3.9. SOME PROPERTIES OF FRAMES

58

Lemma

338

.

For a complete lattice

A

, the map

d

: Up(

A

)

A

preserves

arbitrary meets.

Proof.

Let

S

P

Up(

A

) . We have

d

S

Up(

A

).

d d

S

=

d d

X

S

X

=

d

X

S

d

X

is what we needed to prove.

Lemma

339

.

A complete lattice

A

is a co-frame iff

d

: Up(

A

)

A

preserves

finite joins.

Proof.

. Let

A

be a co-frame. Let

D, D

0

Up(

A

). Obviously

d

(

D

t

D

0

)

w

d

D

and

d

(

D

t

D

0

)

w

d

D

0

, so

d

(

D

t

D

0

)

w

d

D

t

d

D

0

.

Also

l

D

t

l

D

0

=

[

D

t

[

D

0

= (because

A

is a co-frame) =

[

d

t

d

0

d

D, d

0

D

0

.

Obviously

d

t

d

0

D

D

0

, thus

d

D

t

d

D

0

S

(

D

D

0

) =

d

(

D

D

0

)

that is

d

D

t

d

D

0

w

d

(

D

D

0

). So

d

(

D

t

D

0

) =

d

D

t

d

D

0

that is

d

: Up(

A

)

A

preserves binary joins.

It preserves nullary joins since

d

Up(

A

)

Up(

A

)

=

d

Up(

A

)

A

=

A

.

. Suppose

d

: Up(

A

)

A

preserves finite joins. Let

b

A

,

S

P

A

. Let

D

be

the smallest upper set containing

S

(so

D

=

S

h↑i

S

). Then

b

t

l

S

=

l

b

t

[

l

h↑i

S

=

l

b

t

l

[

h↑i

S

= (since

l

preserves finite joins)

l

b

t

[

h↑i

S

=

[

b

[

h↑i

S

=

l

[

a

S

(

b

∩ ↑

a

) =

l

[

a

S

(

b

t

a

) = (since

l

preserves all meets)

[

a

S

l

(

b

t

a

) =

[

a

S

(

b

t

a

) =

l

a

S

(

b

t

a

)

.

Corollary

340

.

If

A

is a co-frame, then the composition

F

=

↑ ◦

d

: Up(

A

)

Up(

A

) is a co-nucleus. The embedding

:

A

Up(

A

) is an isomorphism of

A

onto

the co-frame Fix(

F

).

Proof.

D

w

F

(

D

) follows from theorem

336

.

We have

F

(

F

(

D

)) =

F

(

D

) for all

D

Up(

A

) since

F

(

F

(

D

)) =

d

d

D

=

(because

d

s

=

s

for any

s

) =

d

D

=

F

(

D

).

And since both

d

: Up(

A

)

A

and

preserve finite joins,

F

preserves finite

joins. Thus

F

is a co-nucleus.