background image

(2)

(4)

a

\

b

= min

{

z

A

|

a

b

z

}

. So

a

∪ −

is an upper adjoint of

− \

a

.

(1)

(4)

Because

a

∪ −

preserves all meets.

Corollary 5

Co-brouwerian lattices are distributive.

The following theorem is essentially borrowed from [8]:

Theorem 15

A lattice

A

with least element

0

is co-brouwerian with pseudodif-

ference

\

iff

\

is a binary operation on

A

satisfying the following identities:

1.

a

\

a

= 0

;

2.

a

(

b

\

a

) =

a

b

;

3.

b

(

b

\

a

) =

b

;

4.

(

b

c

)

\

a

= (

b

\

a

)

(

c

\

a

)

.

Proof

We have

c

b

\

a

c

a

a

(

b

\

a

) =

a

b

b

;

c

a

b

c

=

c

(

c

\

a

)

(

a

\

a

)

(

c

\

a

) = (

a

c

)

\

a

b

\

a.

So

c

b

\

a

c

a

b

that is

a

∪ −

is an upper adjoint of

− \

a

. By

a theorem above our lattice is co-brouwerian. By an other theorem above

\

is a pseudodifference.

1. Obvious.

2.

a

(

b

\

a

) =

a

\

{

z

A

|

b

a

z

}

=

\

{

a

z

|

z

A

, b

a

z

}

=

a

b.

3.

b

(

b

\

a

) =

b

T

{

z

A

|

b

a

z

}

=

T

{

b

z

|

z

A

, b

a

z

}

=

b

.

13