background image

CHAPTER 12

Total boundness of reloids

12.1. Thick binary relations

Definition

912

.

I will call

α

-thick

and denote thick

α

(

E

) a

Rel

-morphism

E

when there exists a finite cover

S

of Ob

E

such that

A

S

:

A

×

A

GR

E

.

Definition

913

.

CS(

S

) =

S

A

×

A

A

S

 

for a collection

S

of sets.

Remark

914

.

CS means “Cartesian squares”.

Obvious

915

.

A

Rel

-endomorphism is

α

-thick iff there exists a finite cover

S

of Ob

E

such that CS(

S

)

GR

E

.

Definition

916

.

I will call

β

-thick

and denote thick

β

(

E

) a

Rel

-endomorphism

E

when there exists a finite set

B

such that

h

E

i

B

= Ob

E

.

Proposition

917

.

thick

α

(

E

)

thick

β

(

E

).

Proof.

Let thick

α

(

E

). Then there exists a finite cover

S

of the set Ob

E

such

that

A

S

:

A

×

A

GR

E

. Without loss of generality assume

A

6

=

for every

A

S

. So

A

⊆ h

E

i

{

x

A

}

for some

x

A

for every

A

S

. So

h

E

i

n

x

A

A

S

o

=

[

h

E

i

{

x

A

}

A

S

= Ob

E

and thus

E

is

β

-thick.

Obvious

918

.

Let

X

be a set,

A

and

B

are

Rel

-endomorphisms on

X

and

B

w

A

. Then:

thick

α

(

A

)

thick

α

(

B

);

thick

β

(

A

)

thick

β

(

B

).

Example

919

.

There is a

β

-thick Rel-morphism which is not

α

-thick.

Proof.

Consider the

Rel

-morphism on [0; 1] with the graph on figure

1

:

Γ =

(

x

;

x

)

x

[0; 1]

(

x

; 0)

x

[0; 1]

(0;

x

)

x

[0; 1]

.

y

x

1

0

1

Figure 1.

Thickness counterexample graph

Γ is

β

-thick because

h

Γ

i

{

0

}

= [0; 1].

To prove that Γ is not

α

-thick it’s enough to prove that every set

A

such that

A

×

A

Γ is finite.

171