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Theorem 3.36.

a

n

b

=

d

f

z

2

A

j

z

v

a

^

a

v

b

t

z

g

where

A

is a distributive lattice and

a; b

2

A

.

Proof.

Obviously

f

z

2

A

j

z

v

a

^

a

v

b

t

z

g  f

z

2

A

j

a

v

b

t

z

g

. Thus

d

f

z

2

A

j

z

v

a

^

a

v

b

t

z

g w

a

n

b

.

Let

z

2

A

and

z

0

=

z

u

a

.

a

v

b

t

z

)

a

v

(

b

t

z

)

u

a

,

a

v

(

b

u

a

)

t

(

z

u

a

)

,

a

v

(

b

u

a

)

t

z

0

)

a

v

b

u

z

0

and

a

v

b

t

z

(

a

v

b

u

z

0

. Thus

a

v

b

t

z

,

a

v

b

u

z

0

.

If

z

2 f

z

2

A

j

a

v

b

t

z

g

then

a

v

b

t

z

and thus

z

0

2 f

z

2

A

j

z

v

a

^

a

v

b

t

z

g

:

But

z

0

v

z

thus having

d

f

z

2

A

j

z

v

a

^

a

v

b

t

z

g v

d

f

z

2

A

j

a

v

b

t

z

g

.

Remark 3.37.

If we drop the requirement that

A

is distributive, two formulas for quasidierence

(the denition and the last theorem) fork.

Obvious 3.38.

Dual quasicomplement is the dual of quasicomplement.

Obvious 3.39.

Every pseudocomplement is quasicomplement.

Every dual pseudocomplement is dual quasicomplement.

Every pseudodierence is quasidierence.

Below we will stick to the more general quasies than pseudos. If needed, one can check that a

quasicomplement

a

is a pseudocomplement by the equation

a

a

(and analogously with other

quasies).

Next we will express quasidierence through quasicomplement.

Proposition 3.40.

1.

a

n

b

=

a

n

(

a

u

b

)

for any distributive lattice;

2.

a

#

b

=

a

#(

a

u

b

)

for any distributive lattice with least element.

Proof.

1.

a

v

(

a

u

b

)

t

z

,

a

v

(

a

t

z

)

u

(

b

t

z

)

,

a

v

a

t

z

^

a

v

b

t

z

,

a

v

b

t

z

. Thus

a

n

(

a

u

b

) =

d

f

z

2

A

j

a

v

(

a

u

b

)

t

z

g

=

d

f

z

2

A

j

a

v

b

t

z

g

=

a

n

b

.

2.

a

#(

a

u

b

) =

F

f

z

2

A

j

z

v

a

^

z

u

a

u

b

= 0

g

=

F

f

z

2

A

j

z

v

a

^

(

z

u

a

)

u

a

u

b

= 0

g

=

F

f

z

u

a

j

z

2

A

; z

u

a

u

b

= 0

g

=

F

f

z

2

A

j

z

v

a; z

u

b

= 0

g

=

a

#

b

.

I will denote

Da

the lattice

f

x

2

A

j

x

v

a

g

.

Theorem 3.41.

For

a; b

2

A

where

A

is a distributive lattice with least element

1.

a

n

b

= (

a

u

b

)

+(

D a

)

;

[TODO: least element is not required?]

2.

a

#

b

= (

a

u

b

)

(

D a

)

.

Proof.

1.

(

a

u

b

)

+(

D a

)

=

l

f

c

2

Da

j

c

t

(

a

u

b

) =

a

g

=

l

f

c

2

Da

j

c

t

(

a

u

b

)

w

a

g

=

l

f

c

2

Da

j

(

c

t

a

)

u

(

c

t

b

)

w

a

g

=

l

f

c

2

A

j

c

v

a

^

c

t

b

w

a

g

=

a

n

b:

38

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