prec_{S^{+}})$.

###### Proof.

(i) Let us remind that the members of $T_{S^{+}}$ are clusters of $S^{+}$ ,
which we will denote by $\Gamma,\Delta,\Theta,...$. We will demonstrate the
proof considering the case (Dense), the proofs for the other cases go in the
same manner.

(Dense) $(\forall i,j)(i\prec j\Rightarrow(\exists k)(i\prec k$ and $k\prec
j)$.

($\Rightarrow$) Suppose (Dense) is true in $(T_{S},\prec_{S})$ and let
$\Gamma,\Delta\in T_{S^{+}}$ and $\Gamma\prec_{S^{+}}\Delta$. Then by DM-
compactness there exist $x,y\in T_{S}$ such that $\Gamma=\rho_{S}(x)$, and
$\Delta=\rho_{S}(y)$, so $\rho_{S}(x)\prec_{S^{+}}\rho_{S}(y)$. By Lemma 5.4
(iv) we obtain $x\prec_{S}y$ and by (Dence) there exists $z\in T_{S}$ such
that $x\prec_{S}z\prec_{S}y$. Again by Lemma 5.4 (iv) we obtain
$\rho_{S}(x)\prec_{S^{+}}\rho_{S}(z)\prec_{S^{+}}\rho_{S}(y)$. Because
$\rho_{S}(z)$ is a cluster in $S_{+}$ we put $\Theta=\rho_{S}(z)$ and obtain
$\Gamma\prec_{S^{+}}\Theta\prec_{S^{+}}\Delta$ which shows that (Dense) is
true in $(T_{S^{+}},\prec_{S^{+}})$.

($\Leftarrow$) Suppose (Dense) is true in $(T_{S^{+}},\prec_{S^{+}})$, $x,y\in
T_{S}$ and $x\prec_{S}y$. Then $\rho_{S}(x)\prec_{S^{+}}\rho_{S^{+}}(y)$. By
(Dence) there exists a cluster $\Theta$ (hence there exists $z\in T_{S}$ with
$\rho_{S}(z)=\Theta$) such that
$\rho_{S}(x)\prec_{S^{+}}\rho_{S}(z)\prec_{S^{+}}\rho_{S^{+}}(y)$. This
implies $x\prec_{S}z\prec_{S}y$ which shows that (Dense) is true in
$(T_{S},\prec_{S})$.

(ii) The case of (Tri) $(\forall i,j)(i=j$ or $i\prec j$ or $j\prec i$.

($\Rightarrow$) The proof of this implication is straightforward and requires
neither DM-compactness nor $T0$ property.

($\Leftarrow$) Suppose (Tri) is true in $(T_{S^{+}},\prec_{S^{+}})$ and let
$x,y\in T_{S}$. Then $\rho_{S}(x),\rho_{S}(y)$ are clusters in $S^{+}$. Then
by (Tri) we have $\rho_{S}(x)=\rho_{S}(y)$ or
$\rho_{S}(x)\prec_{S^{+}}\rho_{S}(y)$ or
$\rho_{S}(y)\prec_{S^{+}}\rho_{S}(x)$.

Case 1: $\rho_{S}(x)=\rho_{S}(y)$. Since $\rho_{S}(x)$ and $\rho_{S}(y)$ are
also t-clans then by the assumption that $S$ is a $T0$ space case 1 implies
$x=y$ (by Lemma 5.4 (v)).

Case 2: $\rho_{S}(x)\prec_{S^{+}}\rho_{S}(y)$. By Lemma 5.4 (iv) this implies
$x\prec_{S}y$.

Case 3: $\rho_{S}(y)\prec_{S^{+}}\rho_{S}(x)$. Again by Lemma 5.4 (iv) this
implies $y\prec_{S}x$. Thus, (Tri) is fulfilled in the time structure
$(T_{S},\prec_{S})$.∎

######  Lemma 5.18.

Topological definability. Let $(T_{S},\prec_{S})$ be the time structure of
$S$, $(\Phi)$ be the time condition from the list (RS), (LS), (Up Dir), (Down
Dir), (Circ), (Dens), (Ref), (Lin), (Tri) (Tr) and $(\varphi)$ be the
corresponding time axiom from the list (rs), (ls), (up dir), (down dir),
(circ), (dens), (ref), (lin), (tri), (tr). Then the following conditions are
equivalent (for the case of (Tri) we assume also that $S$ is $T0$):

(i) $(\Phi)$ is true in $(T_{S},\prec_{S})$

(ii) ($\varphi$) is true in $(RC)(S)$.

###### Proof.

$(\Phi)$ is true in $(T_{S},\prec_{S})$ iff (by Lemma 5.17) $(\Phi)$ is true
in the canonical time structure of $S^{+}$, $(T_{S^{+}},\prec_{S^{+}})$ iff
(by Lemma 4.9 ($\varphi$) is true in $S^{+}$ iff (by Lemma 5.16 ) $(\varphi)$
is true in the algebra $RC(S)$. ∎

###  5.5 Canonical DMS for DCA and topological representation
theorem for DCA

Let $A=(B_{A},C^{t}_{A},C^{s}_{A},\mathcal{B}_{A})$ be a DCA. We associate to
DCA in a canonical way a DM-space denoted by $A_{+}$ and called the canonical
DMS of A or the dual DMS of $A$ as follows:

$\bullet$ $A_{+}=_{def}(X^{t}_{A},X^{s}_{A},T_{A},\prec_{A},M_{A})$, where:

$\bullet$ $X^{t}_{A}=$t-$Clans(A)$, $X^{s}_{A}=$s-$Clans(A)$ and
$T_{A}=Clusters(A)$.

$\bullet$ $\prec_{A}$ is the before-after relation in the set $X^{t}_{A}$
defined by (9). The structure $(T_{A},\prec_{A})$ \- the time structure of $A$
is now the time structure of $A_{+}$.

$M_{A}$ is defined as follows and is used to introduce a topology in the set
$X^{t}_{A}$ considering it as a basis of the closed sets in the topology:

$\bullet$ For $a\in B_{A}$ let
$g_{A}(a)=\\{\Gamma\in$t-$Clans(A):a\in\Gamma\\}$ and put

$\bullet$ $M_{A}=\\{g_{A}(a):a\in B_{A}\\}$.

By the topological representation theory of contact algebras (see Section 2.5)
the set $\\{g_{A}(a):a\in B_{A}\\}$ defines a topology in the set $X^{t}_{A}$
and $g_{A}$ is an isomorphic embedding of $B_{A}$ into the algebra
$RC(X^{t}_{A})$ and $M_{A}$ is a Boolean subalgebra of $RC(X^{t}_{A})$
isomorphic to $B_{A}$.

We define the algebra $(A_{+})^{+}$ \- the dual of $A_{+}$ as follows.

$\bullet$
$(A_{+})^{+}=_{def}(M_{A},C^{t}_{A_{+}},C^{s}_{A_{+}},\mathcal{B}_{A_{+}})$.

Having in mind the topological representation theory of contact algebras (see
Section 2.5 and Lemma 4.6 it can be seen that $g_{A}$ is also an isomorphism
from $A$ onto $(A_{+})^{+}$, so $(A_{+})^{+}=g_{A}(B_{A})$ which proves the
following lemma.

######  Lemma 5.19.

$A$ is isomorphic to $(A_{+})^{+}$ and hence $(A_{+})^{+}$ is a DCA.

By definition we have $\rho_{A_{+}}=_{def}\\{g_{A}(a)\in M_{A}:\Gamma\in
g_{A}(a)\\}=\\{g_{A}(a)\in M_{A}:a\in\Gamma\\}$.

######  Lemma 5.20.

(i) For any $\Gamma\in X^{t}_{A}$ $\rho_{A_{+}}(\Gamma)$ is a t-clan in
$(A_{+})^{+}$.

(ii) For any $\Gamma\in X^{s}_{A}$ $\rho_{A_{+}}(\Gamma)$ is a s-clan in
$(A_{+})^{+}$.

(iii) For any $\Gamma\in T_{A}$ $\rho_{A_{+}}(\Gamma)$ is a cluster in
$(A_{+})^{+}$.

###### Proof.

The proof is by a routine verification of the corresponding definitions and
using the results about the clan structure of DCA developed in Section 4.1. As
an example we will demonstrate the proof of (iii).

Let $\Gamma\in T_{A}$. Then $\Gamma$ is a cluster in $A$, so $\Gamma$ is a
t-clan in $A$. By (i) $\rho_{A_{+}}(\Gamma)$ is a t-clan in $(A_{+})^{+}$. We
will show that $\rho_{A_{+}}(\Gamma)$ is a cluster in $(A_{+})^{+}$. Suppose
that for some $a\in B_{A}$, $g_{A}(a)\not\in\rho_{A_{+}}(\Gamma)$. Then
$\Gamma\not\in g_{A}(a)$, so $a\not\in\Gamma$. Then there exists $b\in B_{A}$
such that $b\in\Gamma$ and $a\overline{C}_{A}b$. Then
$g_{A}(b)\in\rho_{A_{+}}(\Gamma)$ and $g_{A}(a)\cap g_{A}(b)=\varnothing$, so
$g_{A}(a)\overline{C^{t}}_{A_{+}}g_{A}(b)$. Note that (iii) verifies the DMS
axiom (S7) for $A_{+}$. ∎

######  Lemma 5.21.

Let $\Gamma,\Delta$ be t-clans in $A$. Then: $\Gamma\prec_{A}\Delta$ iff for
all $g_{A}(a),\newline g_{A}(b)\in M_{A}$: if $\Gamma\in g_{A}(a)$ and
$\Delta\in g_{A}(b)$, then $g_{A}(a)\mathcal{B}_{(A_{+})^{+}}g_{A}(b)$.

