Algebraic General Topology. Vol 1: Paperback / E-book || Axiomatic Theory of Formulas: Paperback / E-book

As an Amazon Associate I earn from qualifying purchases.

Multidimensional Funcoids
by Victor Porton
78640, Shay Agnon 32-29, Ashkelon, Israel
Web: http://www.mathematics21.org
Abstract
First I deﬁne a product of two funcoids. Then I dene multifuncoids and staroids as gen-
eralizations of funcoids. Using staroids I deﬁne a product of an arbitrary (possibly inﬁnite)
family of funcoids and some other products.
1 Draft status
It is a rough draft.
This article is outdated. Read the book instead.
2 Notation
This article presents a ge ne ralization of concepts from [1] and [3].
In this article I will use to denote order in a poset and , to denote meets and joins on a
semilattice. I reserve , , and for set-theoretic supset-relation, intersection, and union.
For a poset A I will denote Least(A) the set of least elements of A. (This set always has eithe r
one or z ero elements.)
With this notation we do not need the conc ept of ﬁlter objects ([4]), we will use the standard
set of ﬁlters, but the order on the lattice of lters will b e opposite the set theoretic inclusion
of ﬁlters.
3 Product of two func oids
3.1 Lemmas
Lemma 1. Let A, B, C are sets, f FCD(A; B), g FCD(B; C), h FCD(A; C). Then
g f
h g
h f
1
.
Proof. See [1].
Lemma 2. Let A, B, C are sets, f RLD(A; B), g RLD(B; C), h RLD(A; C). Then
g f
h g
h f
1
.
Proof. See [1].
Lemma 3. f (A ×
FCD
B) = A ×
FCD
hf iB for elements A A a nd B B of some posets A, B with
least elements and f FCD(A; B).
Proof. hf (A ×
FCD
B)iX =
hf iB if X
A
0 if X A
= hA ×
FCD
hf iBiX .
1
3.2 Deﬁnition
Deﬁnition 4. I will call a quasi-invertible category a partially ordered dagger catego ry such that
it holds
g f
h g
h f
(1)
for every morphisms f Hom(A; B), g Ho m(B; C), h Hom(A; C), where A, B, C are objects
of this category.
Inverting this formula, we get f
g
h
g
f h
. After replacement of variab le s, this
gives: f
g
h g
f h
As it follows from [1], the cate gory of funcoids and the category of reloids are quasi-invertible
(taking f
= f
1
). Moreover by [3] the category of pointfree funcoids between lattices of ﬁlters on
boolean lattices are quasi-invertible.
Deﬁnition 5. The cross-composition product of morphisms f and g of a quasi-invertible category
is the pointfree funcoid Ho m(Src f; Src g) Hom(Dst f ; Dst g) deﬁned by the formulas (for every
a Hom(Src f ; Src g) and b Hom(Dst f ; Dst g)):
f ×
(C)
g
a = g a f
and

f ×
(C)
g
1
b = g
b f .
The cross-composition product is a po intfree funcoid from Hom(Src f ; Src g) to Hom(Dst f ; Dst g).
We need to prove that it is really a pointfree funcoid that is that
b
f ×
(C)
g
a a

f ×
(C)
g
1
b.
This formula means b
g a f
a
g
b f and can be ea sily proved applying the formula (1)
two times.
Proposition 6. a
f ×
(C)
g
b a f
g
b.
Proof. From the lemma.
Proposition 7. a
f ×
(C)
g
b f
a ×
(C)
b
g.
Proof. f
a ×
(C)
b
g f a
b
g a f
g
b a
f ×
(C)
g
b.
Theorem 8.
f ×
(C)
g
= f
×
(C)
g
.
Proof. For every funcoids a Hom(Src f ; Src g) and b Hom(Dst f; D st g) we have:

f ×
(C)
g
b = g
b f = g
b f =
f
×
(C)
g
b.

