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2.1. ORDER THEORY

20

Definition

75

.

I will call

b

complementive

to

a

iff there exists

c

A

such that

b

u

c

=

and

b

t

c

=

a

.

Proposition

76

.

b

is complementive to

a

iff

b

is substractive from

a

and

b

v

a

.

Proof.

. Obvious.

. We deduce

b

v

a

from

b

t

c

=

a

. Thus

a

t

b

=

a

=

b

t

c

.

Proposition

77

.

If

b

is complementive to

a

then (

a

\

b

)

t

b

=

a

.

Proof.

Because

b

v

a

by the previous proposition.

Definition

78

.

Let

A

be a bounded distributive lattice. The

complement

(denoted ¯

a

) of an element

a

A

is such

b

A

that

a

u

b

=

and

a

t

b

=

>

.

Proposition

79

.

If

A

is a bounded distributive lattice then ¯

a

=

> \

a

.

Proof.

b

= ¯

a

b

u

a

=

⊥∧

b

t

a

=

> ⇔

b

u

a

=

⊥∧>t

a

=

a

t

b

b

=

>\

a

.

Corollary

80

.

If

A

is a bounded distributive lattice then exists no more than

one complement of an element

a

A

.

Definition

81

.

An element of bounded distributive lattice is called

comple-

mented

when its complement exists.

Definition

82

.

A distributive lattice is a

complemented lattice

iff every its

element is complemented.

Proposition

83

.

For a distributive lattice (

a

\

b

)

\

c

=

a

\

(

b

t

c

) if

a

\

b

and

(

a

\

b

)

\

c

are defined.

Proof.

((

a

\

b

)

\

c

)

u

c

=

; ((

a

\

b

)

\

c

)

t

c

= (

a

\

b

)

t

c

; (

a

\

b

)

u

b

=

;

(

a

\

b

)

t

b

=

a

t

b

.

We need to prove ((

a

\

b

)

\

c

)

u

(

b

t

c

) =

and ((

a

\

b

)

\

c

)

t

(

b

t

c

) =

a

t

(

b

t

c

).

In fact,

((

a

\

b

)

\

c

)

u

(

b

t

c

) =

(((

a

\

b

)

\

c

)

u

b

)

t

(((

a

\

b

)

\

c

)

u

c

) =

(((

a

\

b

)

\

c

)

u

b

)

t ⊥

=

((

a

\

b

)

\

c

)

u

b

v

(

a

\

b

)

u

b

=

,

so ((

a

\

b

)

\

c

)

u

(

b

t

c

) =

;

((

a

\

b

)

\

c

)

t

(

b

t

c

) =

(((

a

\

b

)

\

c

)

t

c

)

t

b

=

(

a

\

b

)

t

c

t

b

=

((

a

\

b

)

t

b

)

t

c

=

a

t

b

t

c.