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Theorem 57

a

\

b

=

T

{

z

A

|

z

a

a

b

z

}

where

A

is a distributive

lattice and

a, b

A

.

Proof

Obviously

{

z

A

|

z

a

a

b

z

} ⊆ {

z

A

|

a

b

z

}

. Thus

T

{

z

A

|

z

a

a

b

z

} ⊇

a

\

b

.

Let

z

A

and

z

=

z

a

.

a

b

z

a

(

b

z

)

a

a

(

b

a

)

(

z

a

)

a

(

b

a

)

z

a

b

z

and

a

b

z

a

b

z

. Thus

a

b

z

a

b

z

.

If

z

∈ {

z

A

|

a

b

z

}

then

a

b

z

and thus

z

∈ {

z

A

|

z

a

a

b

z

}

.

But

z

z

thus having

T

{

z

A

|

z

a

a

b

z

} ⊆

T

{

z

A

|

a

b

z

}

.

Remark 12

If we drop the requirement that

A

is distributive, two formulas for

quasidifference (the definition and the last theorem) fork.

Obvious 21

Dual quasicomplement is the dual of quasicomplement.

Obvious 22

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 31

1.

a

\

b

=

a

\

(

a

b

)

for any distributive lattice;

2.

a

#

b

=

a

#(

a

b

)

for any distributive lattice with least element.

Proof

1.

a

(

a

b

)

z

a

(

a

z

)

(

b

z

)

a

a

z

a

b

z

a

b

z

. Thus

a

\

(

a

b

) =

T

{

z

A

|

a

(

a

b

)

z

}

=

T

{

z

A

|

a

b

z

}

=

a

\

b

.

2.

a

#(

a

b

) =

S

{

z

A

|

z

a

z

a

b

= 0

}

=

S

{

z

A

|

z

a

(

z

a

)

a

b

= 0

}

=

S

{

z

a

|

z

A

, z

a

b

= 0

}

=

S

{

z

A

|

z

a, z

b

= 0

}

=

a

#

b

.

I will denote

Da

the lattice

{

x

A

|

x

a

}

.

Theorem 58

For

a, b

A

where

A

is a distributive lattice with least element

1.

a

\

b

= (

a

b

)

+(

Da

)

;

44