6.10. ATOMIC FUNCOIDS

118

3

. conforming to Wallman’s disjunction property.

Proof.

By theorem

180

.

Remark

654

.

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

of funcoids see subsections “Separation subsets and full stars” and “Atomically

separable lattices”.

Corollary

655

.

The lattice

FCD

(

A

;

B

) is an atomistic lattice.

Proof.

FiXme

: Should be generalized.

Let

f

FCD

(

A

;

B

). Suppose contrary

to the statement to be proved that

F

atoms

f

@

f

. Then there exists

a

atoms

f

such that

a

u

F

atoms

f

=

FCD

(

A

;

B

)

what is impossible.

Proposition

656

.

atoms(

f

t

g

) = atoms

f

atoms

g

for every funcoids

f, g

FCD

(

A

;

B

) (for every sets

A

,

B

).

Proof.

a

×

FCD

b

6

f

t

g

a

[

f

t

g

]

b

a

[

f

]

b

a

[

g

]

b

a

×

FCD

b

6

f

a

×

FCD

b

6

g

for every atomic filters

a

and

b

.

Theorem

657

.

For every

f, g, h

FCD

(

A

;

B

),

R

P

FCD

(

A

;

B

) (for every

sets

A

and

B

)

1

.

f

u

(

g

t

h

) = (

f

u

g

)

t

(

f

u

h

);

2

.

f

t

d

R

=

d

h

f

ti

R

.

Proof.

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

lattice.

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

))

.