3.3. SEVERAL EQUAL WAYS TO EXPRESS PSEUDODIFFERENCE

42

3.3. Several equal ways to express pseudodifference

Theorem

244

.

For an atomistic co-brouwerian lattice

A

and

a, b

A

the

following expressions are always equal:

1

.

a

\

b

=

d

n

z

A

a

v

b

t

z

o

(quasidifference of

a

and

b

);

2

.

a

#

b

=

d

n

z

A

z

v

a

z

u

b

=

o

(second quasidifference of

a

and

b

);

3

.

d

(atoms

a

\

atoms

b

).

Proof.

Proof of

1

=

3

.

a

\

b

=

l

atoms

a

\

b

= (theorem

163

)

l

A

atoms

a

(

A

\

b

) =

l

A

atoms

a

(

A

if

A /

atoms

b

if

A

atoms

b

!

=

l

A

A

atoms

a, A /

atoms

b

=

l

(atoms

a

\

atoms

b

)

.

Proof of

2

=

3

.

a

\

b

is defined because our lattice is co-brouwerian. Taking the

above into account, we have

a

\

b

=

l

(atoms

a

\

atoms

b

) =

l

z

atoms

a

z

u

b

=

.

So

d

z

atoms

a

z

u

b

=

is defined.

If

z

v

a

z

u

b

=

then

z

0

=

d

x

atoms

z

x

u

b

=

is defined because

z

0

=

z

\

b

(atomisticity taken into account).

z

0

is a lower bound for

z

atoms

a

x

u

b

=

.

Thus

z

0

n

z

A

z

v

a

z

u

b

=

o

and so

d

z

atoms

a

z

u

b

=

is an upper bound of

n

z

A

z

v

a

z

u

b

=

o

.

If

y

is above every

z

0

n

z

A

z

v

a

z

u

b

=

o

then

y

is above every

z

atoms

a

such that

z

u

b

=

and thus

y

is above

d

z

atoms

a

z

u

b

=

.

Thus

d

z

atoms

a

z

u

b

=

is least upper bound of

z

A

z

v

a

z

u

b

=

,

that is

l

z

A

z

v

a

z

u

b

=

=

l

z

atoms

a

z

u

b

=

=

l

(atoms

a

\

atoms

b

)

.