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16. Complementive filter objects and factoring by a filter

Definition 73

Let

A

be a

-semilattice and

A ∈

A

. Then the relation

on

A

is defined by the formula

X, Y

A

: (

X

Y

X

A

A

=

Y

A

A

)

.

Proposition 36

The relation

is an equivalence relation.

Proof

Reflexivity

Obvious.

Symmetry

Obvious

Transitivity

Obvious.

Proposition 37

Let

A

be a distribitive lattice,

A ∈

F

. Then for every

X, Y

A

X

Y

⇔ ∃

A

up

A

:

X

A

A

=

Y

A

A.

Proof

A

up

A

:

X

A

A

=

Y

A

A

⇔ ∃

A

up

A

:

X

F

A

=

Y

F

A

A

up

A

:

X

F

A

F

A

=

Y

F

A

F

A ⇔ ∃

A

up

A

:

X

F

A

=

Y

F

A ⇔

X

F

A

=

Y

F

A ⇔

X

Y

.

On the other hand,

X

F

A

=

Y

F

A ⇔

X

A

A

0

|

A

0

up

A

 

=

Y

A

A

1

|

A

1

up

A

 

⇒ ∃

A

0

, A

1

up

A

:

X

A

A

0

=

Y

A

A

1

⇒ ∃

A

0

, A

1

up

A

:

X

A

A

0

A

A

1

=

Y

A

A

0

A

A

1

⇒ ∃

A

up

A

:

X

A

A

=

Y

A

A

.

Proposition 38

The relation

is a congruence for each of the following:

1. a

-semilattice

A

;

2. a distribitive lattice

A

.

Proof

Let

a

0

, a

1

, b

0

, b

1

A

and

a

0

a

1

and

b

0

b

1

.

1.

a

0

b

0

a

1

b

1

because

(

a

0

b

0

)

∩A

=

a

0

(

b

0

∩A

) =

a

0

(

b

1

∩A

) =

b

1

(

a

0

∩A

) =

b

1

(

a

1

∩A

) = (

a

1

b

1

)

∩A

.

2. Taking the above into account, we need to prove only

a

0

b

0

a

1

b

1

. We

have

(

a

0

b

0

)

∩ A

= (

a

0

∩ A

)

(

b

0

∩ A

) = (

a

1

∩ A

)

(

b

1

∩ A

) = (

a

1

b

1

)

∩ A

.

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