###### Proof.

Let $\Gamma,\Delta$ be t-clans in $A$. Having in mind the relevant definitions
the implication from left to the right is obvious. For the converse
implication suppose that

($\sharp$) For all $g_{A}(a),g_{A}(b)\in M_{A}$, the conditions $\Gamma\in
g_{A}(a)$ and $\Delta\in g_{A}(b)$ imply
$g_{A}(a)\mathcal{B}_{(A_{+})^{+}}g_{A}(b)$

and proceed to show $\Gamma\prec_{A}\Delta$. By (9) this means that for some
$a\in\Gamma$ and $b\in\Delta$ we have $a\mathcal{B}_{A}b$. To this end suppose
$a\in\Gamma$ and $b\in\Delta$. Then $\Gamma\in g_{A}(a)$ and $\Delta\in
g_{A}(b)$. By ($\sharp$) we get $g_{A}(a)\mathcal{B}_{(A_{+})^{+}}g_{A}(b)$
which by the definition of $\mathcal{B}_{(A_{+})^{+}}$ means that for some
t-clans $\Gamma^{\prime},\Delta^{\prime}$ in $A$ we have $\Gamma^{\prime}\in
g_{A}(a)$, $\Delta^{\prime}\in g_{A}(b)$ and
$\Gamma^{\prime}\prec_{A}\Delta^{\prime}$. This implies $a\in\Gamma^{\prime}$
and $b\in\Delta^{\prime}$ and by the definition of
$\Gamma^{\prime}\prec_{A}\Delta^{\prime}$ (see (9)) that $a\mathcal{B}_{A}b$
\- end of the proof. Note that this lemma verifies the DMS axiom (S7) for
$A_{+}$. ∎

######  Lemma 5.22.

Let $A$ be a DCA and $\Gamma\subseteq M_{A}$. Define
$\widehat{\Gamma}=_{def}\\{a\in B_{A}:g_{A}(a)\in\Gamma\\}$. Then the
following conditions are true:

(i) If $\Gamma$ is a t-clan in $(A_{+})^{+}$, then $\widehat{\Gamma}$ is a
t-clan in $A$ and $\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$.

(ii) If $\Gamma$ is an s-clan in $(A_{+})^{+}$, then $\widehat{\Gamma}$ is an
s-clan in $A$ and $\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$.

(iii) If $\Gamma$ is a cluster in $(A_{+})^{+}$, then $\widehat{\Gamma}$ is a
cluster in $A$ and $\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$.

###### Proof.

(i) Let $\Gamma$ be a t-clan in $(A_{+})^{+}$. The verification of grill
properties of $\widehat{\Gamma}$ is easy. Let us prove the t-clan property.
Suppose $a,b\in\widehat{\Gamma}$. Then $g_{A}(a),g_{A}(b)\in\Gamma$. Then
$(g_{A}(a))C^{t}_{(A_{+})^{+}}((g_{A}(b))$. By the definition of
$C^{t}_{(A_{+})^{+}}$ we have $(g_{A}(a))\cap((g_{A}(b))\not=\varnothing$. So
there exists $\Gamma\in t-Clans(A)$ such that $a,b\in\Gamma$, which implies
$aC^{t}_{A}b$.

Let us show the equality $\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$. The
following sequence of equivalencies proves this:

$g_{A}(a)\in\rho_{A_{+}}(\widehat{\Gamma})$ iff $\widehat{\Gamma}\in g_{A}(a)$
iff $a\in\\{b\in B_{A}:g_{A}(b)\in\Gamma\\}$ iff $g_{A}(a)\in\Gamma$.

(ii) Let $\Gamma$ be an s-clan in $(A_{+})^{+}$. We will show the s-clan
property of $\widehat{\Gamma}$. Suppose that $a,b\in\widehat{\Gamma}$. Then
$(g_{A}(a))C^{s}_{(A_{+})^{+}}((g_{A}(b))$. By the definition of
$C^{s}_{(A_{+})^{+}}$, there exists $\Gamma\in s-Clans(A)$ such that
$\Gamma\in g_{A}(a)\cap g_{A}(b)$. This implies $a,b\in\Gamma$ and
consequently $aC^{s}b$. The proof of $\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$
is as in (i).

(iii) Let $\Gamma$ a cluster in $(A_{+})^{+}$. So by it is a t-clan in
$(A_{+})^{+}$ and by (i) $\widehat{\Gamma}$ is a t-clan in $A$. We will show
that $\widehat{\Gamma}$ is a cluster in $A$. Suppose $a\not
in\widehat{\Gamma}$. Then $g_{A}(a)\not\in\Gamma$, hence there exists
$g_{A}(b)\in M_{A}$ such that $g_{A}(b)\in\Gamma$ and
$g_{A}(a)\overline{C^{t}}_{(A_{+})^{+}}g_{A}(b)$. This implies
$b\in\widehat{\Gamma}$ and $g_{A}(a)\cap g_{A}(b)=\varnothing$ which gives
$a\overline{C^{t}}_{A}b$. This shows the cluster property of
$\widehat{\Gamma}$. The proof of $\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$ is as
in (i). ∎

######  Lemma 5.23.

Let $A$ be a DCA, $A_{+}=(X^{t}_{A},X^{s}_{A},T_{A},\prec_{A},M_{A})$ be its
dual space, $\alpha\in RC(X^{t}_{A})$ and $\alpha\cap
X^{t}_{A_{+}}\not=\varnothing$. Then $\alpha\cap
X^{s}_{A_{+}}\not=\varnothing$.

###### Proof.

Suppose $\alpha\in RC(X^{t}_{A})$ and $\alpha\cap X^{t}_{A}\not=\varnothing$.
(remind that $X^{t}_{A}=t-Clans(A)$ and $X^{s}_{A}=s-Clans(A)$). Then there
exists $\Gamma\in X^{t}_{A}$ such that $\Gamma\in\alpha$. Let $F_{\alpha}$ be
the canonical filter of $\alpha$ (see Section 5.14). Then by Lemma 5.14 (ii)
we have $F_{\alpha}\subseteq\rho_{A_{+}}(\Gamma)$. By Lemma 5.20
$\rho_{A_{+}}(\Gamma)$ is a t-clan in $(A_{+})^{+}$. Then there exists an
ultrafilter $U$ in $(A_{+})^{+}$ such that $F_{\alpha}\subseteq
U\subseteq\rho_{A_{+}}(\Gamma)$. $U$ is both a t-clan and an s-clan in
$(A_{+})^{+}$. By Lemma 5.22 (ii) there exists an s-clan $\widehat{U}$ such
that $U=\rho_{A_{+}}(\widehat{U})$. So, we have
$F_{\alpha}\subseteq\rho_{A_{+}}(\widehat{U})$ and $\widehat{U}\in X^{s}_{A}$.
Again by Lemma 5.14 (ii) we get $\widehat{U}\in\alpha$ and consequently
$\alpha\cap X^{s}_{A}\not=\varnothing$. ∎

Note that we have used Lemma 5.14 which presupposes DM-compactness. But as it
was mentioned after the proof of this lemma condition (ii) which we used does
not depend on DM-compactness. Note also that the above lemma verifies the DMS
axiom (S4) for $A_{+}$.

######  Lemma 5.24.

$A_{+}$ is a DMS.

###### Proof.

The proof follows from Lemma 5.20, Lemma 5.23, Lemma 5.21 and Lemma 5.19 which
establish the DMS axioms (S4), (S6), (S7) and (S8) for $A_{+}$. The other
axioms are obviously true. ∎

The following theorem is important.

######  Theorem 5.25.

$A_{+}$ is T0 and DM-compact DMS.

###### Proof.

By Lemma 5.4(v) $A_{+}$ has $T0$ property iff for every two members
$\Gamma,\Delta$ of $X^{t}_{A}(=$t-$Clans(A)$) the following holds: if
$\rho_{A_{+}}(\Gamma)=\rho_{A_{+}}(\Delta)$, then $\Gamma=\Delta$. Suppose
$\rho_{A_{+}}(\Gamma)=\rho_{A_{+}}(\Delta)$ and for the sake of contradiction
that $\Gamma\not=\Delta$, so $\Gamma\not\subseteq\Delta$ or
$\Delta\not\subseteq\Gamma$. Considering the first case this means that there
exists $a$ such that $a\in\Gamma$ and $a\not\in\Delta$.Then by Lemma 5.20
$g_{A}(a)\in\rho_{A_{+}}(\Gamma)$ and $g_{A}(a)\not\in\rho_{A_{+}}(\Delta)$
which shows that $\rho_{A_{+}}(\Gamma)\not=\rho_{A_{+}}(\Delta)$ \- a
contradiction. In a similar way the second case also implies a contradiction.

For DM-compactness we have to show the following three things:

(i) Every t-clan $\Gamma$ of $(A_{+})^{+}$ is a point t-clan,

(ii) Every s-clan of $(A_{+})^{+}$ is a point s-clan,

(iii) Every cluster of $(A_{+})^{+}$ is a point cluster.

Proof of (i). Let $\Gamma$ be a t-clan of $(A_{+})^{+}$. To show that $\Gamma$
is a point t-clan we have to find $\Delta\in X^{t}_{A}$ (= t-Clans(A)) such
that $\Gamma=\rho_{A_{+}}(\Delta)$. Let $\Delta=\widehat{\Gamma}=\\{a\in
B_{A}:g_{A}(a)\in\Gamma\\}$. By Lemma 5.22 (i) $\widehat{\Gamma}$ is a t-clan
in $A$ and hence it is in $X^{t}_{A}$. More over we have
$\rho_{A_{+}}(\widehat{\Gamma})=\Gamma$.

