CORRESPONDENCE BETWEEN LOCAL MAPS AND GLOBAL MAPS IN CELLULAR AUTOMATA

In this paper, we investigate the relationship between local maps and global maps. Let

In Theorem D, in case that S is a discrete topological space, we show that the following two conditions for F are equivalent: (a) F is continuous and shift commutative.
(b) F is a global map of some local map f with respect to some subset This is a generalization of Richardson's Theorem where S and N were assumed to be finite.
Finally, in Theorem E, under the assumption that S is a discrete topological space, we provide two different necessary and sufficient conditions for N S S to contain only continuous local maps.

I B
Let X be a topological space and x an element of X.Then, we use notations: : X N a system of neighborhoods of X, ( ): x N X a system of neighborhoods of x in X, | be a family of topological spaces and be the direct product.Then, we can give forms an open basis for .

O
Thus X becomes a topological space.The topology of X is called the weak topology of the direct product

Cellular automaton
is called a cellular automaton.
We say that n Z to be the cell space and i in n Z a cell, S the state set, and s in S a state, N the neighborhood of 0 in , n Z or simply the neighborhood of 0, and f the local map over , N S or a local map.In the above, it might not be suitable for N to call "neighborhood of 0", since it may not be guaranteed for N to contain 0 of .n Z However, in the present paper, we will follow this well-known naming for N.

C, F and j θ
For n Z and S in , A we call where ( ) ( ).
The following two lemmas are used for the proof of Theorem A.
Proof.We have ( )   Proof.For (a), we show that , , Indeed this is shown as follows: ( ) we have , t s = and so ( ) ( ).
= Thus, F is shift commutative as we have insisted.
For (b), suppose that c, d in C satisfy .
The lemma allows us to define a map ϕ instead of 0 ϕ as follows: Here, applying Lemma 2.2.1, we have two different expressions for ( ).
In the above, ( ) called the global map of f on N.

0 ψ and ψ
Next, we define Note that 0 ψ is well defined, that is, independent of the choice of c with In the above, we also express f, i.e., ( ) if N is already known, and for a global map F, we call the local map of F on N.

Cellular Automaton with a Set S as its State Set
In this section, we treat a cellular automaton where S is a set and N is a subset of .
for any c with .
that is, ϕ and ψ are both bijections and each of them is the inverse of the other.
Proof.Since N is fixed, throughout this section, we write ( ),

Hiroyuki Ishibashi 132
Thus, all what we have to show is that ) as was to be shown.
Next, we show .

Cellular Automaton with a Topological Space S as its State Set
In this section, we prove the following theorem by showing a topological identification, i.e., homeomorphism between

Hiroyuki Ishibashi 134
First, since O is in , Also, by (iii) of ( 2), since h Q is in C O′ for any h in H, we may express Consequently, (3) and (4) give us , , Correspondence between Local Maps and Global Maps … 135 Here, we apply (3.1.3)in Section 3 to have
we have and setting by ( 6), ( 7) and ( 9), we have , , that is, for any H h ∈ and , Therefore, setting by (10), we have O is an open neighborhood of f.So, the rest to be shown is that

( ) ( )
Proof.Firstly, since F is N-reducible, as we have already shown in 3.2, Now, since by (1 we may express N O as follows: we have , (5) For this , Firstly, we note that we can extend each Next, utilizing h c in ( 6) and h P in (2), we define where Now, as desired.
Indeed, we may show that ( ) , by ( 7) and 0 ψ is continuous as was to be shown.□

Local Maps and Global Maps
In this section, we show that ֏ are both continuity preserving maps.
Theorem C. Let S be a topological space.As for ϕ , , C N and , ψ they are those of Theorem A. In particular, assume that S, N are finite.
Then, if As we see above, Theorem C says that ϕ and ψ are both continuity preserving homeomorphisms.The proof for the theorem is straightforward on Propositions 6.1 and 6.2 established below.Proposition 6.1.Let N be a subset of Correspondence between Local Maps and Global Maps … 141 Here, since O′ is in ,

C
O′ we may express it as This shows that we have reduced the case Then, by ( 1) and ( 2), we have .
Here, applying our assumption that F is continuous, we have In particular, by (4), we have ( ) ( ) Further ( 6) and (7) give us  To prove that (b) implies (a), we need the following lemma, in which we show that the discreteness of S together with the continuity of F provide a subset N of Z for which F is N-reducible.
We note that if we choose , Proof.We set automaton with n Z a cell space, S a state set, N a neighborhood of 0 in .
Theorem B, assuming that S is a topological space, we show that there exists a homeomorphism between two topological spaces , we show a bijective correspondence between the set of continuous local maps f and the set of continuous global maps F.

=
to be the configuration set or space, and c in C a configuration, which may be expressed as For the local map f in , N S we call an element F in C C a global map.Definition.A map j θ in C C with n j Z ∈ is said to be a j-shift or simply a shift defined by , ,

Lemma 2 . 2 . 1 .
Let c be an arbitrary element in C.Then, for n N Z ⊆ and ,

3 .
ϕ and ψ Now we introduce two maps ϕ and ψ which will play a central roll in this article.Correspondence between Local Maps and Global Maps

For this , 0 ϕLemma 3 . 1 . 1 .
we have the following lemma which is used in the proof of Theorem C: For any f in , F is Nreducible.

nZ
Further, S and N are arbitrarily chosen and so we do not assume the finiteness for them.For this , A we investigate a bijective correspondence between N S S and [ ] .
Let S be a set, and N be a set with , Correspondence between Local Maps and Global Maps … 131 by n

3 .
Then, ϕ and ψ are homeomorphisms and each of them is the inverse of the other, in other words, (b1) ϕ and ψ are both continuous and We have (b2) by Theorem A. However, (b1) follows from Propositions 5.1 and 5.
are both homeomorphisms, and each of them is the inverse of the other, that is, (b1) A | ϕ and B | ψ are both continuous and (b2)

and thus f 7 .
is continuous.□ Correspondence between Local Maps and Global Maps … 145 Cellular Automaton with S a Discrete Topological SpaceLet us think about the case of a discrete topological space S.Then, our purpose is for any shift commutative and continuous global map F in C C to give a necessary and sufficient condition such that F is a global map of some local map f in ., we show the following: Theorem D. Let S be a discrete topological space.Then, f ϕ (b) F is shift commutative and continuous.

HenceF
S is discrete, λ s is an open set of S. Therefore, can show that F is N-reducible for this N as follows: Suppose that for c, d in C, belongs to λ O by (1) and (2) and so by (3) c is contained in λµ O for some µ in M, and thus by (4), we have is, F is N-deducible.□ Proof of Theorem D. (a) implies (b).By Theorem A, we have is shift commutative.Further, since f is continuous by (a) of Theorem C, implies (a).By Lemma 7.1, F is N-reducible for some subset N of , n Z and so F belongs to [ ] .
F is continuous, by (a) of Theorem C, For sets A, B, we denote the set of maps from A to B by i X A B