f ×
(C)
g
a =
f ×
(C)
g
a = g a f
=

f
×
(C)
g
a.
Theorem 9. Let f , g are morphisms of a quasi-invertible ca tegory where Dst f and Dst g are f.o.
on boolean lattices. Then for every f.o. A
0
F(Src f), B
0
F(Src g)
f ×
(C)
g
(A
0
×
FCD
B
0
) = hf iA
0
×
FCD
hgiB
0
.
Proof. For every atom a
1
×
FCD
b
1
(a
1
atoms
Dst f
, b
1
atoms
Dst g
) of the latt ice of funcoids we have:
a
1
×
FCD
b
1
f ×
(C)
g
(A
0
×
FCD
B
0
) A
0
×
FCD
B
0
f ×
(C)
g
a
1
×
FCD
b
1
(A
0
×
FCD
B
0
) f
g
(a
1
×
FCD
b
1
) hf iA
0
×
FCD
B
0
a
1
×
FCD
hg
ib
1
hf iA
0
a
1
hg
ib
1
B
0
hf iA
0
a
1
hgiB
0
b
1
hf iA
0
×
FCD
hgiB
0
a
1
×
FCD
b
1
. Thus
f ×
(C)
g
(A
0
×
FCD
B
0
) = hf iA
0
×
FCD
hgiB
0
because the lattice FCD(F(Dst f); F(Dst g)) is atomically separable (corollary 64 in [3]).
Proposition 10. A
0
×
FCD
B
0
f ×
(C)
g
A
1
×
FCD
B
1
A
0
[f] A
1
B
0
[g] B
1
for every A
0
F(Src f ),
A
1
F(Dst f ), B
0
F(Src g), B
1
F(Dst g).
Proof. A
0
×
FCD
B
0
f ×
(C)
g
A
1
×
FCD
B
1
A
1
×
FCD
B
1
f ×
(C)
g
(A
0
×
FCD
B
0
)
A
1
×
FCD
B
1
hf iA
0
×
FCD
hgiB
0
A
1
hf iA
0
B
1
hgiB
0
A
0
[f ] A
1
B
0
[g] B
1
.
2 Section 3
4 Function spaces of posets
Deﬁnition 11. Let A
i
is a family of pose ts indexed by some set dom A. We will deﬁne order of
families of posets by the formula
a b i dom A: a
i
b
i
.
I will call this new poset A =
Q
A the function space of posets and the above order product order .
Proposition 12. The function space for posets is also a poset.
Proof.
Reﬂexivity. Obvi ous.
Antisymmetry. Obvious.
Transitivity. Obvious.
Obvious 13. A has least element i each A
i
has a least element. In this case
Least(A) =
Y
idom A
Least(A
i
).
Proposition 14. a
b i dom A: a
i
b
i
for every a, b
Q
A.
Proof. a
b c
Q
A: (c a c b) c
Q
Ai d om A: (c
i
a
i
c
i
b
i
)
i dom Ax
Q
A: (x a
i
x b
i
) i dom A: a
i
b
i
.
Proposition 15.
1. If A
i
are jo in-semilattices then A is a join-semilattice and
A B = λi dom A: Ai Bi. (2)
2. If A
i
are meet-semilattices then A is a meet-semilattice and
A B = λi dom A: Ai Bi. (3)
Proof. It is enough to pr ove the formula (2 ).
It’s obvious tha t λi dom A: Ai Bi A, B.
Let C A, B. Then (for every i dom A) Ci Ai and Ci Bi. Thus Ci Ai Bi that is
C λi dom A: Ai Bi.
Corollary 16. If A
i
are lattices then A is a lattice.
Obvious 17. If A
i
are distributive lattices then A is a distributive lattice.
Obvious 18. If A
i
are (co-)brouwerian lattices then A is a (co-)brouwer ian lat tice.
Proposition 19. If A
i
are boolea n lattices then
Q
A is a boolean lattice.
Proof. We need to prove only that every element a
Q
A has a complement. But this complement
is evidently λi dom a: a
i
.
Proposition 20. If A
i
are lattices then for every S P
Q
A
1.
F
S = λi dom A:
F
{x
i
| x S } wh enever
F
{x
i
| x S } exists;
2.
d
S = λi dom A:
d
{x
i
| x S } wh enever
d
{x
i
| x S } exists.
Proof. It’s enough to prove the ﬁrst formula.
(λi dom A:
F
{x
i
| x S })
i
=
F
{x
i
| x S } x
i
for every x S and i dom A.
Let y x for every x S. Then y
i
x
i
for every i dom A and thus y
i
F
{x
i
| x S} =
(λi dom A:
