background image

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 46

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

connected component.

Proof

Direct implication is obvious. Reverse is proved by contradiction.

7.4

Connectedness regarding funcoids and reloids

Definition 61

S

1

(

µ

) =

RLD

(Ob

µ

;Ob

µ

)

S

1

(

M

)

|

M

up

µ

 

for an endo-

reloid

µ

.

Definition 62

Connectivity reloid

S

(

µ

)

for an endo-reloid

µ

is defined as

follows:

S

(

µ

) =

\ n

RLD

(Ob

µ

;Ob

µ

)

S

(

M

)

|

M

up

µ

o

.

Remark 9

Do not mess the word

connectivity

with the word

connectedness

which means being connected.

1

Proposition 47

S

(

µ

) =

I

RLD

(Ob

µ

)

S

1

(

µ

)

for every endo-reloid

µ

.

Proof

It follows from the theorem about distributivity of

regarding

T

(see

[15]).

Proposition 48

S

(

µ

) =

S

(

µ

)

if

µ

is a principal reloid.

Proof

S

(

µ

) =

T

{

S

(

µ

)

}

=

S

(

µ

).

Definition 63

A filter object

A ∈

F

(Ob

µ

)

is called

connected

regarding an

endo-reloid

µ

when

S

(

µ

(

A ×

RLD

A

))

⊇ A ×

RLD

A

.

Obvious 30.

A filter object

A ∈

Ob

µ

is connected regarding a reloid

µ

iff

S

(

µ

(

A ×

RLD

A

)) =

A ×

RLD

A

.

Definition 64

A filter object

A

is called

connected

regarding an endo-funcoid

µ

when

∀X

,

Y ∈

F

(Ob

µ

)

\

n

0

F

(Ob

µ

)

o

: (

X ∪ Y

=

A ⇒ X

[

µ

]

Y

)

.

1

In some math literature these two words are used interchangeably.

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