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2.1. ORDER THEORY

25

2

.

a

t

(

b

\

a

) =

a

t

l

z

A

b

v

a

t

z

=

l

a

t

z

z

A

, b

v

a

t

z

=

a

t

b.

3

.

b

t

(

b

\

a

) =

b

t

d

n

z

A

b

v

a

t

z

o

=

d

n

b

t

z

z

A

,b

v

a

t

z

o

=

b

.

4

Obviously (

b

t

c

)

\

a

w

b

\

a

and (

b

t

c

)

\

a

w

c

\

a

. Thus

(

b

t

c

)

\

a

w

(

b

\

a

)

t

(

c

\

a

). We have

(

b

\

a

)

t

(

c

\

a

)

t

a

=

((

b

\

a

)

t

a

)

t

((

c

\

a

)

t

a

) =

(

b

t

a

)

t

(

c

t

a

) =

a

t

b

t

c

w

b

t

c.

From this by definition of adjoints: (

b

\

a

)

t

(

c

\

a

)

w

(

b

t

c

)

\

a

.

Theorem

133

.

(

F

S

)

\

a

=

F

n

x

\

a

x

S

o

for all

a

A

and

S

P

A

where

A

is

a co-brouwerian lattice and

F

S

is defined.

Proof.

Because lower adjoint preserves all suprema.

Theorem

134

.

(

a

\

b

)

\

c

=

a

\

(

b

t

c

) for elements

a

,

b

,

c

of a complete

co-brouwerian lattice.

FiXme

: can be generalized for a natural number of elements,

using math induction.

Proof.

a

\

b

=

d

n

z

A

a

v

b

t

z

o

.

(

a

\

b

)

\

c

=

d

n

z

A

a

\

b

v

c

t

z

o

.

a

\

(

b

t

c

) =

d

n

z

A

a

v

b

t

c

t

z

o

.

It is left to prove

a

\

b

v

c

t

z

a

v

b

t

c

t

z

. But this follows from corollary

126

.

2.1.15. Dual pseudocomplement on co-Heyting lattices.

Theorem

135

.

For co-Heyting algebras

> \

b

=

b

+

.

Proof.

> \

b

= min

z

A

> v

b

t

z

= min

z

A

>

=

b

t

z

= min

z

A

b

z

=

b

+

.

Theorem

136

.

(

a

u

b

)

+

=

a

+

t

b

+

for every elements

a

,

b

of a co-Heyting

algebra.

Proof.

a

t

(

a

u

b

)

+

w

(

a

u

b

)

t

(

a

u

b

)

+

w >

. So

a

t

(

a

u

b

)

+

w >

; (

a

u

b

)

+

w

1

\

a

=

a

+

.

We have (

a

u

b

)

+

w

a

+

. Similarly (

a

u

b

)

+

w

b

+

. Thus (

a

u

b

)

+

w

a

+

t

b

+

.