pre-injectivity

pre-injectivity

Let

X

=

∏

i

∈

I

X

i

be a

Cartesian product

.
Call two

elements

x

,

y

∈

X

almost equal

if the set

Δ

⁢

(

x

,

y

)

=

{

i

∈

I

∣

x

i

≠

y

i

}

is finite.
A

function

f

:

X

→

X

is said to be

pre-injective

if it sends distinct almost equal elements into distinct elements,

i.e.

, if

0

<

|

Δ

⁢

(

x

,

y

)

|

<

∞

implies

f

⁢

(

x

)

≠

f

⁢

(

y

)

.

If

X

is finite,

pre-injectivity

is the same as injectivity;
in general, the latter implies the former, but not the other way around.
Moreover, it is not true in general
that a

composition

of pre-injective functions
is itself pre-injective.

A

cellular automaton

is said to be pre-injective if its global function is.
As cellular automata send almost equal

configurations

into almost equal configurations,
the composition of two pre-injective cellular automata is pre-injective.

Pre-injectivity of cellular automata can be characterized
via mutually erasable patterns.
Given a finite subset

E

of

G

,
two patterns

p

1

,

p

2

:

E

→

Q

are

mutually erasable

(briefly, m.e.)
for a cellular automaton

𝒜

=

⟨

Q

,

𝒩

,

f

⟩

on

G

if for any two configurations

c

1

,

c

2

:

G

→

Q

such that

c

i

|

E

=

p

i

and

c

1

|

G

∖

E

=

c

2

|

G

∖

E

one has

F

𝒜

⁢

(

c

1

)

=

F

𝒜

⁢

(

c

2

)

.

Lemma 1

For a cellular automaton

A

=

⟨

Q

,

N

,

f

⟩

,

the following are

equivalent

.

1.

𝒜

has no mutually erasable patterns.

2.

𝒜

is pre-injective.

Proof.

It is immediate that the

negation

of point

1

implies the negation of point

2

.
So let

c

1

,

c

2

:

G

→

Q

be two distinct almost equal configurations such that

F

𝒜

⁢

(

c

1

)

=

F

𝒜

⁢

(

c

2

)

:

it is not restrictive to suppose that

𝒩

is

symmetric

(

i.e.

, if

x

∈

𝒩

then

x

-

1

∈

𝒩

)
and

e

, the

identity element

of

G

, belongs to

𝒩

.
Let

Δ

be a finite subset of

G

such that

c

1

|

G

∖

Δ

=

c

2

|

G

∖

Δ

,

and let

E

=

Δ

𝒩

2

=

{

g

∈

G

∣

∃

z

∈

Δ

,

u

,

v

∈

𝒩

∣

g

=

z

u

v

}

:

(1)

we shall prove that

p

1

=

c

1

|

E

and

p

2

=

c

2

|

E

are mutually erasable.
(They surely are distinct, since

Δ

⊆

E

.)

So let

γ

1

,

γ

2

:

G

→

Q

satisfy

γ

i

|

E

=

p

i

and

γ

2

|

G

∖

E

=

p

i

=

γ

2

|

G

∖

E

.

Let

z

∈

G

.
If

z

∈

Δ

⁢

𝒩

,
then

F

𝒜

⁢

(

γ

1

)

⁢

(

z

)

=

F

𝒜

⁢

(

γ

2

)

⁢

(

z

)

,
because by construction

γ

i

|

z

⁢

𝒩

=

c

i

|

z

⁢

𝒩

;

if

z

∈

G

∖

Δ

⁢

𝒩

,

then

F

𝒜

⁢

(

γ

1

)

⁢

(

z

)

=

F

𝒜

⁢

(

γ

2

)

⁢

(

z

)

as well,
because by construction

γ

1

|

z

⁢

𝒩

=

γ

2

|

z

⁢

𝒩

.

Since

γ

1

and

γ

2

are arbitrary,

p

1

and

p

2

are mutually erasable.

□

References

1

Ceccherini-Silberstein, T. and Coornaert, M. (2010)

Cellular Automata and Groups.

Springer Verlag.

2

Title

pre-injectivity

Canonical name

Preinjectivity

Date of creation

2013-03-22 19:22:04

Last modified on

2013-03-22 19:22:04

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

4

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 37B15

Classification

msc 68Q80

Defines

mutually erasable patterns (cellular automaton)

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