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Conjecture 10

Composition of complete reloids is complete.

Theorem 57

1. For a complete reloid

f

there exists exactly one function

F

F

(Dst

f

)

Src

f

such that

f

=

Src

f

{

α

} ×

RLD

F

(

α

)

|

α

Src

f

 

.

2. For a co-complete reloid

f

there exists exactly one function

F

F

(Src

f

)

Dst

f

such that

f

=

F

(

α

)

×

RLD

Dst

f

{

α

}

|

α

Dst

f

 

.

Proof

We will prove only the first as the second is similar. Let

f

=

Src

f

{

α

} ×

RLD

F

(

α

)

|

α

Src

f

 

=

Src

f

{

α

} ×

RLD

G

(

α

)

|

α

Src

f

 

for some

F, G

F

(Dst

f

)

Src

f

. We need to prove

F

=

G

. Let

β

Src

f

.

f

Src

f

{

β

} ×

RLD

1

F

(Dst

f

)

=

(theorem 40 in [15])

[

RLD

n

Src

f

{

α

} ×

RLD

F

(

α

)

RLD

Src

f

{

β

} ×

1

F

(Dst

f

)

|

α

Src

f

o

=

Src

f

{

β

} ×

RLD

F

(

β

)

.

Similarly

f

∩ ↑

Src

f

{

β

} ×

1

F

(Dst

f

)

=

Src

f

{

β

RLD

G

(

β

). Thus

Src

f

{

β

RLD

F

(

β

) =

Src

f

{

β

} ×

RLD

G

(

β

) and so

F

(

β

) =

G

(

β

).

Definition 54

Completion

and

co-completion

of a reloid

f

RLD

(

A

;

B

)

are defined by the formulas:

Compl

f

= Cor

(

RLD

(

A

;

B

);Compl

RLD

(

A

;

B

))

f

and

CoCompl

f

= Cor

(

RLD

(

A

;

B

);CoCompl

RLD

(

A

;

B

))

f.

Theorem 58

Atoms of the lattice

Compl

RLD

(

A

;

B

)

are exactly direct products

of the form

A

{

α

} ×

RLD

b

where

α

A

and

b

is an atomic f.o. on

B

.

Proof

First, it’s easy to see that

A

{

α

FCD

b

are elements of Compl

RLD

(

A

;

B

).

Also 0

RLD

(

A

;

B

)

is an element of Compl

RLD

.

A

{

α

} ×

RLD

b

are atoms of Compl

FCD

because these are atoms of

RLD

.

It remains to prove that if

f

is an atom of Compl

RLD

(

A

;

B

) then

f

=

A

{

α

} ×

RLD

b

for some

α

A

and an atomic f.o.

b

on

B

.

Suppose

f

is a non-empty complete reloid. Then

A

{

α

} ×

RLD

b

f

for

some

α

A

and atomic f.o.

b

on

B

. If

f

is an atom then

f

=

A

{

α

} ×

FCD

b

.

Obvious 27.

Compl

RLD

(

A

;

B

) is an atomistic lattice.

Proposition 30

Compl

f

=

f

|

Src

f

{

α

}

|

α

Src

f

 

for every reloid

f

.

49