12.4. CONNECTEDNESS REGARDING FUNCOIDS AND RELOIDS

228

Do not mess the word

connectivity

with the word

connectedness

which means

being connected.

1

Proposition

1205

.

S

(

µ

) = 1

RLD

Ob

µ

t

S

1

(

µ

) for every endoreloid

µ

.

Proof.

By the proposition

607

.

Proposition

1206

.

S

(

µ

) =

S

(

µ

) and

S

1

(

µ

) =

S

1

(

µ

) if

µ

is a principal reloid.

Proof.

S

(

µ

) =

d

{

S

(

µ

)

}

=

S

(

µ

);

S

1

(

µ

) =

d

{

S

1

(

µ

)

}

=

S

1

(

µ

).

Definition

1207

.

A filter

A ∈

F

(Ob

µ

) is called

connected

regarding an en-

doreloid

µ

when

S

1

(

µ

u

(

A ×

RLD

A

))

w A ×

RLD

A

.

Obvious

1208

.

A filter

A ∈

F

(Ob

µ

) is connected regarding an endoreloid

µ

iff

S

1

(

µ

u

(

A ×

RLD

A

)) =

A ×

RLD

A

.

Definition

1209

.

A filter

A ∈

F

(Ob

µ

) is called

connected

regarding an end-

ofuncoid

µ

when

∀X

,

Y ∈

F

(Ob

µ

)

\ {⊥

F

(Ob

µ

)

}

: (

X t Y

=

A ⇒ X

[

µ

]

Y

)

.

Proposition

1210

.

Let

A

be a typed set of type

U

. The filter

A

is connected

regarding an endofuncoid

µ

on

U

iff

X, Y

T

(Ob

µ

)

\ {⊥

T

(Ob

µ

)

}

: (

X

t

Y

=

A

X

[

µ

]

Y

)

.

Proof.

. Obvious.

. It follows from co-separability of filters.

Theorem

1211

.

The following are equivalent for every typed set

A

of type

U

and

Rel

-endomorphism

µ

on a set

U

:

1

.

A

is connected regarding

µ

.

2

.

A

is connected regarding

RLD

µ

.

3

.

A

is connected regarding

FCD

µ

.

Proof.

1

2

.

S

1

(

RLD

µ

u

(

A

×

RLD

A

)) =

S

1

(

RLD

(

µ

u

(

A

×

A

))) =

RLD

S

1

(

µ

u

(

A

×

A

))

.

So

S

1

(

RLD

µ

u

(

A

×

RLD

A

))

w

A

×

RLD

A

RLD

S

1

(

µ

u

(

A

×

A

))

w↑

RLD

(

A

×

A

) =

A

×

RLD

A.

1

3

It follows from the previous proposition.

Next is conjectured a statement more strong than the above theorem:

Conjecture

1212

.

Let

A

be a filter on a set

U

and

F

be a

Rel

-endomorphism

on

U

.

A

is connected regarding

FCD

F

iff

A

is connected regarding

RLD

F

.

1

In some math literature these two words are used interchangeably.