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3.2. QUASIDIFFERENCE AND QUASICOMPLEMENT

39

Proof.

Direct implication is the above proposition. Let’s prove the reverse

implication.

Let “atoms” be injective. Consider an element

a

of our poset. Let

b

=

d

atoms

a

. Obviously

b

v

a

and thus atoms

b

atoms

a

. But if

x

atoms

a

then

x

v

b

and thus

x

atoms

b

. So atoms

a

= atoms

b

. By injectivity

a

=

b

that

is

a

=

d

atoms

a

.

Theorem

229

.

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

is atomistic.

Proof.

Suppose the contrary that is

a

A

d

atoms

a

. Then, because our lattice

is separable, there exists

c

A

such that

c

u

a

6

=

and

c

u

d

atoms

a

=

. There

exists atom

d

v

c

such that

d

v

c

u

a

.

d

u

d

atoms

a

v

c

u

d

atoms

a

=

. But

d

atoms

a

. Contradiction.

Theorem

230

.

Let

A

be an atomic meet-semilattice with least element. Then

the following statements are equivalent:

1

.

A

is separable.

2

.

A

is strongly separable.

3

.

A

is atomically separable.

4

.

A

conforms to Wallman’s disjunction property.

5

.

a, b

A

: (

a

@

b

⇒ ∃

c

A

\ {⊥}

: (

c

a

c

v

b

)).

Proof.

1

2

4

5

Proved above.

3

5

Let our semilattice be atomically separable. Let

a

@

b

. Then atoms

a

atoms

b

and there exists

c

atoms

b

such that

c /

atoms

a

.

c

6

=

and

c

v

b

, from which (taking into account that

c

is an atom)

c

v

b

and

c

u

a

=

. So our semilattice conforms to the formula

5

.

5

3

Let formula

5

hold. Then for any elements

a

@

b

there exists

c

6

=

such

that

c

v

b

and

c

u

a

=

. Because

A

is atomic there exists atom

d

v

c

.

d

atoms

b

and

d /

atoms

a

. So atoms

a

6

= atoms

b

and atoms

a

atoms

b

.

Consequently atoms

a

atoms

b

.

Theorem

231

.

Any atomistic poset is strongly separable.

Proof.

?x

v

?y

atoms

x

v

atoms

y

x

v

y

because atoms

x

=

?x

atoms

A

.

3.2. Quasidifference and Quasicomplement

I’ve got quasidifference and quasicomplement (and dual quasicomplement) re-

placing max and min in the definition of pseudodifference and pseudocomplement

(and dual pseudocomplement) with

d

and

d

. Thus quasidifference and (dual)

quasicomplement are generalizations of their pseudo- counterparts.

Remark

232

.

Pseudocomplements

and

pseudodifferences

are standard termi-

nology.

Quasi-

counterparts are my neologisms.

Definition

233

.

Let

A

be a poset,

a

A

.

Quasicomplement

of

a

is

a

=

l

c

A

c

a

.

Definition

234

.

Let

A

be a poset,

a

A

.

Dual quasicomplement

of

a

is

a

+

=

l

c

A

c

a

.