The proofs of (ii) and (iii) are similar by using Lemma 5.22 (ii) and (iii). ∎

######  Theorem 5.26.

Topological representation theorem for DCA. Let $A$ be a DCA. Then the
following conditions for $A$ are true:

(i) $(A_{+})^{+}$ is a stable subalgebra of the algebra $RC(A_{+})$.

(ii) The algebra $RC(A_{+})$ is a DCA.

(iii) The function $g_{A}$ is a stable isomorphic embedding of $A$ into
$RC(A_{+})$.

(iv) If Ax is a time axiom, then Ax is true in $A$ iff Ax is true in
$RC(A_{+})$.

###### Proof.

(i) By Theorem 5.25 $A_{+}$ is a DM-compact DMS and hence by Lemma 5.15 (i)
$(A_{+})^{+}$ is a stable Boolean subalgebra of $RC(A_{+})$.

(ii) follows from (i) and Lemma 5.15 (ii).

(iii) By Lemma 5.19 $g_{A}$ is an isomorphism from $A$ onto $(A_{+})^{+}$ and
hence by (i) $g_{A}$ is a stable isomorphic embedding of $A$ into $RC(A_{+})$.

(iv) follows from Lemma 5.25 and Lemma 5.16. ∎

###  5.6 Contact algebra as a special case of dynamic contact algebra

Let $A=(B_{A},C_{A})$ be a contact algebra. By Lemma 4.2 $A$ can be considered
as a DCA algebra on the base of the following definable relations: $a,b\in
B_{A}$:

(1) $aC^{t}_{A}b\Leftrightarrow_{def}aC^{max}_{A}b\Leftrightarrow a\not=0$ and
$b\not=0$.

(2) $a\mathcal{B}_{A}b\Leftrightarrow_{def}aC_{A}^{t}b$.

(3) $aC^{s}_{A}b\Leftrightarrow_{def}aC_{A}b$.

Let $A=(B_{A},C^{s}_{A},C^{t}_{A},\mathcal{B}_{A})$ be a DCA which satisfies
the above conditions. Then it is obviously equivalent to the contact algebra
$(B_{A},C_{A})$. Condition (3) is just giving another name of $C^{s}_{A}$, and
conditions (1) and (2) can be relaxed correspondingly to the following:

$(1^{\prime})$ If $a\not=0$ and $b\not=0$, then $aC^{t}_{A}b$,

$(2^{\prime})$ If $a\not=0$ and $b\not=0$, then $a\mathcal{B}_{A}b$.

Obviously $(1^{\prime})$ implies (1) and $(2^{\prime})$ implies (2). Hence if
a DCA satisfies $(1^{\prime})$ and $(2^{\prime})$, t5hen it is equivalent to
the contact algebra $(B_{A},C^{s})$. Condition (1) then makes t-clans to
coincide with grills, and in this case to have only one cluster, denote it by
$t_{0}$ (the only time point of $A$ which is just the union of all
ultrafilters in $A$). Condition (2) implies that $\mathcal{B}_{A}=C^{t}_{A}$
which makes the relation $\prec_{A}$ to be the universal relation between
grills and especially for $t_{0}$ to have $t_{0}\prec_{A}t_{0}$. This suggests
the following formal definition.

######  Definition 5.27.

We say that $A$ is a trivial DCA if it satisfies the conditions $(1^{\prime})$
and $(2^{\prime})$.

Thus for the dual space $A_{+}$ of a trivial DCA we have that
$T_{A}=\\{t_{0}\\}$ is a singleton set and that $t_{0}$ is the only time point
of $A$. This suggests to consider this as a characteristic property of a DMS
corresponding in some sense to a trivial DCA and to adopt the following formal
definition.

######  Definition 5.28.

We say that $S$ is a trivial DMS if the set $T_{S}=\\{t_{0}\\}$ is a singleton
with a single time point $t_{0}$ and $t_{0}\prec_{S}t_{0}$

######  Lemma 5.29.

Let $S$ be a $T0$ and DM-compact space. Then the following two conditions are
equivalent:

(i) $S$ is trivial DMS.

(ii) The dual algebra $S^{+}$ is a trivial DCA.

###### Proof.

. (i)$\Rightarrow$(ii). Suppose that $S$ is trivial DMS. First we will show
that the DCA algebra $S^{+}$ has at most one cluster. Note that it has
clusters. Let $\Gamma,\Delta$ be two clusters. By DM-compactness there is
$x\in T_{S}$ such that $\rho_{S}(x)=\Gamma$ and $y\in T_{S}$ such that
$\rho_{S}(y)=\Delta$. But $T_{S}$ is a singleton, so $x=y$ which implies
$\Gamma=\rho_{S}(x)=\rho_{S}(y)=\Delta$. So we have only one cluster, say
$\Gamma_{0}$.

In order to show (ii) it is sufficient that the following is true for
arbitrary regular closed sets $\alpha,\beta\in RC(X^{t}_{S}$:

If $\alpha\not=\varnothing$ and $\beta\not=\varnothing$, then $\alpha
C^{t}_{S}\beta$ and then $\alpha\mathcal{B}_{S}\beta$.

Suppose $\alpha\not=\varnothing$ and $\beta\not=\varnothing$, then there exist
$x\in\alpha$ and $y\in\beta$. Now we will apply the properties of canonical
filters (see Lemma 5.14 from Section 5.4). Conditions $x\in\alpha$ and
$y\in\beta$ imply $F_{\alpha}\subseteq\rho_{S}(x)$ and
$F_{\beta}\subseteq\rho_{S}(y)$. $\rho_{S}(x)$ and $\rho_{S}(y)$ are t-clans
in $S^{+}$ and can be extended into clusters. But there is only one cluster
$\Gamma_{0}=\rho_{S}(z)$ for some $z\in T_{S}$. Hence
$F_{\alpha}\subseteq\rho_{S}(z)$ and $F_{\beta}\subseteq\rho_{S}(z)$. Then by
the properties of canonical filters we get $z\in\alpha$ and $z\in\beta$, so
$\alpha\cap\beta\not=\varnothing$ which shows $\alpha C^{t}_{S}\beta$. Because
$z$ is the only element of $T_{S}$ we have $z\prec_{S}z$ which also shows that
$\alpha\mathcal{B}_{S}\beta$.

(ii)$\Rightarrow$(i) Let $S^{+}$ be a trivial DCA. We mentioned that the
condition (1) makes t-clans to coincide with grills. Because there exists only
one maximal grill - the union of all ultrafilters, then there exists only one
cluster, say $\Gamma_{0}$. By DM-compactness there exists $x\in T_{S}$ such
that $\rho_{S}(x)=\Gamma_{0}$. We will show that $T_{S}$ is a singleton.
Suppose that $y\in T_{S}$. By axiom S8 of DMS $\rho_{S}(y)$ is a cluster an
because we have only one cluster $\Gamma_{0}$ we have
$\rho_{S}(y)=\Gamma_{0}$. So $\rho_{S}(x)=\rho_{S}(y)$. Because $S$ is a $T0$
space this equality implies $x=y$. ∎

######  Theorem 5.30.

New topological representation theorem for contact algebras.  Let
$A=(B_{A},C_{A})$ be a contact algebra. Consider it as a trivial DCA. Then the
following conditions are true.

(i) The regular set-algebra $RC(A_{+})$ is a trivial DCA.

(ii) The function $g_{A}$ is a stable isomorphic embedding of $A$ into
$RC(A_{+})$.

###### Proof.

The Theorem is a consequence of of Theorem 5.26 \- Topological representation
theorem for DCA. Condition (iii) of the theorem says that the function $g_{A}$
is a stable isomorphic embedding of $A$ into $RC(A_{+})$. This proves our
condition (ii). Let us note that it is easy to see that the lifting Lemma 5.2
is true for the formulas ($1^{\prime}$) and ($2^{\prime}$). This implies that
the conditions ($1^{\prime}$) and ($2^{\prime}$) are true in $RC(A_{+})$, so
$RC(A_{+})$ is a trivial DCA and this proves our condition (i). ∎

##  6 Topological duality theory for DCA

In this section we extend the topological representation of DCAs to a
topological duality theory of DCAs in terms of DMSes. We assume basic
knowledge of category theory: categories, morphisms, functors and natural
isomorphisms (see, for instance, Chapter I from [52]). Since DCA is a
generalization of contact algebra, and DMS is a generalization of
mereotopological space, the developed duality theory in this section will
generalize the duality theory for contact algebras and mereotopological spaces
presented by Goldblatt and Griece in [34] and some proofs below will be the
same as in [34]. Other topological dualities for contact and precontact
algebras are presented in [20] and it is possible to generalize them for DCAs,
but in this paper we follow the scheme of [34] for two purposes: first,
because the corresponding notion of DMS fits quite well to the topological
representation theory for DCS-s, and second, because the proofs in this case
are more short.

###  6.1 The categories DCA and DMS

######  Definition 6.1.

The category DCA consists of the class of all DCAs supplied with the following
morphisms, called DCA-morphisms.

Let $A_{i}=(B_{A_{i}},C^{s}_{A_{i}},C^{t}_{A_{i}},\mathcal{B}_{A_{i}})$,
$i=1,2$ be two DCAs. Then $f:A_{1}\longrightarrow A_{2}$ is a DCA-morphism if
it is a mapping $f:B_{A_{1}}\longrightarrow B_{A_{2}}$ which satisfies the
following conditions:

(f 1) $f$ is a Boolean homomorphism from $B_{A_{1}}$ into $B_{A_{2}}$.

For all $a,b\in B_{A_{1}}$:

(f 2) if $f(a)C_{A_{2}}^{s}f(b)$, then $aC^{s}_{A_{1}}b$,

(f 3) if $f(a)C_{A_{2}}^{t}f(b)$, then $aC^{t}_{A_{1}}b$,

(f 4) if $f(a)\mathcal{B}_{A_{2}}^{s}f(b)$, then $a\mathcal{B}^{s}_{A_{1}}b$.

