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3.1. STRAIGHT MAPS AND SEPARATION SUBSETS

30

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

174

.

Every boolean lattice is 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

=

.

3.1.3. Atomically Separable Lattices.

Proposition

175

.

“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

97

).

Definition

176

.

I will call

atomically separable

such a poset that “atoms” is

an injection.

Proposition

177

.

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

178

.

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

179

.

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

A

such that

c

u

a

6

=

and

c

u

F

atoms

a

=

. There

exists atom

d

v

c

such that

d

v

c

u

a

.

d

u

F

atoms

a

v

c

u

F

atoms

a

=

. But

d

atoms

a

. Contradiction.

Theorem

180

.

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.

4

.

a, b

A

: (

a

@

b

⇒ ∃

c

A

\ {⊥}

: (

c

a

c

v

b

)).

Proof.

1

3

4

Proved above.