background image

Proof.

"

Ob

A

is connected regarding a reloid

f

i

A

is connected regarding every

F

2

xyGR

f

that

is when (taken into account that connectedness for

"

RLD

F

is the same as connectedness of

"

FCD

F

)

8

F

2

xyGR

f

8X

;

Y 2

F

(

Ob

f

)

n

0

F

(

Ob

f

)

 

: (

X t Y

=

"

Ob

f

A

) X

[

"

FCD

F

]

Y

)

,

8X

;

Y 2

F

(

Ob

f

)

n

0

F

(

Ob

f

)

 

8

F

2

xyGR

f

: (

X t Y

=

"

Ob

f

A

) X

[

"

FCD

F

]

Y

)

,

8X

;

Y 2

F

(

Ob

f

)

n

0

F

(

Ob

f

)

 

: (

X t Y

=

"

Ob

f

A

) 8

F

2

xyGR

f

:

X

[

"

FCD

F

]

Y

)

,

8X

;

Y 2

F

(

Ob

f

)

n

0

F

(

Ob

f

)

 

: (

X t Y

=

"

Ob

f

A

) X

[(

FCD

)

f

]

Y

)

that is when the set

"

Ob

f

A

is connected regarding the funcoid

(

FCD

)

f

.

Conjecture 11.34.

A set

A

is connected regarding an endofuncoid

i for every

a; b

2

A

there

exists a totally ordered set

P

A

such that min

P

=

a

, max

P

=

b

and

8

q

2

P

n f

b

g

:

f

x

2

P

j

x

6

q

g

[

]

f

x

2

P

j

x > q

g

:

Weaker condition:

8

q

2

P

n f

b

g

:

f

x

2

P

j

x

6

q

g

[

]

f

x

2

P

j

x > q

g _ 8

q

2

P

n f

a

g

:

f

x

2

P

j

x < q

g

[

]

f

x

2

P

j

x

>

q

g

:

11.5 Algebraic properties of

S

and

S

Theorem 11.35.

S

(

S

(

f

)) =

S

(

f

)

for every endoreloid

f

.

Proof.

S

(

S

(

f

)) =

d

f"

RLD

S

(

R

)

j

R

2

xyGR

S

(

f

)

g v

d

f"

RLD

S

(

R

)

j

R

2 f

S

(

F

)

j

F

2

xyGR

f

gg

=

d

f"

RLD

S

(

S

(

R

))

j

R

2

xyGR

f

g

=

d

f"

RLD

S

(

R

)

j

R

2

xyGR

f

g

=

S

(

f

)

.

So

S

(

S

(

f

))

v

S

(

f

)

. That

S

(

S

(

f

))

w

S

(

f

)

is obvious.

Corollary 11.36.

S

(

S

(

f

)) =

S

(

S

(

f

)) =

S

(

f

)

for every endoreloid

f

.

Proof.

Obviously

S

(

S

(

f

))

w

S

(

f

)

and

S

(

S

(

f

))

w

S

(

f

)

.

But

S

(

S

(

f

))

v

S

(

S

(

f

)) =

S

(

f

)

and

S

(

S

(

f

))

v

S

(

S

(

f

)) =

S

(

f

)

.

Conjecture 11.37.

S

(

S

(

f

)) =

S

(

f

)

for

1. every endoreloid

f

;

2. every endofuncoid

f

.

Conjecture 11.38.

For every endoreloid

f

1.

S

(

f

)

S

(

f

) =

S

(

f

)

;

2.

S

(

f

)

S

(

f

) =

S

(

f

)

;

3.

S

(

f

)

S

(

f

) =

S

(

f

)

S

(

f

) =

S

(

f

)

.

Conjecture 11.39.

S

(

f

)

S

(

f

) =

S

(

f

)

for every endofuncoid

f

.

11.5 Algebraic properties of

S

and

S

151