$A_{1}$ is the domain of $f$ and $A_{2}$ the codomain of $f$.

We define $f_{+}=_{def}f^{-1}$ acting on t-clans of $A_{2}$ as follows: for
$\Gamma\in t$-$Clans(A_{2})$, $f^{-1}(\Gamma)=_{def}\\{a\in
B_{A_{1}}:f(a)\in\Gamma\\}$.

A DCA-morphism $f:A_{1}\longrightarrow A_{2}$ is a DCA-isomorphism (in the
sense of category theory) if there is a DCA-morphism $g:A_{2}\longrightarrow
A_{1}$ such that the compositions $f\circ g$ and $g\circ f$ are the identity
morphism of their domains. It is a well known fact that this definition is
equivalent to the standard algebraic definition of isomorphism in universal
algebra.

######  Definition 6.2.

The category DMS consists of the class of all DMSes equipped with suitable
morphisms called DMS morphism. The definition is as follows. Let
$S_{i}=(X^{t}_{S_{i}},X^{s}_{S_{i}},T_{S_{i}},\prec_{S_{i}},M_{S_{i}})$,
$i=1,2$ be two DMSes. A DMS-morphism is a mapping

$\theta$: $X^{t}_{S_{1}}\longrightarrow X^{t}_{S_{2}}$ such that:

($\theta$ 1) if $x\in X^{s}_{S_{1}}$, then $\theta(x)\in X^{s}_{S_{2}}$,

($\theta$ 2) If $x\prec_{S_{1}}y$, then $\theta(x)\prec_{S_{2}}\theta(y)$.

Let $a\subseteq X^{t}_{S_{2}}$ and $\theta^{-1}(a)=_{def}\\{x\in
X^{t}_{S_{1}}:\theta(x)\in a\\}$. We define $\theta^{+}=_{def}\theta^{-1}$.

The next two requirements for $\theta$ are the following:

($\theta$ 3) If $a\in M_{S_{2}}$ then $\theta^{-1}(a)\in M_{S_{1}}$ and

($\theta$ 4) the map $\theta^{-1}:M_{S_{2}}\longrightarrow M_{S_{1}}$ is a
Boolean algebra homomorphism from $(M_{2})$ into $(M_{1})$.

Note that in $M_{S}$ the join operation is a set theoretical union of regular
closed sets. Since meets in Boolean algebra is definable by the join and the
complement *, for the condition ($\theta$ 4) it is sufficient to assume that
$\theta^{-1}$ preserves complement.

A DMS-morphism $\theta:S_{1}\longrightarrow S_{2}$ is a DMS-isomorphism if
there exists a converse DMS-morphism $\eta:S_{2}\longrightarrow S_{1}$ such
that the compositions $\theta\circ\eta$ and $\eta\circ\theta$ are identity
morphisms in the corresponding domains.

The following lemma states an equivalent definition of DMS-isomorphism.
Similar statement for mereotopological isomorphism is Theorem 2.2 from [34].

######  Lemma 6.3.

Let $S,S^{\prime}$ be DM-spaces and $\theta:S\mapsto S^{\prime}$ be a DMS-
morphism from $S$ into $S^{\prime}$. Let $a\subseteq X^{t}_{S}$ and define
$\theta[a]=\\{\theta(x):x\in a\\}$. Then the following two conditions are
equivalent:

(i) $\theta$ is a DMS-isomorphism from $S$ onto $S^{\prime}$.

(ii) $\theta$ is a DMS-morphism which is is a bijection from $X^{t}_{S}$ onto
$X^{t}_{S^{\prime}}$ satisfying the following conditions:

(1) If $\theta(x)\in X^{s}_{S^{\prime}}$, then $x\in X^{s}_{S}$.

(2) If $\theta(x)\prec_{S^{\prime}}\theta(y)$, then $x\prec_{S}y$.

(3) If $a\in M_{S}$, then $\theta[a]\in M_{S^{\prime}}$.

###### Proof.

(i)$\Rightarrow$(ii) Suppose that $\theta$ is a DMS isomorphism from $S$ onto
$S^{\prime}$. Then obviously $\theta$ is a bijection with converse $\eta$ such
that $\theta$ is a DMS-morphisms from $S$ onto $S^{\prime}$ and $\eta$ is a
DMS-morphism from $S^{\prime}$ onto $S$ such that the composition
$\theta\circ\eta$ is the identity in $S^{\prime}$ and $\eta\circ\theta$ is the
identity in $S$. To show (1) let $\theta(x)\in X^{s}_{S^{\prime}}$. Then
$x=\eta(\theta(x))\in X^{s}_{S}$, because $\eta$ is a DMS-morphism from
$S^{\prime}$ onto $S$. In a similar way we show (2). To show (3) let $a\in
M_{S}$. Then $\eta^{-1}(a)\in M_{S^{\prime}}$, because $\eta$ is a DMS-
morphism from $S^{\prime}$ onto $S$. This means that for any $x\in
X^{t}_{S^{\prime}}$ and $a\in M_{S}$ the following holds: $x\in\eta^{-1}(a)$
iff $\eta(x)\in a$ iff (by the definition of $\theta[a]$)
$\theta(\eta(x))\in\theta[a]$ iff (because $\theta(\eta(x))=x$)
$x\in\theta[a]$. This shows that $\theta[a]=\eta^{-1}(a)$, which shows that
$\theta[a]\in M_{S^{\prime}}$.

(i)$\Leftarrow$(ii) Suppose that $\theta$ is a DMS-morphism from $S$ into
$S^{\prime}$ and that (ii) is true. Conditions (1), (2) and (3) imply that
$\eta$ satisfy conditions ($\theta 1)$, ($\theta 2)$ and ($\theta 3)$ for DMS-
morphism. Since $\theta$ is a DMS morphism, it follows that the map
$a\mapsto\theta^{-1}(a)$ is a Boolean homomorphism from $M_{S^{\prime}}$ to
$M_{S}$. Because $\theta$ is a bijection, it follows that for its converse
$\eta$, the map $a\mapsto\eta^{-1}(a)$ is a Boolean homomorphism from $M_{S}$
to $M_{S^{\prime}}$, which shows that the condition ($\theta 4)$ is also
fulfilled. So $\eta$ is a DMS morphism from $S^{\prime}$ to $S$. Because
$\theta$ and $\eta$ are converses to each other, their compositions are the
identity mappings in the corresponding domains. So, $\theta$ is a DMS-
isomorphism from $S$ onto $S^{\prime}$. ∎

Let $f:A_{1}\longrightarrow A_{2}$ and $g:A_{2}\longrightarrow A_{3}$ be two
DCA-morphisms. The composition $h=f\circ g$ is a mapping
$h:B_{A_{1}}\longrightarrow B_{A_{3}}$ acting as follows; for $a\in
B_{A_{1}}$: $h(a)=g(f(a))$. In a similar way we define composition for DMS
morphisms.

The following lemma has an easy proof.

######  Lemma 6.4.

(i) The composition of two DCA-morphisms is a DCA-mor phism. The identity
mapping $1_{A}$ on each DCA $A$ is a DCA-morphism. Hence DCA is indeed a
category.

(ii) The composition of two DMS-morphisms is a DMS-morphism. The identity
mapping $1_{S}$ on each DMS $S$ is a DMS-morphism. Hence DMS is indeed a
category.

It follows from Lemma 6.4 that DCA and DMS are indeed categories.

We denote by DMS∗ the full subcategory of DMS of all T0 and DM-compact DMSes.

We introduce two contravariant functors

$\Phi$: DCA$\rightarrow$DMS, and $\Psi$: DMS$\rightarrow$DCA as follows:

(I) For a given DCA $A$ we put $\Phi(A)=A_{+}$ and for a DCA-morphism
$f:A\longrightarrow A^{\prime}$ we put $\Phi(f)=f_{+}$ and prove that $f_{+}$
is a DMS-morphism from $(A^{\prime})_{+}$ into $A$.

(II) For a given DMS $S$ we put $\Psi(S)=S^{+}$ and for a DMS-morphism
$\theta:S\longrightarrow S^{\prime}$ we put $\Psi(\theta)=\theta^{+}$ and
prove that $\theta^{+}:$ is a DMS morphism from $(S^{\prime})^{+}$ into $S$.

(III) We show that for each DCA $A$ the mapping $g_{A}(a)=\\{\Gamma\in
t$-$Clans(A):a\in\Gamma\\}$, $a\in B_{A}$ is a natural isomorphism (in the
sense of category theory (see [52] Chapter I, 4.)) from $A$ to
$\Psi(\Phi(A))=(A_{+})^{+}$.

(IV) We show that for each T0 and DM-compact DMS $S$ the mapping
$\rho_{S}(x)=\\{a\in M_{S}:x\in a\\}$, $x\in X^{t}_{S}$, is a natural
isomorphism from $S$ to $\Phi(\Psi(S)=(S^{+})_{+}$.

All this shows that the category DCA is dually equivalent to the category DMS∗
of T0 an DM-compact DMS. The realization of (I)-(IV) is given in the next
subsection.

###  6.2 Facts for DCA-morphisms and DMS-morphisms

######  Lemma 6.5.

Every DMS-morphism is a continuous mapping.

###### Proof.

Let $\theta:S\longrightarrow S^{\prime}$ be a DMS-morpism. Since $\theta^{-1}$
maps $M_{S^{\prime}}$ (which is the closed basis of the topology of
$S^{\prime}$) into $M_{S}$, then $\theta$ is continuous. ∎

######  Lemma 6.6.

Let $f:A\longrightarrow A^{\prime}$ be a DCA-morphism. Then:

(i) If $\Gamma$ is a t-clan in $A^{\prime}$ then $f^{-1}(\Gamma)=_{def}\\{a\in
B_{A}:f(a)\in\Gamma\\}$ is a t-clan in $A$.

