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3.1. STRAIGHT MAPS AND SEPARATION SUBSETS

38

8

.

a, b

A

: (

a

@

b

⇒ ∃

c

A

\ {⊥}

: (

c

a

c

v

b

)).

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

6

=

and

c

u

a

=

. But

c

u

b

6

=

. So

?a

6

=

?b

.

2

7

Let property

2

hold. Let

a

6v

b

. Then

?a

6v

?b

that is it there exists

c

?a

such that

c /

?b

, in other words

c

u

a

6

=

and

c

u

b

=

. Let

d

=

c

u

a

.

Then

d

v

a

and

d

6

=

and

d

u

b

=

. So disjunction property of Wallman

holds.

7

8

Obvious.

8

7

Let

b

6v

a

. Then

a

u

b

@

b

that is

a

0

@

b

where

a

0

=

a

u

b

. Consequently

c

A

\ {⊥}

: (

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

=

. Thus Wallman’s disjunction property holds.

Proposition

223

.

Every boolean lattice is strongly separable.

Proof.

Let

a, b

A

where

A

is a boolean lattice an

a

6

=

b

. Then

a

u

¯

b

6

=

or

¯

a

u

b

6

=

because otherwise

a

u

¯

b

=

and

a

t

¯

b

=

>

and thus

a

=

b

. Without loss

of generality assume

a

u

¯

b

6

=

. Then

a

u

c

6

=

and

b

u

c

=

for

c

=

a

u

¯

b

6

=

,

that is our lattice is separable.

It is strongly separable by theorem

222

.

3.1.3. Atomically Separable Lattices.

Proposition

224

.

“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

(corollary

109

).

Definition

225

.

I will call

atomically separable

such a poset that “atoms” is

an injection.

Proposition

226

.

a, b

A

: (

a

@

b

atoms

a

atoms

b

) iff

A

is atomically

separable for a poset

A

.

Proof.

. Obvious.

. Let

a

6

=

b

for example

a

6v

b

. Then

a

u

b

@

a

; atoms

a

atoms(

a

u

b

) =

atoms

a

atoms

b

and thus atoms

a

6

= atoms

b

.

Proposition

227

.

Any atomistic poset is atomically separable.

Proof.

We need to prove that atoms

a

= atoms

b

a

=

b

. But it is obvious

because

a

=

l

atoms

a

and

b

=

l

atoms

b.

Theorem

228

.

A complete lattice is atomistic iff it is atomically separable.