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CHAPTER 13

Total boundness of reloids

13.1. Thick binary relations

Definition

1231

.

I will call

α

-thick

and denote thick

α

(

E

) a

Rel

-

endomorphism

E

when there exists a finite cover

S

of Ob

E

such that

A

S

:

A

×

A

GR

E

.

Definition

1232

.

CS(

S

) =

S

A

×

A

A

S

 

for a collection

S

of sets.

Remark

1233

.

CS means “Cartesian squares”.

Obvious

1234

.

A

Rel

-endomorphism is

α

-thick iff there exists a finite cover

S

of Ob

E

such that CS(

S

)

GR

E

.

Definition

1235

.

I will call

β

-thick

and denote thick

β

(

E

) a

Rel

-

endomorphism

E

when there exists a finite set

B

such that

h

GR

E

i

B

= Ob

E

.

Proposition

1236

.

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

GR

E

i

{

x

A

}

for some

x

A

for every

A

S

. So

h

GR

E

i

n

x

A

A

S

o

=

[

h

GR

E

i

{

x

A

}

A

S

= Ob

E

and thus

E

is

β

-thick.

Obvious

1237

.

Let

X

be a set,

A

and

B

be

Rel

-endomorphisms on

X

and

B

w

A

. Then:

thick

α

(

A

)

thick

α

(

B

);

thick

β

(

A

)

thick

β

(

B

).

Example

1238

.

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].

232