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17.12 Multireloids

Denition 17.164.

I will call a

multireloid

of the form

A

=

A

i

2

n

, where every each

A

i

is a set, a

pair

(

f

;

A

)

where

f

is a lter on the set

Q

A

.

Denition 17.165.

I will denote Obj

(

f

;

A

) =

A

and GR

(

f

;

A

) =

f

for every multireloid

(

f

;

A

)

.

I will denote

RLD

(

A

)

the set of multireloids of the form

A

.

The multireloid

"

RLD

(

A

)

F

for a relation

F

is dened by the formulas:

Obj

"

RLD

(

A

)

F

=

A

and GR

"

RLD

(

A

)

F

=

"

Q

A

F :

Let

a

be a multireloid of the form

A

and dom

A

=

n

.

Let every

f

i

be a reloid with Src

f

i

=

A

i

.

The star-composition of

a

with

f

is a multireloid of the form

i

2

dom

A

:

Dst

f

i

dened by the

formulas:

arity StarComp

(

a

;

f

) =

n

;

GR StarComp

(

a

;

f

) =

l

(

"

RLD

(

A

)

GR StarComp

(

A

;

F

)

j

A

2

GR

a; F

2

Y

i

2

n

GR

f

i

)

;

Obj

m

StarComp

(

a

;

f

) =

i

2

n

:

Dst

f

i

:

Theorem 17.166.

Multireloids with above dened compositions form a quasi-invertible category

with star-morphisms.

Proof.

We need to prove:

1. StarComp

(

StarComp

(

m

;

f

);

g

) =

StarComp

(

m

;

i

2

arity

m

:

g

i

f

i

);

2. StarComp

(

m

;

i

2

arity

m

:

id

Obj

m

i

) =

m

;

3.

b

/

StarComp

(

a

;

f

)

,

a

/

StarComp

(

b

;

f

y

)

(the rest is obvious).

Really,

1. Using properties of generalized lter bases, StarComp

(

StarComp

(

a

;

f

);

g

)

=

d

"

RLD

StarComp

(

B

;

G

)

j

B

2

GR StarComp

(

a

;

f

)

; G

2

Q

i

2

n

GR

g

i

 

=

d

"

RLD

StarComp

(

StarComp

(

A

;

F

);

G

)

j

A

2

GR

a; F

2

Q

i

2

n

f

i

; G

2

Q

i

2

n

g

i

 

=

d

"

RLD

StarComp

(

A

;

G

F

)

j

A

2

GR

a; F

2

Q

i

2

n

f

i

; G

2

Q

i

2

n

g

i

 

=

d

"

RLD

StarComp

(

A

;

H

)

j

A

2

GR

a; H

2

Q

i

2

n

i

2

n

:

g

i

f

i

 

=

StarComp

(

a

;

i

2

n

:

g

i

f

i

)

.

2. StarComp

(

m

;

i

2

arity

m

:

id

Obj

m

i

) =

d

"

RLD

(

A

)

StarComp

(

A

;

H

)

j

A

2

GR

m; H

2

Q

i

2

arity

m

GR id

Obj

m

i

 

=

d

"

RLD

(

A

)

StarComp

(

A

;

i

2

arity

m

:

H

i

)

j

A

2

GR

m; H

2

Q

i

2

arity

m

GR id

Obj

m

i

 

=

d

"

RLD

(

A

)

StarComp

(

A

;

i

2

arity

m

:

id

X

i

)

j

A

2

GR

m;

X

2

Q

i

2

arity

m

Obj

m

i

 

=

d

"

RLD

(

A

)

(

A

\

Q

X

)

j

A

2

GR

m; X

2

Q

i

2

arity

m

Obj

m

i

 

=

d

"

RLD

(

A

)

A

j

A

2

GR

m

 

=

m

.

3. Using properties of generalized lter bases,

b

/

StarComp

(

a

;

f

)

, 8

A

2

GR

a; B

2

GR

b; F

2

Q

i

2

n

GR

f

i

:

B

/

StarComp

(

A

;

F

)

, 8

A

2

GR

a; B

2

GR

b; F

2

Q

i

2

n

GR

f

i

:

B

/

DQ

(

C

)

F

E

A

, 8

A

2

GR

a; B

2

GR

b;

F

2

Q

i

2

n

GR

f

i

:

A

/

D Q

(

C

)

F

¡

1

E

B

, 8

A

2

GR

a; B

2

GR

b; F

2

Q

i

2

n

GR

f

i

:

A

/

StarComp

(

B

;

F

y

)

,

a

/

StarComp

(

b

;

f

y

)

.

Denition 17.167.

Let

f

be a multireloid of the form

A

. Then for

i

2

dom

A

Pr

i

RLD

f

=

l

h"

A

i

ih

Pr

i

i

GR

f :

17.12 Multireloids

227