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3.2. QUASIDIFFERENCE AND QUASICOMPLEMENT

41

2

.

a

#(

a

u

b

) =

l

z

A

z

v

a

z

u

a

u

b

=

=

l

z

A

z

v

a

(

z

u

a

)

u

a

u

b

=

=

l

z

u

a

z

A

, z

u

a

u

b

=

=

l

z

A

z

v

a, z

u

b

=

=

a

#

b.

I will denote

Da

the lattice

n

x

A

x

v

a

o

.

Theorem

242

.

For

a, b

A

where

A

is a distributive lattice

1

.

a

\

b

= (

a

u

b

)

+(

Da

)

;

2

.

a

#

b

= (

a

u

b

)

(

Da

)

if

A

has least element.

Proof.

1

.

(

a

u

b

)

+(

Da

)

=

l

c

Da

c

t

(

a

u

b

) =

a

=

l

c

Da

c

t

(

a

u

b

)

w

a

=

l

c

Da

(

c

t

a

)

u

(

c

t

b

)

w

a

=

l

c

A

c

v

a

c

t

b

w

a

=

a

\

b.

2

.

(

a

u

b

)

(

Da

)

=

l

c

Da

c

u

a

u

b

=

=

l

c

A

c

v

a

c

u

a

u

b

=

=

l

c

A

c

v

a

c

u

b

=

=

a

#

b.

Proposition

243

.

(

a

t

b

)

\

b

v

a

for an arbitrary complete lattice.

Proof.

(

a

t

b

)

\

b

=

d

n

z

A

a

t

b

v

b

t

z

o

.

But

a

v

z

a

t

b

v

b

t

z

. So

n

z

A

a

t

b

v

b

t

z

o

n

z

A

a

v

z

o

.

Consequently, (

a

t

b

)

\

b

v

d

n

z

A

a

v

z

o

=

a

.