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Theorem 15.77.

Let

A

,

B

be sets of lters over boolean lattices. Then

FCD

(

A

;

B

)

is atomic.

Proof.

Let

f

2

FCD

(

A

;

B

)

and

f

=

/ 0

FCD

(

A

;

B

)

. Then dom

f

=

/ 0

A

, thus exists

a

2

atoms dom

f

. So

h

f

i

a

=

/ 0

B

thus exists

b

2

atoms

h

f

i

a

. Finally the atomic pointfree funcoid

a

FCD

b

v

f

.

Theorem 15.78.

Let

A

,

B

be sets of lters over boolean lattices. Then the poset

FCD

(

A

;

B

)

is

separable.

Proof.

Let

f ; g

2

FCD

(

A

;

B

)

,

f

@

g

. Then taking into account the theorem

6.65

exists

a

2

atoms

A

such that

h

f

i

a

@

h

g

i

a

. By corollary

4.138

B

is atomically separable. So exists

b

2

atoms

B

such

that

h

f

i

a

u

b

= 0

B

and

b

v h

g

i

a

. For every

x

2

atoms

A

h

f

i

a

u h

a

FCD

b

i

a

=

h

f

i

a

u

b

=0

B

;

x

=

/

a

) h

f

i

x

u h

a

FCD

b

i

x

=

h

f

i

x

u

0

B

= 0

B

:

Thus

h

f

i

x

u h

a

b

i

x

= 0

B

and consequently

f

u

(

a

FCD

b

) = 0

FCD

(

A

;

B

)

.

h

a

FCD

b

i

a

=

b

v h

g

i

a;

x

=

/

a

) h

a

FCD

b

i

x

= 0

B

v h

g

i

x:

Thus

h

a

FCD

b

i

x

v h

g

i

x

and consequently

a

FCD

b

v

g

.

So the lattice of pointfree funcoids is separable by the theorem

3.14

.

Corollary 15.79.

Let

A

,

B

be sets of lters over boolean lattices. The poset

FCD

(

A

;

B

)

is:

1. separable;
2. atomically separable;
3. conforming to Wallman's disjunction property.

Proof.

By the theorem

3.21

.

Remark 15.80.

For more ways to characterize (atomic) separability of the lattice of pointfree

funcoids see subsections Separation subsets and full stars and Atomically separable lattices.

Corollary 15.81.

Let

(

A

;

Z

0

)

and

(

B

;

Z

1

)

be primary ltrators over boolean lattices. The poset

FCD

(

A

;

B

)

is an atomistic lattice.

Proof.

By the corollary

15.34

FCD

(

A

;

B

)

is a complete lattice. Let

f

2

FCD

(

A

;

B

)

. Suppose

contrary to the statement to be proved that

F

atoms

f

@

f

. Then there exists

a

2

atoms

f

such

that

a

u

F

atoms

f

= 0

FCD

(

A

;

B

)

what is impossible.

Proposition 15.82.

Let

A

,

B

be sets of lters over boolean lattices.

atoms

(

f

t

g

) =

atoms

f

[

atoms

g

for every

f ; g

2

FCD

(

A

;

B

)

.

Proof.

(

a

FCD

b

)

u

(

f

t

g

) =

/

; ,

a

[

f

t

g

]

b

,

a

[

f

]

b

_

a

[

g

]

b

,

(

a

FCD

b

)

u

f

=

/ 0

FCD

(

A

;

B

)

_

(

a

FCD

b

)

u

g

=

/ 0

FCD

(

A

;

B

)

for every

a

2

atoms

A

and

b

2

atoms

B

(used the corollary

15.69

and

theorem

15.35

).

Theorem 15.83.

Let

(

A

;

Z

0

)

and

(

B

;

Z

1

)

be primary ltrators over boolean lattices. For every

f ; g; h

2

FCD

(

A

;

B

)

,

R

2

P

FCD

(

A

;

B

)

:

1.

f

u

(

g

t

h

) = (

f

u

g

)

t

(

f

u

h

)

;

2.

f

t

d

R

=

d

h

f

t i

R

.

Proof.

We will take into account that the lattice of funcoids is an atomistic lattice (corollary

15.81

).

1. atoms

(

f

u

(

g

t

h

)) =

atoms

f

\

atoms

(

g

t

h

) =

atoms

f

\

(

atoms

g

[

atoms

h

) = (

atoms

f

\

atoms

g

)

[

(

atoms

f

\

atoms

h

) =

atoms

(

f

u

g

)

[

atoms

(

f

u

h

) =

atoms

((

f

u

g

)

t

(

f

u

h

))

.

15.10 Atomic pointfree funcoids

191