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Theorem 2.122.

A lattice

A

with least element

0

is co-brouwerian with pseudodierence

n

i

n

is a binary operation on

A

satisfying the following identities:

1.

a

n

a

= 0

;

2.

a

t

(

b

n

a

) =

a

t

b

;

3.

b

t

(

b

n

a

) =

b

;

4.

(

b

t

c

)

n

a

= (

b

n

a

)

t

(

c

n

a

)

.

Proof.

(

.

We have

c

w

b

n

a

)

c

t

a

w

a

t

(

b

n

a

) =

a

t

b

w

b

;

c

t

a

w

b

)

c

=

c

t

(

c

n

a

)

w

(

a

n

a

)

t

(

c

n

a

) = (

a

t

c

)

n

a

w

b

n

a

.

So

c

w

b

n

a

,

c

t

a

w

b

that is

a

t ¡

is an upper adjoint of

¡n

a

. By a theorem above

our lattice is co-brouwerian. By another theorem above

n

is a pseudodierence.

)

.

1. Obvious.
2.

a

t

(

b

n

a

) =

a

t

l

f

z

2

A

j

b

v

a

t

z

g

=

l

f

a

t

z

j

z

2

A

; b

v

a

t

z

g

=

a

t

b:

3.

b

t

(

b

n

a

) =

b

t

d

f

z

2

A

j

b

v

a

t

z

g

=

d

f

b

t

z

j

z

2

A

; b

v

a

t

z

g

=

b

.

4. Obviously

(

b

t

c

)

n

a

w

b

n

a

and

(

b

t

c

)

n

a

w

c

n

a

. Thus

(

b

t

c

)

n

a

w

(

b

n

a

)

t

(

c

n

a

)

. We have

(

b

n

a

)

t

(

c

n

a

)

t

a

=

((

b

n

a

)

t

a

)

t

((

c

n

a

)

t

a

) =

(

b

t

a

)

t

(

c

t

a

) =

a

t

b

t

c

w

b

t

c:

From this by denition of adjoints:

(

b

n

a

)

t

(

c

n

a

)

w

(

b

t

c

)

n

a

.

Theorem 2.123.

(

F

S

)

n

a

=

F

f

x

n

a

j

x

2

S

g

for all

a

2

A

and

S

2

P

A

where

A

is a co-

brouwerian lattice and

F

S

is dened.

Proof.

Because lower adjoint preserves all suprema.

Theorem 2.124.

(

a

n

b

)

n

c

=

a

n

(

b

t

c

)

for elements

a

,

b

,

c

of a complete co-brouwerian lattice.

Proof.

a

n

b

=

d

f

z

2

A

j

a

v

b

t

z

g

.

(

a

n

b

)

n

c

=

d

f

z

2

A

j

a

n

b

v

c

t

z

g

.

a

n

(

b

t

c

) =

d

f

z

2

A

j

a

v

b

t

c

t

z

g

.

It is left to prove

a

n

b

v

c

t

z

,

a

v

b

t

c

t

z

.

Let

a

n

b

v

c

t

z

. Then

a

t

b

v

b

t

c

t

z

by the lemma and consequently

a

v

b

t

c

t

z

.

Let

a

v

b

t

c

t

z

. Then

a

n

b

v

(

b

t

c

t

z

)

n

b

v

c

t

z

by a theorem above.

2.1.15 Dual pseudocomplement on co-Heyting lattices

Proposition 2.125.

For co-Heyting algebras

1

n

b

=

b

+

.

2.1 Order theory

29