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Backward Funcoids
by Victor Porton
Email: porton@narod.ru
Web: http://www.mathematics21.org
December 15, 2014
This is a preliminary partial draft.
Fix a family A of posets.
Deﬁnition 1. Let f be a staroid of ﬁlters F(A
i
) on boolean lattices A
i
. Backward funcoid for the
argument k 2 dom A of f is the funcoid Back(f ; k) deﬁned by the formula (for every X 2 A
k
)
hBack(f ; k)iX =
(
L 2
Y
i2dom A
F(A
i
) j X 2 hf i
k
L
)
:
Proposition 2. Backward funcoid is properly deﬁned.
Proof. hBack(f ; k)i
(X t Y ) = fL 2
Q
A j X t Y 2 hf i
k
Lg = fL 2
Q
A j X 2 hf i
k
L _
Y 2 hf i
k
Lg = fL 2
Q
A j X 2 hf i
k
Lg [ fL 2
Q
A j Y 2 hf i
k
Lg = hBack(f; k)i
X [ hBack(f;
k)i
Y .
Obvious 3. Backward funcoid is co-complete.
Proposition 4. If f is a principal staroid then Back(f;k) is a complete funcoid. [TODO: generalize
for boolean lattices?]
Proof. ??
Proposition 5. f can be restored from Back(f ; k) (for every ﬁxed k).
Proof. ??
Proposition 6. f 7! Back(f ; k) is an order isomorphism Strd
A
! FCD
¡
A
k
; Strd
i2(dom A)nfk g
.
Proof. ??
1