F
{x
i
| x S })
i
that is y λi dom A:
F
{x
i
| x S }.
Thus
F
S = λi dom A:
F
{x
i
| x S } by the deﬁnition of join.
Function spaces of posets 3
Corollary 21. If A
i
are complete lattices then A is a complete lat tice.
Proposition 22. If each A
i
is a separable poset with least element (for some index set n) then
Q
A is a sepa rable poset.
Proof. Let a
b. Then i dom A: a
i
b
i
. So x A
i
: (x
a
i
x b
i
) (or vic e versa).
Take y = (((dom A) \ {i}) × {0}) {(i; x)}. Then y
a and y b.
Obvious 23. If every A
i
is a poset with least element 0
i
, then the set of atoms of
Q
A is
{({k} × atoms
A
k
) (λi (d om A) \ {k}: 0
i
) | k dom A}.
Proposition 24. If every A
i
is an atomistic poset with least element 0
i
, then
Q
A is an atomistic
poset.
Proof. x
i
=
F
atoms x
i
for every x
i
A
i
. Thus
x = λi dom x: x
i
=
G
idom x
atoms x
i
=
G
idom x
λj dom x:
x
i
if j = i
0
i
if j
i.
Take join two times.
Corollary 25. If A
i
are atomistic complete lattices, then
Q
A is atomically separable.
Proof. Proposition 14 in [4].
Proposition 26. L e t (A
in
; Z
in
) is a family of ﬁltrators. Then (
Q
A;
Q
Z) is a ﬁltra tor.
Proof. We need to prove that
Q
Z is a sub-poset of
Q
A. First
Q
Z
Q
A because Z
i
A
i
for each i n.
Let A, B
Q
Z and A
Q
Z
B. Then i n: A
i
Z
i
B
i
; consequently i n: A
i
A
i
B
i
that is
A
Q
A
B.
Proposition 27. L e t (A
in
; Z
in
) is a family of ﬁltrators.
1. Th e ﬁltrator (
Q
A;
Q
Z) is (ﬁnitely) j oin-closed if every (A
i
; Z
i
) is (ﬁ nitely) join- closed.
2. Th e ltrator (
Q
A;
Q
Z) is (ﬁnitely) meet-closed if every (A
i
; Z
i
) is (ﬁnitely) meet-closed.
Proof. Let every (A
i
; Z
i
) is ﬁnitely join-closed. Let A, B
Q
Z. Then A
Q
Z
B = λ n:
A
i
Z
i
B
i
= λ n: A
i
A
i
B
i
= A
Q
A
B.
Let now every (A
i
; Z
i
) is ﬁnitely jo in-closed. Let S P
Q
Z. Then
F
Q
Z
S = λi dom A:
F
Z
i
{x
i
| x S } = λi dom A:
F
A
i
{x
i
| x S } =
F
Q
A
S.
The rest follows fr om symmetry.
Proposition 28 . If each (A
i
; Z
i
) where i n (for some index set n) is a down-aligned ﬁltrator with
separable core (for some index set n) then (
Q
A;
Q
Z) is with separable core.
Proof. Let a
b. Then i n: a
i
b
i
. So x Z
i
: (x
a
i
x b
i
) (or vice versa).
Take y = ((n \ {i}) × {0}) {(i; x)}. Then we have y
a and y b and y Z.
Proposition 29. Let every A
i
is a bounded lattice. Every (A
i
; Z
i
) is a ce ntral ltrator iﬀ (
Q
A;
Q
Z) is a central ﬁltrator.
Proof. x Z(
Q
A) y
Q
A:
x y = 0
Q
A
x y = 1
Q
A
y
Q
Ai dom A:
(x
i
y
i
= 0
A
i
x
i
y
i
= 1
A
i
) i dom Ay A
i
: (x
i
y
i
= 0
A
i
x
i
y
i
= 1
A
i
) i dom A:
x
i
Z(A
i
).
Proposition 30. For e very element a of a product ﬁltrator (
Q
A;
Q
Z):
1. up a =
Q
idom a
up a
i
;
4 Section 4
2. down a =
Q
idom a
down a
i
.
Proof. We will prove only t he ﬁr st as the second is dual.
up a = {c
Q
Z | c a} = {c
Q
Z | i dom a: c
i
a
i
} = {c
Q
Z | i dom a:
c
i
up a
i
} =
Q
idom a
up a
i
.
Proposition 31. If every (A
i
; Z
i
) is a ﬁltered complete lattice ﬁltrator, then (
Q
A;
Q
Z) is a
ﬁltered complete lattice ﬁltrator.
Proof. That
Q
