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1. separable;

2. atomically separable;

3. conforming to Wallman’s disjunction property.

Proof

By the theorem 22 in [15].

Remark 2

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

of funcoids see [15], subsections “Separation subsets and full stars” and “Atom-
ically separable lattices”.

Corollary 8

The lattice

FCD

(

A

;

B

)

is an atomistic lattice.

Proof

Let

f

FCD

(

A

;

B

). Suppose contrary to the statement to be proved

that

S

atoms

f

f

. Then it exists

a

atoms

f

such that

a

S

atoms

f

=

0

FCD

(

A

;

B

)

what is impossible.

Proposition 20

atoms(

f

g

) = atoms

f

atoms

g

for every funcoids

f, g

FCD

(

A

;

B

)

(for every small sets

A

and

B

).

Proof

a

×

FCD

b

6≍

f

g

a

[

f

g

]

b

a

[

f

]

b

a

[

g

]

b

a

×

FCD

b

6≍

f

a

×

FCD

b

6≍

g

for every atomic filter objects

a

and

b

.

Theorem 25

For every

f, g, h

FCD

(

A

;

B

)

,

R

P

FCD

(

A

;

B

)

(for every

small sets

A

and

B

)

1.

f

(

g

h

) = (

f

g

)

(

f

h

)

;

2.

f

T

R

=

T

h

f

∪i

R

.

Proof

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

lattice.

1. atoms (

f

(

g

h

)) = atoms

f

atoms(

g

h

) = atoms

f

(atoms

g

atoms

h

) =

(atoms

f

atoms

g

)

(atoms

f

atoms

h

) = atoms(

f

g

)

atoms(

f

h

) =

atoms ((

f

g

)

(

f

h

)).

2. atoms (

f

T

R

) = atoms

f

atoms

T

R

= atoms

f

T

h

atoms

i

R

=

T

h

(atoms

f

)

∪i h

atoms

i

R

=

T

h

atoms

i h

f

∪i

R

= atoms

T

h

f

∪i

R

. (Used the following equality.)

h

(atoms

f

)

∪i h

atoms

i

R

=

{

(atoms

f

)

A

|

A

∈ h

atoms

i

R

}

=

{

(atoms

f

)

A

| ∃

C

R

:

A

= atoms

C

}

=

{

(atoms

f

)

(atoms

C

)

|

C

R

}

=

{

atoms(

f

C

)

|

C

R

}

=

{

atoms

B

| ∃

C

R

:

B

=

f

C

}

=

{

atoms

B

|

B

∈ h

f

∪i

R

}

=

h

atoms

i h

f

∪i

R.

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