Let now

C

up

h↑

f

iA

. Then

Base

(

A

)

h

f

1

i

C

⊇ h↑

f

1

ih↑

f

iA ⊇ A

and thus

h

f

1

i

C

up

A

.

Corollary 20.

f

Mor

GreFunc

1

(

A

;

B

)

⇔ B ⊆ h↑

f

iA

for every

Set

-morphism

f

from Base

(

A

)

to

Base

(

B

)

.

Corollary 21.

f

Mor

GreFunc

2

(

A

;

B

)

⇔ B

=

h↑

f

iA

for every

Set

-morphism

f

from Base

(

A

)

to

Base

(

B

)

.

Corollary 22.

A

>

2

B

iff it exists a

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

such that

B

=

h↑

f

iA

.

Corollary 23.

up

B ⊇

f

∗A ⇔ B ⊆ h↑

f

iA

.

Corollary 24.

A

>

1

B

iff it exists a

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

such that

B ⊆ h↑

f

iA

.

Proposition 25.

For a bijective

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

the following are equivalent:

1. up

B

=

f

∗A

.

2.

C

Base

(

B

): (

C

up

B ⇔ h

f

1

i

C

up

A

)

.

3.

C

Base

(

A

): (

h

f

i

C

up

B ⇔

C

up

A

)

.

4.

h

f

i|

up

A

is a bijection from up

A

to up

B

.

5.

h

f

i|

up

A

is a function onto up

B

.

6.

B

=

h↑

f

iA

.

7.

f

Mor

GreFunc

2

(

A

;

B

)

.

8.

f

Mor

FuncBij

(

A

;

B

)

.

Proof.

(1)

(2).

up

B

=

f

∗A ⇔

up

B

=

{

C

P

Base

(

B

)

| h

f

1

i

C

up

A} ⇔ ∀

C

Base

(

B

):

(

C

up

B ⇔ h

f

1

i

C

up

A

)

.

(2)

(3).

Because

f

is a bijection.

(2)

(5).

For every

C

up

B

we have

h

f

1

i

C

up

A

and thus

h

f

i|

up

A

h

f

1

i

C

=

h

f

ih

f

1

i

C

=

C

. Thus

h

f

i|

up

A

is onto up

B

.

(4)

(5).

Obvious.

(5)

(4).

We need to prove only that

h

f

i|

up

A

is an injection. But this follows from the fact

that

f

is a bijection.

(4)

(3).

We have

C

Base

(

A

): ((

h

f

i|

up

A

)

C

up

B ⇔

C

up

A

)

and consequently

C

Base

(

A

): (

h

f

i

C

up

B ⇔

C

up

A

)

.

(6)

(1).

From the last corollary.

(1)

(7).

Obvious.

(7)

(8).

Obvious.

Corollary 26.

The following are equivalent for every f.o.

A

and

B

:

1.

A

is directly isomorphic to a f.o.

B

.

2. There are a bijective

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

such that for every

C

P

Base

(

B

)

C

up

B ⇔ h

f

1

i

C

up

A

3. There are a bijective

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

such that for every

C

P

Base

(

B

)

h

f

i

C

up

B ⇔

C

up

A

.

4. There are a bijective

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

such that

h

f

i|

up

A

is a bijection

from up

A

to up

B

.

4

Section 3