(ii) If $\Gamma$ is an s-clan in $A^{\prime}$ then
$f^{-1}(\Gamma)=_{def}\\{a\in B_{A}:f(a)\in\Gamma\\}$ is an s-clan in $A$.

###### Proof.

The proof consists of a routine check of the corresponding definitions of
t-clan and s-clan. ∎

######  Lemma 6.7.

(i) Let $A,A^{\prime}$ be two DCAs and $f:A\longrightarrow A^{\prime}$ be a
DCA-morphism. Then $f_{+}$ is a DMS-morphism from $(A^{\prime})_{+}$ to
$A_{+}$.

(ii)The mapping $g_{A}(a)=\\{\Gamma\in t$-$Clans(A):a\in\Gamma\\}$, $a\in
B_{A}$ is a natural DCA-isomorphism of $A$ onto $\Psi(\Phi(A))=(A_{+})^{+}$.

###### Proof.

(i) Remind that
$(A^{\prime})_{+}=(t$-$Clans(A^{\prime}),s$-$Clans(A^{\prime}),Clusters(A^{\prime})$,
$\prec_{A^{\prime}},\newline M_{A^{\prime}})$. If $\Gamma\in
t$-$Clans(A^{\prime})$, then by Lemma 6.6 $f^{-1}(\Gamma)$ is a t-clan of $A$
and similarly for the case when $\Gamma$ is an s-clan. This shows that the
condition ($\theta 1$) for DMS-morphisms is fulfilled. For the condition
($\theta 2$) let $\Gamma\prec_{A^{\prime}}\Delta$, $\Gamma,\Delta\in
t$-$Clans(A^{\prime})$. We have to show that
$f^{-1}(\Gamma)\prec_{A}f^{-1}(\Delta)$. By the definition of $\prec_{A}$ for
clans (see (9)) this means the following. Let $a\in f^{-1}(\Gamma)$, $b\in
f^{-1}(\Delta)$. Then $f(a)\in\Gamma$ and $f(b)\in\Delta$. But
$\Gamma\prec_{A^{\prime}}\Delta$, so $f(a)\mathcal{B}_{A^{\prime}}f(b)$, which
by (f 4) implies $a\mathcal{B}_{A}b$. This shows that $\Gamma\prec_{A}\Delta$.

The next step is to verify the condition ($\theta 3$) of DMS-morphisms, namely
that $(f_{+})^{+}$ maps the members of $M_{A^{\prime}}$ into the members of
$M_{A}$. Note that the members of $M_{A}$ are of the form $g_{A}(a)$ for $a\in
B_{A}$ and that $g_{A}(a)=\\{\Gamma\in t-Clans(A):a\in\Gamma\\}$ and similarly
for the members of $M_{A^{\prime}}$. In order to verify ($\theta 3$) we will
show that for any $a\in B_{A}$ the following equality holds which indeed shows
that $(f_{+})^{+}$ maps $M_{A}$ into $M_{A^{\prime}}$:

$(f_{+})^{+}(g_{A}(a))=g_{A^{\prime}}(f(a))$ (13)

To show (13) note that $(f_{+})^{+}(g_{A}(a))$ is a subset of
t-Clans$(A^{\prime})$. So let $\Gamma\in t$-$Clans(A^{\prime})$. Then the
following sequence of equivalences proves (13):

$\Gamma\in(f_{+})^{+}(g_{A}(a))$ iff $\Gamma\in(f^{-1})^{-1}(g_{A}(a))$ iff
$f^{-1}(\Gamma)\in g_{A}(a)$ iff $a\in f^{-1}(\Gamma)$ iff $f(a)\in\Gamma$ iff
$\Gamma\in g_{A^{\prime}}(f(a))$.

Now we verify the condition ($\theta 4$) of DMS-morphisms: $(f_{+})^{+}$
preserves the Boole-an complement. We show this by applying (13) and the facts
that $f$ and $g_{A^{\prime}}$ acts as Boolean homomorphisms:

$(f^{+})_{+}((g_{A}(a))^{*})$=$(f^{+})_{+}(g_{A}(a^{*}))$=$g_{A^{\prime}}f(a^{*})$=$(f^{+})_{+}(g_{A}(a^{*}))$=

$((f^{+})_{+}(g_{A}(a)))^{*}$.

(ii) The statement that $g_{A}$ is a natural isomorphism in the sense of
category theory means the following: first, that $g_{A}$ is indeed an
isomorphism from $A$ onto $A_{+}$ (this is the Theorem 5.19) and second, that
for any DCA-morphism $f:A\longrightarrow A^{\prime}$, the following equality
should be true: $g_{A^{\prime}}\circ f=(f_{+})^{+}\circ g_{A}$. By the
definition of the composition $\circ$ for DCA-morphisms this equality is
equivalent to the following: for any $a\in B_{A}$ the following holds:

$g_{A^{\prime}}(f(a))=(f_{+})^{+}(g_{A}(a))$, which is just (13). ∎

######  Lemma 6.8.

Let $S,S^{\prime}$ be two DMS-s and $\theta:S\longrightarrow S^{\prime}$ be a
DMS-morphism from $S$ to $S^{\prime}$. Then $\theta^{+}$ is a DCA-morphism
from $(S^{\prime})^{+}$ to $S^{+}$.

###### Proof.

We have to verify that $\theta^{+}=\theta^{-1}$ satisfies the conditions
(f1)-(f4) for DCA-morphism. Condition (f1) is fulfilled by the condition
($\theta 4$) for DMS-morphisms. For condition (f2) suppose that for some
$a,b\in M_{S^{\prime}}$, $\theta^{-1}(a)C^{t}_{S}\theta^{-1}(b)$ and proceed
to show $aC^{t}_{S^{\prime}}b$. This implies that there exists $x\in
X^{t}_{S}$ such that $x\in\theta^{-1}(a)$ and $x\in\theta^{-1}(b)$. From here
we obtain $\theta(x)\in a$, $\theta(x)\in b$ and $\theta(x)\in
X^{t}_{S^{\prime}}$ (by condition ($\theta 1$) for DMS morphism) which yields
$aC^{t}_{S^{\prime}}b$. In a similar way one can verify condition (f3).

For (f4) suppose $\theta^{-1}(a)\mathcal{B}_{S}\theta^{-1}(b)$ and proceed to
show that $a\mathcal{B}_{S^{\prime}}b$. Then there exist $x,y\in X^{t}_{S}$
such that $x\prec_{S}y$, $x\in\theta^{-1}(a)$, $y\in\theta^{-1}(b)$. This
implies $\theta(x)\in a$, $\theta(y)\in b$, and by ($\theta 1$) and ($\theta
2$) that $\theta(x),\theta(y)\in X^{t}_{S^{\prime}}$ and
$\theta(x)\prec_{S^{\prime}}\theta(y)$. This implies
$a\mathcal{B}_{S^{\prime}}b$. ∎

Before the formulation of the next statement let us see what is $(S^{+})_{+}$
for a DMS $S$. $S^{+}$ is the dual of $S$ which is the DCA algebra
$(M_{S},C^{t}_{S},C^{s}_{S},\mathcal{B}_{S})$ (see Definition 5.3). Then
$(S^{+})_{+}$ is the dual space of the algebra $S^{+}$ which is
$(S^{+})_{+}=(X^{t}_{S^{+}},X^{s}_{S^{+}},T_{S^{+}},\prec_{S^{+}},M_{S^{+}})$,
where $X^{t}_{S^{+}}$ is the set of t-clans of $S^{+}$, $X^{s}_{S^{+}}$ is the
set of s-clans of $S^{+}$, $T_{S^{+}}$ is the set of clusters of $S^{+}$,
$\prec_{S^{+}}$ is the relation defined by (9) between t-clans, and
$M_{S^{+}}$ is the set $\\{g_{S^{+}}(a):a\in M_{S}\\}$, where
$g_{S^{+}}(a)=_{def}\\{\Gamma\in t-clans(S^{+}):a\in\Gamma\\}$ (see Section
5.5).

######  Lemma 6.9.

(i) Let $S$ be a DMS. Then $\rho_{S}$ is a DMS-morphism from $S$ to
$(S^{+})_{+}$.

(ii) Let $S$ be a DM-compact DMS and let for $a\subseteq X^{t}_{S}$,
$\rho_{S}[a]=_{def}\\{\rho_{S}(x):x\in a\\}$. Then for $a\in M_{S}$:
$\rho_{S}[a]=g_{S^{+}}(a)$ (for the function $g_{A}$ for a DCA A see Section
5.5).

(iii) If $S$ is $T0$ and DM-compact, then $\rho_{S}$ is a DMS-isomorphism from
$S$ onto $(S^{+})_{+}$.

(iv) If $S$ is a $T0$ and DM-compact DMS, then $\rho_{S}$ is a natural
isomorphism from $S$ to $\Phi(\Psi(S))=(S^{+})_{+}$.

