background image

Conjecture 2.

b

/

Anch

(

A

)

StarComp

(

a

;

f

)

, 8

A

2

GR

a; B

2

GR

b; i

2

n

:

A

i

[

f

i

]

B

i

for anchored

relations

a

and

b

on powersets.

It's conequence:

Conjecture 3.

b

/

Anch

(

A

)

StarComp

(

a

;

f

)

,

a

/

Anch

(

A

)

StarComp

(

b

;

f

y

)

for anchored relations

a

and

b

on powersets.

Conjecture 4.

b

/

pStrd

(

A

)

StarComp

(

a

;

f

)

,

a

/

pStrd

(

A

)

StarComp

(

b

;

f

y

)

for pre-staroids

a

and

b

on powersets.

Proposition 5.

Anchored relations with objects being posets with above dened star-morphisms

is a 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

.

(the rest is obvious).

Really,

L

2

GR StarComp

(

StarComp

(

m

;

f

);

g

)

,

(

i

2

arity

m

:

h

g

i

¡

1

i

L

i

)

2

GR StarComp

(

m

;

f

)

,

(

i

2

n

:

h

f

i

¡

1

i

(

i

2

n

:

h

g

i

¡

1

i

L

i

)

i

)

2

GR

m

,

(

i

2

arity

m

:

h

f

i

¡

1

ih

g

i

¡

1

i

L

i

)

2

GR

m

,

(

i

2

arity

m

:

h

(

g

i

f

i

)

¡

1

i

L

i

)

2

GR

m

,

L

2

GR StarComp

(

m

;

i

2

arity

m

:

g

i

f

i

)

;

L

2

GR StarComp

(

m

;

i

2

arity

m

:

id

Obj

m

i

)

,

(

i

2

n

:

h

id

Obj

m

i

i

L

i

)

2

GR

m

,

(

i

2

arity

m

:

h

id

Obj

m

i

i

L

i

)

2

GR

m

,

(

i

2

arity

m

:

L

i

)

2

GR

m

,

L

2

GR

m

.

Conjecture 6.

StarComp

(

a

t

b

;

f

) =

StarComp

(

a

;

f

)

t

StarComp

(

b

;

f

)

for anchored relations

a

,

b

of a form

A

, where every

A

i

is a distributive lattice, and an indexed family

f

of pointfree funcoids

with Src

f

i

=

A

i

.

[TODO: Put conjectures from this article to agt-open-problems.pdf]

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