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Proof.

(1)

,

(2)

,

(3)

,

(4)

,

(5)

,

(6).

By the above theorem.

(8)

)

(4).

Let property (8) hold. Let

a

@

b

. Then it exists element

c

v

b

such that

c

=

/ 0

and

c

u

a

= 0

. But

c

u

b

=

/ 0

. So

?a

=

/

?b

.

(2)

)

(7).

Let property (2) hold. Let

a

v

b

. Then

?a

v

? b

that is it there exists

c

2

?a

such

that

c

2

/

?b

, in other words

c

u

a

=

/ 0

and

c

u

b

= 0

. Let

d

=

c

u

a

. Then

d

v

a

and

d

=

/ 0

and

d

u

b

= 0

. So disjunction property of Wallman holds.

(7)

)

(8).

Obvious.

(8)

)

(7).

Let

b

v

a

. Then

a

u

b

@

b

that is

a

0

@

b

where

a

0

=

a

u

b

. Consequently

9

c

2

A

n f

0

g

:

(

c

a

0

^

c

v

b

)

. We have

c

u

a

=

c

u

b

u

a

=

c

u

a

0

. So

c

v

b

and

c

u

a

= 0

. Thus Wallman's

disjunction property holds.

Proposition 3.15.

Every boolean lattice is separable.

Proof.

Let

a; b

2

A

where

A

is a boolean lattice an

a

=

/

b

. Then

a

u

b

 =

/ 0

or

a

u

b

=

/ 0

because

otherwise

a

u

b

 = 0

and

a

t

b

 = 1

and thus

a

=

b

. Without loss of generality assume

a

u

b

 =

/ 0

. Then

a

u

c

=

/ 0

and

b

u

c

= 0

for

c

=

a

u

b

 =

/ 0

.

3.1.3 Atomically Separable Lattices

Proposition 3.16.

atoms is a straight monotone map (for any meet-semilattice).

Proof.

Monotonicity is obvious. The rest follows from the formula

atoms

(

a

u

b

) =

atoms

a

\

atoms

b

(the corollary

2.87

).

Denition 3.17.

I will call

atomically separable

such a poset that atoms is an injection.

Proposition 3.18.

8

a; b

2

A

: (

a

@

b

)

atoms

a

atoms

b

)

i

A

is atomically separable for a poset

A

.

Proof.

(

.

Obvious.

)

.

Let

a

=

/

b

for example

a

v

b

. Then

a

u

b

@

a

; atoms

a

atoms

(

a

u

b

) =

atoms

a

\

atoms

b

and

thus atoms

a

=

/

atoms

b

.

Proposition 3.19.

Any atomistic poset is atomically separable.

Proof.

We need to prove that atoms

a

=

atoms

b

)

a

=

b

. But it is obvious because

a

=

G

atoms

a

and

b

=

G

atoms

b:

Theorem 3.20.

If a lattice with least element is atomic and separable then it is atomistic.

Proof.

Suppose the contrary that is

a

A

F

atoms

a

. Then, because our lattice is separable, there

exists

c

2

A

such that

c

u

a

=

/ 0

and

c

u

F

atoms

a

= 0

. There exists atom

d

v

c

such that

d

v

c

u

a

.

d

u

F

atoms

a

v

c

u

F

atoms

a

= 0

. But

d

2

atoms

a

. Contradiction.

Theorem 3.21.

Let

A

be an atomic meet-semilattice with least element. Then the following

statements are equivalent:

1.

A

is separable.

2.

A

is atomically separable.

3.

A

conforms to Wallman's disjunction property.

3.1 Straight maps and separation subsets

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