###### Proof.

(i) We have to verify whether $\rho_{S}$ satisfies the conditions ($\theta
1$)-($\theta 4$) for DMS-morphisms. By Lemma 5.4 $\rho_{S}(x)$ is a t-clan in
$S^{+}$ for $x\in X^{t}_{S}$ and an s-clan in $S^{+}$ for $x\in X^{s}_{S}$.
This verifies the conditions ($\theta 1$) and ($\theta 2$) for DMS-morphisms.
Condition ($\theta 2$) is guaranteed by axiom (7) for DMS and Lemma 5.4 (iv).
For condition ($\theta 3$) we have to show that $(\rho_{S})^{-1}$ transforms
the members from $M_{S^{+}}$ into the members from $M_{S}$ (recall that the
members of $M_{S^{+}}$ are of the form $g_{S^{+}}(a)$, $a\in M_{S}$, see the
text before the lemma). This can be seen from the following equality

$(\rho_{S})^{-1}(g_{S^{+}}(a))=a$ (14)

Indeed, for $x\in X^{t}_{S}$ we have:

$x\in(\rho_{S})^{-1}(g_{S^{+}}(a))$ iff $\rho_{S}(x)\in g_{S^{+}}(a)$ iff
$a\in\rho_{S}(x)$ iff $x\in a$.

For condition ($\theta 4$) we have to show that $(\rho_{S})^{-1}$ preserves
Boolean complement. The following sequence of equalities proves this:
$(\rho_{S})^{-1}(g_{S^{+}}(a^{*}))=a^{*}=((\rho_{S})^{-1}(g_{S^{+}}(a)))^{*}$,
which is true on the base of (14).

(ii) Suppose $a\in M_{S}$ and let us show first $\rho_{S}[a]\subseteq
g_{S^{+}}(a)$:

$\rho_{S}(x)\in\rho_{S}[a]$ $\Rightarrow$ $x\in a$ $\Rightarrow$
$a\in\rho_{S}(x)$ $\Rightarrow$ $\rho_{S}(x)\in g_{S^{+}}(a)$ (because
$\rho_{S}(x)$ is a t-clan in the DCA algebra $S^{+}$). For the converse
inclusion, let $\Gamma$ be a t-clan in $S^{+}$. The by DM-compactness there
exists $x\in X^{t}_{S}$ such that $\Gamma=\rho_{S}(x)$. Then for $a\in M_{S}$:

$\Gamma\in g_{S^{+}}(a)$ $\Rightarrow$ $a\in\Gamma$ $\Rightarrow$
$a\in\rho_{S}(x)$ and $x\in a$ $\Rightarrow$ $\rho_{S}(x)\in\rho_{S}[a]$
$\Rightarrow$ $\Gamma\in\rho_{S}[a]$.

(iii) Let $S$ be $T0$ and DM-compact. Then by Lemma 5.7Then $\rho_{S}$ is a
one-one mapping from $X^{t}_{S}$ onto the set of all t-clans of $S^{+}$, which
are the points of $(S^{+})_{+}$. By (i) $\rho_{S}$ is a DMS-morphism from $S$
to $(S^{+})_{+}$. So in order to show that $\rho_{S}$ is a DMS-isomorphism
from $S$ onto $(S^{+})_{+}$ we have to see if $\rho_{S}$ satisfies the
conditions (1), (2) and (3) of Lemma 6.3 (ii).

For condition (1) suppose $\rho_{S}(x)\in X^{s}_{S^{+}}$. Then $\rho_{S}(x)$
is a t-clan in $M_{S}$. By DM-compactness there exists $y\in X^{s}_{S}$ such
that $\rho_{S}(x)=\rho_{S}(y)$. By $T0$ condition this implies $x=y$, so $x\in
X^{s}_{S}$.

For condition (2) suppose $\rho_{S}(x)\prec_{S^{+}}\rho_{S}(y)$. Then by Lemma
5.4 and axiom (S7) for DMS we obtain $x\prec_{S}y$.

For condition (3) suppose $a\in M_{S}$ and proceed to show that $\theta[a]\in
M_{(S^{+})_{+}}$. By (ii) $\theta[a]=g_{S^{+}}(a)$ and since $g_{S^{+}}(a)\in
M_{(S^{+})_{+}}$ we get $\theta[a]\in M_{(S^{+})_{+}}$.

Thus the conditions (1), (2) and (3) are fulfilled which proves that
$\rho_{S}$ is a DMS-isomorphism from $S$ onto $(S^{+})_{+}$.

(iv) Let $S$ be a T0 and DM-compact DMS. In order $\rho_{S}$ to be a natural
isomorphism from $S$ to $(S^{+})_{+}$ it has to satisfy the following two
conditions: first, $\rho_{S}$ have to be a DMS-isomorphism - this is
guaranteed by (iii), and second, for every DMS morphism $\theta:S\Rightarrow
S^{\prime}$: the following equality should be true:
$\theta\circ\rho_{S^{\prime}}=\rho_{S}\circ(\theta^{+})_{+}$. This equality is
equivalent to the following condition: for $x\in X^{t}_{S}$

$(\theta^{+})_{+}(\rho_{S}(x)=\rho_{S^{\prime}}(\theta(x))$ (15)

The following sequence of equivalencies proves (15). For $a\in
M_{S^{\prime}}$:

$a\in(\theta^{+})_{+}(\rho_{S}(x))$ iff $a\in(\theta^{+})^{-1}(\rho_{S}(x))$
iff $\Theta^{+}(a)\in\rho_{S}(x)$ iff $x\in\theta^{+}(a)$ iff
$x\in\theta^{-1}(a)$ $\theta(x)\in a$ iff $a\in\rho_{S^{\prime}}(\theta(x))$.
∎

As applications of the developed theory we establish some isomorphism
correspondences between the objects of the two categories. The isomorphism
between two objects will be denoted by the symbol $\cong$.

######  Lemma 6.10.

Let $A,A^{\prime}$ be two DCAs. Then the following conditions are equivalent:

(i) $A\cong A^{\prime}$,

(ii) $A_{+}\cong(A^{\prime})_{+}$,

(iii) $(A_{+})^{+}\cong((A^{\prime})_{+})^{+}$

###### Proof.

(i)$\Leftrightarrow$(iii).By Lemma 5.19 we have $A\cong(A_{+})^{+}$ and
$A^{\prime}\cong(A^{\prime}_{+})^{+}$. This makes obvious the equivalence
(i)$\Leftrightarrow$(iii).

(i)$\Rightarrow$(ii) Suppose $A\cong A^{\prime}$, then there exists a on-one
mapping $f$ from $A$ onto $A^{\prime}$ with a converse mapping $h$ such that
$f:A\mapsto A^{\prime}$ is a DCA morphism from $A$ onto $A^{\prime}$ and
$h:A^{\prime}\mapsto A$ is a DCA- morphism from $A^{\prime}$ onto $A$ such
that the composition $f\circ h$ is the identity mapping in $A^{\prime}$ and
the composition $h\circ f$ is the identity mapping in $A$. Then by Lemma 6.7
$f_{+}$ is a DMS-morphism from $A^{\prime}_{+}$ onto $A_{+}$ and $h_{+}$ is a
DMS-morphism from $A_{+}$ onto $A^{\prime}_{+}$.

We shall show the following:

(1) The composition $f_{+}\circ h_{+}$ is the identity in $A^{\prime}_{+}$,
and

(2) The composition $h_{+}\circ f_{+}$ is the identity in $A_{+}$.

Then, by the definition of DMS- isomorphism this will imply that both $f_{+}$
and $h_{+}$ are DMS-isomorphisms in the corresponding directions.

Note that the members of $A_{+}$ are the t-clans of $A$ and similarly for
$A^{\prime}_{+}$.

To show (1) let $\Gamma$ be a point of the space $A^{\prime}_{+}$, i.e.
$\Gamma$ is a t-clan in $A^{\prime}$. We shall show that $(f_{+}\circ
h_{+})(\Gamma)=\Gamma$ which will prove (1). This is seen from the following
sequence of equivalencies where $a$ is an arbitrary element of
$B_{A^{\prime}}$:

$a\in(f_{+}\circ h_{+})(\Gamma$ ) iff $a\in(f_{+}(h_{+}(\Gamma))$ iff $a\in
f^{-1}(h_{+}(\Gamma))$ iff $f(a)\in h_{+}(\Gamma)$ iff $f(a)\in
h^{-1}(\Gamma)$ iff $h(f(a))\in\Gamma$ iff $a\in\Gamma$.

Here we use that $h(f(a))=a$ for $a\in B_{A^{\prime}}$ because $h$ is the
converse of the one-one mapping $f$ from $B_{A}$ onto $B_{A^{\prime}}$.

In a similar way we show (2).

(ii)$\Rightarrow$(iii ) The proof is similar to the above one. Suppose
$A_{+}\cong(A^{\prime})_{+}$, then there exists a one-one mapping $\theta$ and
its converse $\eta$ such that $\theta$ is a DMS-morphism from $A_{+}$ onto
$(A^{\prime})_{+}$ and $\eta$ is a DMS-morphism from $(A^{\prime})_{+}$ onto
$A_{+}$. Then by Lemma 6.8 $\theta^{+}$ is a DCA-morphism from
$(A^{\prime}_{+})^{+}$ into $(A_{+})^{+}$ and $\eta^{+}$ is a DCA-morphism
from $(A^{\prime}_{+})^{+}$ into $(A_{+})^{+}$. We shall show that both
$\theta^{+}$ and $\eta^{+}$ are DCA-isomorphisms in the corresponding
directions by showing that their compositions are identities in the
corresponding domains. Let us note that the domain of $\theta^{+}$ is the
members of the algebra $(A^{\prime}_{+})^{+}$ which are of the form
$g_{A^{\prime}}(a)$, $a\in_{B_{A^{\prime}}}$, and similarly for the members of
$(A_{+})^{+}$. Namely we will show the following two things:

(3) $(\theta^{+}\circ\eta^{+})(g_{A^{\prime}}(a))=g_{A^{\prime}}(a)$ for any
$a\in B_{A^{\prime}}$,

(4) $(\eta^{+}\circ\theta^{+})(g_{A^{\prime}}(a))=g_{A^{\prime}}(a)$ for any
$a\in B_{A}$,

To show (3) note that $g_{A^{\prime}}(a)=\\{\Gamma\in
t-clans(A^{\prime}):a\in\Gamma$. So let $\Gamma\in t-clans(A^{\prime})$. Then
the following sequence of equivalents proves (3):

$\Gamma\in(\theta^{+}\circ\eta^{+})(g_{A^{\prime}}(a))$ iff
$\Gamma\in(\theta^{+}(eta^{+}(g_{A^{\prime}}(a))))$ iff
$\Gamma\in(\theta^{-1}(eta^{+}(g_{A^{\prime}}(a))))$ iff
$\theta(\Gamma)\in(eta^{+}(g_{A^{\prime}}(a)))$ iff
$\theta(\Gamma)\in(eta^{-1}(g_{A^{\prime}}(a)))$ iff $\eta(\theta(\Gamma))\in
g_{A^{\prime}}(a)$ iff $\Gamma\in g_{A^{\prime}}(a)$.

