3.3. QUASIDIFFERENCE AND QUASICOMPLEMENT

33

I will denote quasicomplement and dual quasicomplement for a specific poset

A

as

a

(

A

)

and

a

+(

A

)

.

Definition

193

.

Let

a, b

A

where

A

is a distributive lattice.

Quasidifference

of

a

and

b

is

a

\

b

=

l

z

A

a

v

b

t

z

.

Definition

194

.

Let

a, b

A

where

A

is a distributive lattice.

Second quasid-

ifference

of

a

and

b

is

a

#

b

=

G

z

A

z

v

a

z

b

.

Theorem

195

.

a

\

b

=

d

n

z

A

z

v

a

a

v

b

t

z

o

where

A

is a distributive lattice and

a, b

A

.

Proof.

Obviously

n

z

A

z

v

a

a

v

b

t

z

o

n

z

A

a

v

b

t

z

o

. Thus

d

n

z

A

z

v

a

a

v

b

t

z

o

w

a

\

b

.

Let

z

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

)

(

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

n

z

A

a

v

b

t

z

o

then

a

v

b

t

z

and thus

z

0

z

A

z

v

a

a

v

b

t

z

.

But

z

0

v

z

thus having

d

n

z

A

z

v

a

a

v

b

t

z

o

v

d

n

z

A

a

v

b

t

z

o

.

Remark

196

.

If we drop the requirement that

A

is distributive, two formulas

for quasidifference (the definition and the last theorem) fork.

Obvious

197

.

Dual quasicomplement is the dual of quasicomplement.

Obvious

198

.

Every pseudocomplement is quasicomplement.

Every dual pseudocomplement is dual quasicomplement.

Every pseudodifference is quasidifference.

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 quasidifference through quasicomplement.

Proposition

199

.

1

.

a

\

b

=

a

\

(

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

\

(

a

u

b

) =

d

n

z

A

a

v

(

a

u

b

)

t

z

o

=

d

n

z

A

a

v

b

t

z

o

=

a

\

b

.