
Conjecture 2. b /
Anch(A)
StarComp(a; f ) , 8A 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 defined star-morphisms
is a category with star morphisms.
Proof. We need to prove:
1. StarComp(StarComp(m; f ); g) = StarComp(m; λi 2 arity m: g
i
◦ f
i
);
2. StarComp(m; λi 2 arity m: id
Obj
m
i
) = m.
(the rest is obvious).
Really, L 2 GR StarComp(StarComp(m; f ); g) , (λi 2 arity m: hg
i
¡1
iL
i
) 2 GR StarComp(m; f ) ,
(λi 2 n: hf
i
¡1
i(λi 2 n: hg
i
¡1
iL
i
)
i
) 2 GR m , (λi 2 arity m: hf
i
¡1
ihg
i
¡1
iL
i
) 2 GR m , (λi 2 arity m:
h(g
i
◦ f
i
)
¡1
iL
i
) 2 GR m , L 2 GR StarComp(m; λi 2 arity m : g
i
◦ f
i
);
L 2 GR StarComp(m; λi 2 arity m: id
Obj
m
i
) , (λi 2 n: hid
Obj
m
i
iL
i
) 2 GR m , (λi 2 arity m:
hid
Obj
m
i
iL
i
) 2 GR m , (λi 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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