We have just used that $\eta(\theta(\Gamma))=\Gamma$, because $\eta$ is the
converse of the one-one mapping $\theta$ from $X^{t}_{A_{+}}=t-Calans(A)$ onto
$X^{t}_{(A^{\prime})_{+}}=t-clans(A^{\prime})$. The proof of (4) is similar. ∎

######  Lemma 6.11.

Let $S,S^{\prime}$ be two DMSes. Then the following conditions are equivalent:

(i) $S\cong S^{\prime}$,

(ii) $S^{+}\cong(S^{\prime})^{+}$,

(iii) $(S^{+})_{+}\cong((S^{\prime})^{+})_{+}$.

###### Proof.

The proof is analogous to the proof of Lemma 6.10 ∎

As a corollary from Lemma 6.10 and Lemma 6.11 we obtain the following addition
to the topological representation theorem for DCAs.

######  Corollary 6.12.

There exists a bijective correspondence between the class of all, up to DCA-
isomorphism DCAs, and the class of all, up to DMS-isomorphism DMSes; namely,
for every DCA-algebra $A$ the corrseponding DMS of $A$ is $A_{+}$ \- the
canonical DM-space of $A$; and for every DMS $S$ the corresponding DCA of $S$
is $S^{+}$ – The canonical DC-algebra of $S$.

###  6.3 Topological duality theorem for DCAs

In this section we prove the third important theorem of this paper.

######  Theorem 6.13.

Topological duality theorem for DCAs. The category DCA of all dynamic contact
algebras is dually equivalent to the category DMS∗ of all $T0$ and DM-compact
DMSes.

###### Proof.

The proof follows from Lemma 6.7, Lemma 6.8 and Lemma 6.9. ∎

The above theorem has several consequences to some important subcategories of
DCA and DMS. The first example is the following. Let $Ax$ be a subset of the
set of temporal axioms (rs), (ls), (up dir), (down dir), (circ), (dens),
(ref), (lin), (tri), (tr). Consider the class of all DCAs satisfying the
axioms from $Ax$. It is easy to see that this class forms a full subcategory
of the category of all DCAs under the DCA-morphism. Denote this subcategory by
DCA(Ax). Let $\widehat{Ax}$ be the subset of the corresponding to Ax time
condition from the list (RS), (LS), (Up Dir), (Down Dir), (Circ), (Dens),
(Ref), (Lin), (Tri), (Tr). Consider the class of all $T0$ and DM-compact DMSes
which satisfy the axioms $\widehat{Ax}$. It is easy to see that this class is
a full subcategory of the category DMS∗ of all T0 and DM-compact dynamic
mereotopological spaces. Denote this subcategory by DMS$(\widehat{Ax})^{*}$

######  Theorem 6.14.

The category DCA(Ax) of all dynamic contact algebras satisfying Ax is dually
equivalent to the category DMS$(\widehat{Ax})^{*}$ of all $T0$ and DM-compact
DMSes satisfying $\widehat{Ax}$.

###### Proof.

Let $S$ be a $T0$ and DM-compact DMS. It follows by Lemma 5.18 that $S$
satisfies $\widehat{Ax}$ iff $S^{+}$ satisfies Ax. Now the theorem is a
corollary of Theorem 6.13. ∎

Another subcategory of DCA is the class of all trivial DCAs with the same
morphisms. Denote it by DCAtrivial. The corresponding subcategory of DMS∗ with
the same morphisms is the class of all trivial $T0$ and DM-compact DMSes.
Denote it by DMS∗trivial. The following theorem is also an obvious consequence
of Theorem 6.13

######  Theorem 6.15.

The category DCAtrivial is dually isomorphic to the category
DMS∗trivial.

######  Remark 6.16.

Note that the category of contact algebras can be identified in an obvious way
with the category of trivial DCAs by enriching contact algebras with some
definable relations. Having in mind Lemma 5.10, Lemma 5.11 and Corollary 5.12
it can be shown that the category of mereocompact and $T0$ mereotopological
spaces from [34] can also be identified with the category of $T0$ and DM-
compact trivial DMS. This implies that the duality theorem for contact
algebras from [34] can be derived from Theorem 6.15.

##  7 Concluding remarks

Overview. The aim of this paper is to present with some details a version of
point-free theory of space and time based on a special representative example
of a dynamic contact algebra (DCA). The axioms of the algebra are true
sentences from a concrete point-based model, the snapshot model, developed in
Section 3. Theorem 4.15 \- the Representation theorem for DCA by snapshot
models snows that the chosen axioms are enough to code the intuition based on
snapshot construction which can be considered as the cinematographic model of
spacetime. In Section 4 we introduced topological models of DCAs giving them
another intuition based on topology. These models are based on the notion of
Dynamic mereotopological space (DMS). Let us note that the abstract definition
of DCA can be considered as a ‘dynamic generalization’ of contact algebra,
which in a sense is a certain point-free theory of space called also a
mereotopology. In this relation contact algebras can be considered as a
‘static mereotopology’ while dynamic contact algebras can be considered as a
‘dynamic mereotopology’. Let us note that topological models of contact
algebras, which are considered as the standard models of this notion, contain
one type of points, which are just the ‘space points’ while dynamic
mereotopological spaces contain several kinds of points: partial time points,
time points and space points, which in turn are also partial time points. Time
points realize the time contact, while space points realize the space contact.
The fact that each space point is a partial time point says that _space_ in
this model is reduced to _time_ , a feature quite similar to the Robb’s
axiomatic treating of Minkowskian spacetime geometry in which space is reduced
to time (see [61] and the discussion in Section 1.1). Another common feature
of both snapshot and topological models is that the properties of the
underline time structure correspond to the validity of time axioms which are
point-free conditions for dynamic regions formulated by the relations of time
contact $C^{t}$ and precedence relation $\mathcal{B}$. Because regions are
observable things, then recognizing which time axioms they satisfy we may
conclude which abstract properties satisfies the corresponding time structure.

Discussions and some open problems. Time contact relation $aC^{t}b$, and
precedence relation $a\mathcal{B}b$ between two dynamic regions $a$ and $b$ in
snapshot models are defined by the predicate ‘existence’ defined in Boolean
algebras as follows: $E(a)$ iff $a\not=0$. One may ask if this predicate is a
good one. It has the following disadvantage - there are too many existing
regions and only one non-existing - the zero region. For instance we can not
see the zero region, but we can see on the sky a non-existing star - see
Remark 3.2. What we see is different from 0 but this does not mean that it is
existing at the moment of observation. So the adopted in this paper definition
for ‘existence’ is approximate one and we need a more exact definition
corresponding to what we intuitively mean by ‘actual existence’. This is a
problem discussed in our papers [74, 75] in which we introduce an axiomatic
definition and corresponding models of predicate ‘actual existence’ (denoted
by $AE(a)$) and a corresponding relation between regions called ‘actual
contact’. The predicate $E(a)$ satisfies the axioms of $AE(a)$ and is the
simplest one, but $AE$ is more general - it is possible for some region $a$ to
have $a\not=0$ but not $AE(a)$ like ‘non-existing stars’ discussed in Remark
3.2. One of our future plans is to reconstruct the theory of the present paper
on the base of the more realistic predicates of actual existence and actual
contact.

Another subject of discussion is the relation $aC^{t}b$ called ‘time contact’
which is a kind of simultaneity relation. Special relativity theory (SR) also
studies a kind of simultaneity relation and states that it is not absolute and
is relative to the observer. Is it possible to relate these two notions? In
general these two relations are different because in our system this is a
relation between regions and in SR it is between events, which are not regions
but space-time points. Nevertheless we will try to find some correspondence.
By event in SR one normally assume a space point, taken from our ordinary
space, with attached time-point (a date), taken from a clock attached to the
space point with the assumption that all attached clocks work synchronously
(the possibility to have synchronized clocks in all points of our space is
explained by Einstein in [29] by a special synchronization procedure). So
events are pairs $(A,t)$, where $A$ is a space point and $t$ is a real number
interpreted as a date. According to Einstein’s natural definition, two events
$(A_{1},t_{1})$, $(A_{2},t_{2})$ are simultaneous if $t_{1}=t_{2}$ which shows
that simultaneity is an equivalence relation. Note that Einstein did not give
formal definition of ‘event’, but in the terminology of Minkowski spacetime,
which is a formal explication of SR spacetime, events are just spacetime
points and two spacetime points are simultaneous if they have equal time
coordinates. In our system we do not introduce the notion of event but in the
abstract DCA an (approximate) analog of event can be identified with a pair
$(U,\Gamma)$ where $U$ is an ultrafilter and $\Gamma$ is a cluster containing
$U$ \- $U$ is a space point and $\Gamma$ is a time point (see Section 5.5).
Let $(U_{i},\Gamma_{i})$, i=1,2 be two events in DCA. Then, according to the
simultaneity relation between events it can be easily seen that
$(U_{1},\Gamma_{1})$ is simultaneous with $(U_{2},\Gamma_{2})$ iff
$U_{1}R^{t}U_{2}$ which is just the canonical relation between ultrafilters
corresponding to the contact relation $C^{t}$. Note that $R^{t}$ is also an
equivalence relation as the simultaneity relation in SR is. So an analog of SR
simultaneity relation in our theory is the relation $R^{t}$ considered between
‘events’ in the sense of DCA.

