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12.6. IRREFLEXIVE RELOIDS

230

Proof.

S

(

S

(

f

)) =

RLD

l

R

up

S

(

f

)

S

(

R

)

v

RLD

l

R

S

(

F

)

F

up

f

 

S

(

R

) =

RLD

l

R

up

f

S

(

S

(

R

)) =

RLD

l

R

up

f

S

(

R

) =

S

(

f

)

.

So

S

(

S

(

f

))

v

S

(

f

). That

S

(

S

(

f

))

w

S

(

f

) is obvious.

Corollary

1224

.

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

1225

.

S

(

S

(

f

)) =

S

(

f

) for

1

. every endoreloid

f

;

2

. every endofuncoid

f

.

Conjecture

1226

.

S

(

f

)

S

(

f

) =

S

(

f

) for every endoreloid

f

.

Theorem

1227

.

S

(

f

)

S

(

f

) =

S

(

f

)

S

(

f

) =

S

(

f

)

S

(

f

) =

S

(

f

) for

every endoreloid

f

.

Proof.

2

It is enough to prove

S

(

f

)

S

(

f

) =

S

(

f

) because

S

(

f

)

v

S

(

f

)

S

(

f

)

v

S

(

f

)

S

(

f

) and likewise for

S

(

f

)

S

(

f

).

S

(

µ

)

S

(

µ

) =

d

RLD

F

up

S

(

µ

)

(

F

F

) = (see below) =

d

RLD

X

up

µ

(

S

(

X

)

S

(

X

)) =

d

RLD

X

up

µ

S

(

X

) =

S

(

µ

).

F

up

S

(

µ

)

F

up

d

F

F

up

µ

S

(

F

)

(by properties of filter bases)

X

up

µ

:

F

w

S

(

X

)

⇒ ∃

X

up

µ

:

F

F

w

S

(

X

)

S

(

X

) thus

RLD

l

F

up

S

(

µ

)

F

F

w

RLD

l

X

up

µ

(

S

(

X

)

S

(

X

));

X

up

µ

S

(

X

)

up

S

(

µ

)

⇒ ∃

F

up

S

(

µ

) :

S

(

X

)

S

(

X

)

w

F

F

thus

RLD

l

F

up

S

(

µ

)

F

F

v

RLD

l

X

up

µ

(

S

(

X

)

S

(

X

))

.

Conjecture

1228

.

S

(

f

)

S

(

f

) =

S

(

f

) for every endofuncoid

f

.

12.6. Irreflexive reloids

Definition

1229

.

Endoreloid

f

is irreflexive iff

f

1

Ob

f

.

Proposition

1230

.

Endoreloid

f

is irreflexive iff

f

v > \

1.

Proof.

By theorem

604

.

Obvious

1231

.

f

\

1 is an irreflexive endoreloid if

f

is an endoreloid.

Proposition

1232

.

S

(

f

) =

S

(

f

t

1) if

f

is an endoreloid, endofuncoid, or

endorelation.

2

Can be more succintly proved considering

µ

7→

S

(

µ

) as a pointfree funcoid?