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Proof

See [6].

Remark 4

See [7] for a more strong result.

Theorem 6

The center of a bounded distributive lattice constitutes its sublat-

tice.

Proof

Let

A

be a bounded distributive lattice and

Z

(

A

) is its center. Let

a, b

Z

(

A

). Consequently ¯

a,

¯

b

Z

(

A

). Then ¯

a

¯

b

is the complement of

a

b

because

(

a

b

)

a

¯

b

) = (

a

b

¯

a

)

(

a

b

¯

b

) = 0

0 = 0

and

(

a

b

)

a

¯

b

) = (

a

¯

a

¯

b

)

(

b

¯

a

¯

b

) = 1

1 = 1

.

So

a

b

is complemented, analogously

a

b

is complemented.

Theorem 7

The center of a bounded distributive lattice constitutes a boolean

lattice.

Proof

Because it is a distributive complemented lattice.

2.5. Galois connections

See [1] and [5] for more detailed treatment of Galois connections.

Definition 17

Let

A

and

B

be two posets. A

Galois connection

between

A

and

B

is a pair of functions

f

= (

f

;

f

)

with

f

:

A

B

and

f

:

B

A

such

that:

x

A

, y

B

: (

f

x

B

y

x

A

f

y

)

.

f

is called

upper adjoint

of

f

and

f

is called

lower adjoint

of

f

.

Theorem 8

A pair

(

f

;

f

)

of functions

f

:

A

B

and

f

:

B

A

is a

Galois connection iff both of the following:

1.

f

and

f

are monotone.

2.

x

A

f

f

x

and

f

f

y

B

y

for every

x

A

and

y

B

.

Proof

2.

x

A

f

f

x

since

f

x

B

f

x

;

f

f

y

B

y

since

f

y

A

f

y

.

1. Let

a, b

A

and

a

A

b

. Then

a

A

b

A

f

f

b

. So by definition

f

a

f

b

that is

f

is monotone. Analogously

f

is monotone.

f

x

B

y

f

f

x

A

f

y

x

A

f

y

. The other direction is analogous.

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