2.3. Difference and complement

Definition 8

Let

A

be a distributive lattice with least element

0

. The

differ-

ence

(denoted

a

\

b

) of elements

a

and

b

is such

c

A

that

b

c

= 0

and

a

b

=

b

c

. I will call

b

substractive

from

a

when

a

\

b

exists.

Theorem 4

If

A

is a distributive lattice with least element

0

, there exists no

more than one difference of elements

a, b

A

.

Proof

Let

c

and

d

are both differences

a

\

b

. Then

b

c

=

b

d

= 0 and

a

b

=

b

c

=

b

d

. So

c

=

c

(

b

c

) =

c

(

b

d

) = (

c

b

)

(

c

d

) = 0

(

c

d

) =

c

d.

Analogously,

d

=

d

c

. Consequently

c

=

c

d

=

d

c

=

d

.

Definition 9

I will call

b

complementive

to

a

when there exists

c

A

such

that

b

c

= 0

and

b

c

=

a

.

Proposition 2

b

is complementive to

a

iff

b

is substractive from

a

and

b

a

.

Proof

Obvious.

We deduce

b

a

from

b

c

=

a

. Thus

a

b

=

a

=

b

c

.

Proposition 3

If

b

is complementive to

a

then

(

a

\

b

)

b

=

a

.

Proof

Because

b

a

by the previous proposition.

Definition 10

Let

A

be a bounded distributive lattice. The

complement

(de-

noted

¯

a

) of element

a

A

is such

b

A

that

a

b

= 0

and

a

b

= 1

.

Proposition 4

If

A

is a bounded distributive lattice then

¯

a

= 1

\

a

.

Proof

b

= ¯

a

b

a

= 0

b

a

= 1

b

a

= 0

1

a

=

a

b

b

= 1

\

a

.

Corollary 2

If

A

is a bounded distributive lattice then exists no more than one

complement of an element

a

A

.

Definition 11

An element of bounded distributive lattice is called

comple-

mented

when its complement exists.

Definition 12

A distributive lattice is a

complemented lattice

iff every its

element is complemented.

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