background image

11.5. ALGEBRAIC PROPERTIES OF

S

AND

S

170

Proof.

S

(

S

(

f

)) =

l

RLD

S

(

R

)

R

xyGR

S

(

f

)

v

l

RLD

S

(

R

)

R

n

S

(

F

)

F

xyGR

f

o

=

l

RLD

S

(

S

(

R

))

R

xyGR

S

(

f

)

=

l

RLD

S

(

R

)

R

xyGR

S

(

f

)

=

S

(

f

)

.

So

S

(

S

(

f

))

v

S

(

f

). That

S

(

S

(

f

))

w

S

(

f

) is obvious.

Corollary

908

.

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

909

.

S

(

S

(

f

)) =

S

(

f

) for

1

. every endoreloid

f

;

2

. every endofuncoid

f

.

Conjecture

910

.

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

911

.

S

(

f

)

S

(

f

) =

S

(

f

) for every endofuncoid

f

.