12.4. CONNECTEDNESS REGARDING FUNCOIDS AND RELOIDS

227

2

.

S

1

(

µ

(

A

×

A

))

A

×

A

.

3

.

S

1

(

µ

(

A

×

A

)) =

A

×

A

.

4

.

A

is connected regarding

µ

.

Proof.

1

2

Let for every

a, b

A

there is a nonzero-length path between

a

and

b

in

A

through

µ

. Then

a

(

S

1

(

µ

A

×

A

))

b

for every

a, b

A

. It is possible

only when

S

1

(

µ

(

A

×

A

))

A

×

A

.

3

1

For every two vertices

a

and

b

we have

a

(

S

1

(

µ

A

×

A

))

b

. So (by the

previous) for every two vertices

a

and

b

there exists a nonzero-length path

from

a

to

b

.

3

4

Suppose

¬

(

X

[

µ

(

A

×

A

)]

Y

) for some

X, Y

P

f

\ {∅}

such that

X

Y

=

A

. Then by a lemma

¬

(

X

[(

µ

(

A

×

A

))

n

]

Y

) for every

n

Z

+

.

Consequently

¬

(

X

[

S

1

(

µ

(

A

×

A

))]

Y

). So

S

1

(

µ

(

A

×

A

))

6

=

A

×

A

.

4

3

If

h

S

1

(

µ

(

A

×

A

))

i

{

v

}

=

A

for every vertex

v

then

S

1

(

µ

(

A

×

A

)) =

A

×

A

. Consider the remaining case when

V

def

=

h

S

1

(

µ

(

A

×

A

))

i

{

v

} ⊂

A

for some vertex

v

. Let

W

=

A

\

V

. If card

A

= 1 then

S

1

(

µ

(

A

×

A

))

id

A

=

A

×

A

; otherwise

W

6

=

. Then

V

W

=

A

and so

V

[

µ

]

W

what

is equivalent to

V

[

µ

(

A

×

A

)]

W

that is

h

µ

(

A

×

A

)

i

V

W

6

=

.

This is impossible because

h

µ

(

A

×

A

)

i

V

=

h

µ

(

A

×

A

)

i

h

S

1

(

µ

(

A

×

A

))

i

V

=

(

µ

(

A

×

A

))

2

(

µ

(

A

×

A

))

3

∪ · · · ∪

V

⊆ h

S

1

(

µ

(

A

×

A

))

i

V

=

V.

2

3

Because

S

1

(

µ

(

A

×

A

))

A

×

A

.

Corollary

1199

.

A set

A

is connected regarding a binary relation

µ

iff it is

connected regarding

µ

(

A

×

A

).

Definition

1200

.

A

connected component

of a set

A

regarding a binary relation

F

is a maximal connected subset of

A

.

Theorem

1201

.

The set

A

is partitioned into connected components (regarding

every binary relation

F

).

Proof.

Consider the binary relation

a

b

a

(

S

(

F

))

b

b

(

S

(

F

))

a

.

is

a symmetric, reflexive, and transitive relation. So all points of

A

are partitioned

into a collection of sets

Q

. Obviously each component is (strongly) connected. If

a set

R

A

is greater than one of that connected components

A

then it contains

a point

b

B

where

B

is some other connected component. Consequently

R

is

disconnected.

Proposition

1202

.

A set is connected (regarding a binary relation) iff it has

one connected component.

Proof.

Direct implication is obvious. Reverse is proved by contradiction.

12.4. Connectedness regarding funcoids and reloids

Definition

1203

.

Connectivity reloid

S

1

(

µ

) =

d

RLD

M

up

µ

S

1

(

M

) for an en-

doreloid

µ

.

Definition

1204

.

S

(

µ

) for an endoreloid

µ

is defined as follows:

S

(

µ

) =

RLD

l

M

up

µ

S

(

M

)

.