Theorem 15.73.

Let

(

A

;

Z

0

)

and

(

B

;

Z

1

)

be primary ltrators over boolean lattices. If

A 2

A

then

FCD

is a complete homomorphism of the lattice

A

to a the lattice

FCD

(

A

;

B

)

, if also

A

=

/ 0

A

then it is an order embedding.

Proof.

Let

S

2

P

A

,

X

2

Z

0

,

x

2

atoms

A

.

G

hA

FCD

i

S

X

=

G

fhA

FCD

Bi

X

j B 2

S

g

=

( F

S

if

X

u

A

A

=

/ 0

A

0

B

if

X

u

A

A

= 0

A

=

FCD

G

S

X:

Thus

F

hA

FCD

i

S

=

FCD

F

S

by theorem

15.25

.

l

hA

FCD

i

S

x

=

l

fhA

FCD

Bi

x

j B 2

S

g

=

(

d

S

if

x

u

A

A

=

/ 0

A

0

B

if

x

u

A

A

= 0

A

=

FCD

l

S

x:

Thus

d

hA

FCD

i

S

=

FCD

d

S

by theorem

15.54

.

If

A

=

/ 0

A

then obviously the function

FCD

is injective.

Proposition 15.74.

Let

A

be a meet-semilattice with least element and

B

be a poset with least

element. If

a

is an atom of

A

,

f

2

FCD

(

A

;

B

)

then

f

j

a

=

a

FCD

h

f

i

a

.

Proof.

Let

X 2

A

.

X u

a

=

/ 0

A

) h

f

j

a

iX

=

h

f

i

a;

X u

a

= 0

A

) h

f

j

a

iX

= 0

B

:

Proposition 15.75.

f

(

FCD

B

) =

FCD

h

f

iB

for elements

A 2

A

and

B

2

B

of some posets

A

,

B

,

C

with least elements and

f

2

FCD

(

B

;

C

)

.

Proof.

Let

X 2

A

,

Y 2

B

.

h

f

(

FCD

B

)

iX

=

h

f

iB

if

/

A

0

if

X  A

=

hA

FCD

h

f

iBiX

.

h

(

f

(

FCD

B

))

¡

1

iY

=

h

(

FCD

A

)

f

¡

1

iY

=

(

A

if

h

f

¡

1

iY

/

B

0

if

h

f

¡

1

iY  B

=

A

if

/

h

f

iB

0

if

Y  h

f

iB

=

hh

f

iB

FCD

AiY

=

h

(

FCD

h

f

iB

)

¡

1

iY

.

15.10 Atomic pointfree funcoids

Theorem 15.76.

Let

A

,

B

be sets of lters over boolean lattices. A

f

2

FCD

(

A

;

B

)

is an atom

of the poset

FCD

(

A

;

B

)

i there exist

a

2

atoms

A

and

b

2

atoms

B

such that

f

=

a

FCD

b

.

Proof.

A

and

B

are atomic by the theorem

4.135

.

)

.

Let

f

be an atom of the poset

FCD

(

A

;

B

)

. Let's get elements

a

2

atoms dom

f

and

b

2

atoms

h

f

i

a

. Then for every

X 2

A

A

a

) h

a

FCD

b

iX

= 0

B

v h

f

iX

;

/

A

a

) h

a

FCD

b

iX

=

b

v h

f

iX

:

So

a

FCD

b

v

f

; because

f

is atomic we have

f

=

a

FCD

b

.

(

.

Let

a

2

atoms

A

,

b

2

atoms

B

,

f

2

FCD

(

A

;

B

)

. If

b

B

h

f

i

a

then

:

(

a

[

f

]

b

)

,

f

u

(

a

FCD

b

) =

0

FCD

(

A

;

B

)

(because

A

and

B

are bounded meet-semilattices); if

b

v h

f

i

a

then

8X 2

A

:

(

/

a

) h

f

iX w

b

)

,

f

w

a

FCD

b

. Consequently

f

u

(

a

FCD

b

) = 0

FCD

(

A

;

B

)

_

f

w

a

FCD

b

;

that is

a

FCD

b

is an atomic pointfree funcoid.

190

Pointfree funcoids