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CHAPTER 16

Funcoids are filters

The motto of this chapter is: “Funcoids are filters on a (boolean) lattice.”

16.1. Rearrangement of collections of sets

Let

Q

be a set of sets.

Let

be the relation on

S

Q

defined by the formula

a

b

⇔ ∀

X

Q

: (

a

X

b

X

)

.

Proposition

1369

.

is an equivalence relation on

S

Q

.

Proof.

Reflexivity. Obvious.

Symmetry. Obvious.

Transitivity. Let

a

b

b

c

. Then

a

X

b

X

c

X

for every

X

Q

.

Thus

a

c

.

Definition

1370

.

Rearrangement

R

(

Q

) of

Q

is the set of equivalence classes

of

S

Q

for

.

Obvious

1371

.

S

R

(

Q

) =

S

Q

.

Obvious

1372

.

/

R

(

Q

).

Lemma

1373

.

card

R

(

Q

)

2

card

Q

.

Proof.

Having an equivalence class

C

, we can find the set

f

P

Q

of all

X

Q

such that

a

X

, for every

a

C

.

b

a

⇔ ∀

X

Q

: (

a

X

b

X

)

⇔ ∀

X

Q

: (

X

f

b

X

)

.

So

C

=

b

S

Q

b

a

can be restored knowing

f

. Consequently there are no more than

card

P

Q

= 2

card

Q

classes.

Corollary

1374

.

If

Q

is finite, then

R

(

Q

) is finite.

Proposition

1375

.

If

X

Q

,

Y

R

(

Q

) then

X

Y

6

=

∅ ⇔

Y

X

.

Proof.

Let

X

Y

6

=

and

x

X

Y

. Then

y

Y

x

y

⇔ ∀

X

0

Q

: (

x

X

0

y

X

0

)

(

x

X

y

X

)

y

X

for every

y

. Thus

Y

X

.

Y

X

X

Y

6

=

because

Y

6

=

.

Proposition

1376

.

If

∅ 6

=

X

Q

then there exists

Y

R

(

Q

) such that

Y

X

X

Y

6

=

.

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