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3.2. QUASIDIFFERENCE AND QUASICOMPLEMENT

40

I will denote quasicomplement and dual quasicomplement for a specific poset

A

as

a

(

A

)

and

a

+(

A

)

.

Definition

235

.

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

236

.

Let

a, b

A

where

A

is a distributive lattice.

Second quasid-

ifference

of

a

and

b

is

a

#

b

=

l

z

A

z

v

a

z

b

.

Theorem

237

.

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

)

a

v

(

b

u

a

)

t

z

0

a

v

b

t

z

0

and

a

v

b

t

z

a

v

b

t

z

0

. Thus

a

v

b

t

z

a

v

b

t

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

238

.

If we drop the requirement that

A

is distributive, two formulas

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

Obvious

239

.

Dual quasicomplement is the dual of quasicomplement.

Obvious

240

.

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

241

.

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

.