An analog of our before-after relation $\prec$ between events in SR is
$(A_{1},t_{1})\prec(A_{2},t_{2})$ iff $t_{1}<t_{2}$. This relation, like
simultaneity, is not absolute and is relative to the observer. Note also that
it is different from the Robb’s causal relation ‘before’ taken as the unique
basic relation between events in the axiomatic presentation of Minkowski
geometry [61]). The natural analog of the above relation between DCAs ‘events’
is
$(U_{1},\Gamma_{1})\prec(U_{2},\Gamma_{2})\Leftrightarrow_{def}\Gamma_{1}\prec\Gamma_{2}$.
But we have $\Gamma_{1}\prec\Gamma_{2}$ iff $U_{1}\prec U_{2}$ which shows
that the relation coincides with the canonical relation $\prec$ between
ultrafilters corresponding to the precontact relation $\mathcal{B}$. This
shows that the canonical relation $\prec$ between ultrafilters which is used
to characterize $\mathcal{B}$ is not an analog of the Robb’s causal relation
(let us denote it by $\prec_{Robb}$) which has a special definition in
Minkowski spacetime by means of its metric. An analog of this definition in
Einstein’s SR is the following: $(A_{1},t_{1})\prec_{Robb}(A_{2},t_{2})$ iff
$|A_{1}A_{2}|\leq|t_{1}-t_{2}|$ and $t_{1}<t_{2}$. This relation is stronger
than the relation $\prec$. It will be nice to have an abstract version of DCA
containing stronger than $\mathcal{B}$ precontact relation corresponding to
causality. We put this problem to the list of our future plans.

Comparing the presented in this paper theory with SR we see that there is
another feature which differs the corresponding theories: RS considers many
observers and can prove that some relations between events like simultaneity
are relative to corresponding observer, while a given DCA $A$ is based on only
one observer, denote it by $O(A)$ (this observer can be identified with an
abstract person describing the standard dynamic model of space which is
isomorphic to $A$). So, because we have only one observer in our formalism, we
can not give formal proofs whether the basic relations between regions are
relative or not to the observer. Hence, building a theory like DCA
incorporating many observers is the next open problem.

One possibility for a theory with many observers describing one and the same
reality is to consider a family of DCAs with some relations between them. Let
$A$ and $A^{\prime}$ be two DCAs from such a set. Examples of possible
relations between them are, for instance, the following:

(1) The observers $O(A)$ and $O(A^{\prime})$ are at rest to each other, they
have synchronous clocks, and have some possibilities to communicate. For
instance if we have two observers with equal cameras who are at rest to each
other and are filming the same reality with their cameras. The communication
is when on of them can point out to the other some of the observed objects.

(2) The observers $O(A)$ and $O(A^{\prime})$ are not at rest to each other but
have synchronous clocks and some possibilities to communicate. A situation
similar to the above but one of the observers is moving with respect to the
other.

Is it possible to find a meaningful abstract characterizations of such
relations by using some morphism like relations between the algebras $A$ and
$A^{\prime}$? An example of a set of DCAs with some morphisms between them is
the category DCA considered as a small category (the class od DCAs is a set).
Then a natural question is what are saying the DCA-morphisms between the
algebras considered as algebras produced by observers describing one and the
same reality. For instance, what is the meaning of the condition on DCA-
morphism $f:A\longrightarrow A^{\prime}$:

($\sharp$) If $f(a)C_{A^{\prime}}^{t}f(b)$, then $aC^{t}_{A}b$

If we interpret $f$ as a way for the observer $O(A)$ to point out some regions
to the observer $O(A^{\prime})$, then $(\sharp)$ says that if $O(A^{\prime})$
sees that the pointed regions are in a time contact, then the same has been
seen by $A$. Similar interpretation have the other conditions on DCA-morphisms
concerned $C^{s}$ and $\mathcal{B}$. This means that $O(A^{\prime})$ is seen
the reality in the same way as $O(A)$ from which we may conclude that
observers are at rest to each other. So an open problem is to study small
categories of DCAs with different kinds of meaningful morphisms between them.

Let us finish this section by formulating one more open problem. The
axiomatization of Minkowski geometry presented by Robb [61] is point-based:
the primitive concepts are points and the binary relation ‘before’ on points
satisfying some axioms. The problem is to present a point-free
characterization of Minkowskian geometry similar to DCA eventually with more
spatio-temporal primitive relations between regions and probably by
axiomatizing some special regions in this geometry, for instance, Minkowski’s
light cones. An analogous result for Euclidean geometry is the Tarski result
in [66], where he presented an abstract axiomatization of Euclidean balls.
Euclidean balls are the regions in Euclidean geometry from which it is
possible to extract the Euclidean metrics. In Minkowskian geometry light cones
coded in some way Minkowskian metrics. Similar proposal for a point-free
characterization of affine geometry was proposed by Whitehead in [83] by an
abstract characterization of the set of convex regions (called by Whitehead
‘ovals’).

## Acknowledgments

The author is sponsored by Contract DN02/15/19.12.2016 with Bulgarian NSF.
Project title: Space, Time and Modality: Relational, Algebraic and Topological
Models.

Thanks are due to my colleagues Georgi Dimov, Tinko Tinchev and Philippe
Balbiani for the collaboration and many stimulating discussions related to the
field, and to my former Ph.D. students Vladislav Nenchev and Tatyana Ivanova
for their excellent dissertations and to Veselin Petrov for introducing me to
Whitehead’s philosophy.

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## Appendix: Short review of papers on RBTS

In this appendix we present a short, probably incomplete review of papers on
RBTS appeared after 1977 and not discussed in [69]. The papers are classified
in several groups.

(I) Results on mereology. First I want to mention here some papers devoted to
a detailed analysis of results obtained by Polish logicians in the field of
mereology and RBTS. The book Metamereology [59] extends some results on
mereology, the paper [36] is devoted to a detailed analysis of Grzegorczyk
point-free theory of space [35] and the paper [37] \- to a full analysis of
Tarski geometry of solids ([66]).

(II) Further results on contact and precontact algebras. The papers [25, 11]
contain some technical results on contact algebras. The paper [19] transfers
the notion of dimension from topology to the corresponding notion of some
classes of contact algebras and the paper [68] extends contact algebras with
connectedness predicates and studies the corresponding quantifier-free logics.
The paper [18] characterizes contact algebras on Euclidean spaces. The papers
[23, 24] presented topological representation theorem for precontact algebras
and new representation theorems for some classes of contact algebras.

(III) Duality theory of contact and precontact algebras and some related
systems. There are many papers generalizing De Vries duality theorem [76]
mainly with applications to topology: [6], [7], [14], [15], [16], [17], [9] \-
for Boolean algebras with quasi-modal operators which are equivalent to
precontact algebras, [10] \- for subordination Tarski algebras with
application to De Vries duality. A paper about duality theory for contact and
precontact algebras is [20] which include also some generalizations of the
Stone Duality Theorem. Another duality theorem for contact algebras is based
on mereotopological spaced is presented in [34].

(IV) Generalizations of contact algebras. The paper [58] contains a
generalization of contact algebra based only on the standard mereological
relations part-of, overlap and underlap plus standard mereotopological
relations of contact, dual contact and non-tangential inclusion and studies
also a modal logic based on these relations. The paper [42] studies
generalizations of contact algebras based on distributive lattices with three
basic mereotopological relations of contact, dual contact and non-tangential
inclusion taken as primitive relations. Representation theorems for extended
contact algebras based on equivalence relations is in the paper [3].
Generalization of contact algebra based on non-distributive lattices is
presented in [39, 77, 78].

Another generalization of contact algebra is the notion of sequent algebra
which presents Tarski and Scott consequence relations as mereotopological
relations - see [73] and [41]. In standard models with regular closed subsets
of a topological space Tarski consequence relation $a_{1},\dots,a_{n}\vdash b$
is defined as $a_{1}\cap,\ldots,\cap a_{n}\newline \subseteq b$, which makes
possible to define n-ary contact by
$C_{n}(a_{1},\dots,a_{n})\Leftrightarrow_{def}a_{1},\ldots,\newline
a_{n}\not\vdash 0$ and ordinary contact as
$aCb\Leftrightarrow_{def}a,b\not\vdash 0$. Generalizations of contact algebras
with predicates of actual existence and actual contact are subject of [74,
75]. In standard contact algebras the predicate of existence is defined as
follows: $E(a)\Leftrightarrow_{def}a\not=0$. This is a quite weak predicate,
because the only non-existing region is $0$. The generalization is to relax
this definition as follows: take a fixed grill $\Gamma$ (see Definition 2.14)
and define $E(a)\Leftrightarrow_{def}a\in\Gamma$. Another line of
generalizations is to consider Boolean algebras with contact relation and
measure - see [50] and [51].

(V) Modal and Quantifier-free logics based on contact and precontact algebras.
Modal logics based on mereological and mereotopological relations arising from
contact algebras or topology are presented in [53] and [58]. Papers on
quantifier-free logics in the style of [4] related to contact algebras and
their extensions and generalizations are [68] for logics with connectedness
predicates, [46] \- studying them form computational point of view, [42],[40],
[41] \- for logics based on extended contact algebras. Quantifier-free logics
related to contact algebras with measure are [50] and [51].