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15. COUNTER-EXAMPLES ABOUT FUNCOIDS AND RELOIDS

255

Proof.

Consider

h

= id

FCD

Ω(

N

)

. By proved above

h

=

f

u

g

where

f

=

1

FCD

N

=

FCD

(

N

,

N

)

id

N

,

g

=

FCD

(

N

,

N

)

(

N

×

N

\

id

N

).

We have id

N

,

N

×

N

\

id

N

GR

h

.

So

(

RLD

)

out

h

=

RLD

l

up

h

=

RLD

(

N

,

N

)

l

GR

h

v↑

RLD

(

N

,

N

)

(id

N

(

N

×

N

\

id

N

)) =

RLD

(

N

,

N

)

;

and thus (

RLD

)

out

h

=

RLD

(

N

,

N

)

.

Example

1350

.

There exists a funcoid

h

such that (

FCD

)(

RLD

)

out

h

6

=

h

.

Proof.

It follows from the previous example.

Example

1351

.

(

RLD

)

in

(

FCD

)

f

6

=

f

for some convex reloid

f

.

Proof.

Let

f

= 1

RLD

N

. Then (

FCD

)

f

= 1

FCD

N

. Let

a

be some non-trivial ultra-

filter on

N

. Then (

RLD

)

in

(

FCD

)

f

w

a

×

RLD

a

6v

1

RLD

N

and thus (

RLD

)

in

(

FCD

)

f

6v

f

.

Example

1352

.

There exist composable funcoids

f

and

g

such that

(

RLD

)

out

(

g

f

)

A

(

RLD

)

out

g

(

RLD

)

out

f.

Proof.

f

= id

FCD

Ω(

N

)

and

g

=

>

F

(

N

)

×

FCD

N

{

α

}

for some

α

N

. Then

(

RLD

)

out

f

=

RLD

(

N

,

N

)

and thus (

RLD

)

out

g

(

RLD

)

out

f

=

RLD

(

N

,

N

)

.

We have

g

f

= Ω(

N

)

×

FCD

N

{

α

}

.

(

RLD

)

out

(Ω(

N

)

×

FCD

N

{

α

}

) = Ω(

N

)

×

RLD

N

{

α

}

by properties of funcoidal

reloids.

Thus (

RLD

)

out

(

g

f

) = Ω(

N

)

×

RLD

N

{

α

} 6

=

RLD

(

N

,

N

)

.

Conjecture

1353

.

For every composable funcoids

f

and

g

(

RLD

)

out

(

g

f

)

w

(

RLD

)

out

g

(

RLD

)

out

f.

Example

1354

.

(

FCD

) does not preserve binary meets.

Proof.

(

FCD

)(1

RLD

N

u

(

>

RLD

(

N

,

N

)

\

1

RLD

N

)) = (

FCD

)

RLD

(

N

,

N

)

=

FCD

(

N

,

N

)

.

On the other hand,

(

FCD

)1

RLD

N

u

(

FCD

)(

>

RLD

(

N

,

N

)

\

1

RLD

N

) =

1

FCD

N

u ↑

FCD

(

N

,

N

)

(

N

×

N

\

id

N

) = id

FCD

Ω(

N

)

6

=

FCD

(

N

,

N

)

(used proposition

1061

).

Corollary

1355

.

(

FCD

) is not an upper adjoint (in general).

Considering restricting polynomials (considered as reloids) to ultrafilters, it is

simple to prove that each that restriction is injective if not restricting a constant

polynomial. Does this hold in general? No, see the following example:

Example

1356

.

There exists a monovalued reloid with atomic domain which

is neither injective nor constant (that is not a restriction of a constant function).

Proof.

(based on [

31

]) Consider the function

F

N

N

×

N

defined by the for-

mula (

x, y

)

7→

x

.

Let

ω

x

be a non-trivial ultrafilter on the vertical line

{

x

} ×

N

for every

x

N

.

Let

T

be the collection of such sets

Y

that

Y

(

{

x

} ×

N

)

ω

x

for all but

finitely many vertical lines. Obviously

T

is a filter.

Let

ω

atoms

T

.

For every

x

N

we have some

Y

T

for which (

{

x

} ×

N

)

Y

=

and thus

N

×

N

(

{

x

} ×

N

)

u

ω

=

F

(

N

×

N

)

.