3.10

Atomic funcoids

Theorem 22

An

f

FCD

(

A

;

B

)

is an atom of the lattice

FCD

(

A

;

B

)

(for

small sets

A

,

B

) iff it is funcoidal product of two atomic filter objects.

Proof

Let

f

FCD

(

A

;

B

) be an atom of the lattice

FCD

(

A

;

B

). Let’s get elements

a

atoms dom

f

and

b

atoms

h

f

i

a

. Then for every

X ∈

F

(

A

)

X ≍

a

a

×

FCD

b

X

= 0

F

(

B

)

⊆ h

f

i X

,

X 6≍

a

a

×

FCD

b

X

=

b

⊆ h

f

i X

.

So

a

×

FCD

b

f

; because

f

is atomic we have

f

=

a

×

FCD

b

.

Let

a

atoms 1

F

(

A

)

,

b

atoms 1

F

(

B

)

,

f

FCD

(

A

;

B

). If

b

≍ h

f

i

a

then

¬

(

a

[

f

]

b

),

f

a

×

FCD

b

; if

b

⊆ h

f

i

a

then

∀X ∈

F

(

A

) : (

X 6≍

a

⇒ h

f

i X ⊇

b

),

f

a

×

FCD

b

. Consequently

f

a

×

FCD

b

f

a

×

FCD

b

; that is

a

×

FCD

b

is an atom.

Theorem 23

The lattice

FCD

(

A

;

B

)

is atomic (for every small sets

A

,

B

).

Proof

Let

f

is a non-empty funcoid from

A

to

B

. Then dom

f

6

= 0

F

(

A

)

, thus

by the theorem 47 in [15] there exists

a

atoms dom

f

. So

h

f

i

a

6

= 0

F

(

B

)

thus

exists

b

atoms

h

f

i

a

. Finally the atomic funcoid

a

×

FCD

b

f

.

Theorem 24

The lattice

FCD

(

A

;

B

)

is separable (for every small sets

A

,

B

).

Proof

Let

f, g

FCD

(

A

;

B

),

f

g

. Then exists

a

atoms 1

F

(

A

)

such

that

h

f

i

a

⊂ h

g

i

a

. So because the lattice

F

(

B

) is atomically separable then

exists

b

atoms 1

F

(

B

)

such that

h

f

i

a

b

= 0

F

(

B

)

and

b

⊆ h

g

i

a

. For every

x

atoms 1

F

(

A

)

h

f

i

a

a

×

FCD

b

a

=

h

f

i

a

b

= 0

F

(

B

)

,

x

6

=

a

⇒ h

f

i

x

a

×

FCD

b

x

=

h

f

i

x

0

F

(

B

)

= 0

F

(

B

)

.

Thus

h

f

i

x

a

×

FCD

b

x

= 0

F

(

B

)

and consequently

f

a

×

FCD

b

.

a

×

FCD

b

a

=

b

⊆ h

g

i

a,

x

6

=

a

a

×

FCD

b

x

= 0

F

(

B

)

⊆ h

g

i

x.

Thus

a

×

FCD

b

x

⊆ h

g

i

x

and consequently

a

×

FCD

b

g

.

So the lattice

FCD

(

A

;

B

) is separable by the theorem 19 in [15].

Corollary 7

The lattice

FCD

(

A

;

B

)

is:

27