A is a complete lattice is already proved above. We have for every a
Q
A
d
Q
A
up a = λi dom A:
d
{x
i
| x up a} = λi dom A:
d
{x | x up a
i
} = λi dom A:
d
up a
i
= λi dom A: a
i
= a.
Obvious 32. If every (A
i
; Z
i
) is a preﬁltered co mplete lattice ﬁltrator, then (
Q
A;
Q
Z) is a
preﬁltered complete lattice ﬁlt rato r.
Proposition 33. Let A
i
is a non -empty poset. Every (A
i
; Z
i
) is a semiﬁltered complete lattice
ﬁltrator iﬀ (
Q
A;
Q
Z) is a semiﬁltered complete lattice ﬁltrator.
Proof. up a up b λi dom A: up a
i
up b
i
λi dom A: a
i
b
i
a b for e very a, b
Q
A
(used the fact that up a
i
0 because up is injective).
Proposition 34. L e t (A
i
; Z
i
) a re ﬁl trators and each Z
i
is a complete lattice. Fo r a
Q
A:
1. Cor a = λi dom a: Cor a
i
;
2. Cor
a = λi dom a: Cor
a
i
.
Proof. We will prove only t he ﬁr st, because the second is dual.
Cor a =
d
Q
Z
upa = λi dom a:
d
Z
i
{x
i
| x up a} = λi doma:
d
Z
i
{x | x upa
i
}= λi doma:
d
Z
i
up a
i
= λi dom a: Cor a
i
.
Proposition 35. If each (A
i
; Z
i
) is a ﬁltrator with (co-)separable core, then (
Q
A;
Q
Z) is a
ﬁltrator with (co-)separable core.
Proof. We will prove only f or separable core, as co-separable core is dual.
x
Q
A
y i dom A: x
i
A
i
y
i
i dom AX up x
i
: X
A
i
y
i
X up xi dom A:
X
i
A
i
y
i
X up x: X
Q
A
y for every x, y
Q
A.
Obvious 36.
1. If each (A
i
; Z
i
) is a down-aligned ltrator, then (
Q
A;
Q
Z) is a down-aligned ltrator.
2. If each (A
i
; Z
i
) is an up-aligned ﬁltrator, then (
Q
A;
Q
Z) is an up-aligned ﬁltrator.
Proposition 37. If every b
i
is substractive from a
i
where a and b are n-indexed families of
distributive lattices with le ast elements (where n is an index set), then a \ b = λi n: a
i
\ b
i
.
Proof. We need to prove (λi n: a
i
\ b
i
) b = 0 and a b = b (λi n: a
i
\ b
i
).
Really, (λi n: a
i
\ b
i
) b = λi n: (a
i
\ b
i
) b
i
= 0 and b (λi n: a
i
\ b
i
) = λ i n:
b
i
(a
i
\ b
i
) = λi n: b
i
a
i
= a b.
Proposition 38. If every A
i
is a distributive lattice, then a \
b = λ i dom A: a
i
\
b
i
for every a,
b
Q
A whenever every a
i
\
b
i
is de ﬁned.
Proof. We need to prove that λi dom A: a
i
\
b
i
=
d
{z
Q
A | a b z }.
To prove it is enough to show a
i
\
b
i
=
d
{z
i
| z
Q
A, a b z} that is a
i
\
b
i
=
d
{z A
i
| a
i
b
i
z} what is true by deﬁnition.
Proposition 39. If every A
i
is a dis tributive lattice with least element, then a#b = λi dom A:
a
i
#b
i
for every a, b
Q
A whenever every a
i
#b
i
is de ﬁned.
Function spaces of posets 5
Proof. We need to prove that λi dom A: a
i
#b
i
=
d
{z
Q
A | z a z b}.
To prove it is enough to show a
i
#b
i
=
d
{z
i
| z
Q
A, z a z b} that is a
i
#b
i
=
d
{z A
i
| z a
i
j dom A: z
j
b
j
} that is a
i
#b
i
=
d
{z A
i
| z a
i
z b
i
} (take z
i
= 0 for
j
i) what is true by deﬁnition.
Proposition 40. Let every A
i
is a poset with least element and a
i
is deﬁned. Then a
= λi n: a
i
.
Proof. We need to prove that λi dom A: a
i
=
F
{c A | c a}. To prove this it is enough to show
that a
i
=
F
{c
i
| c A, c a} th at is a
i
=
F
{c
i
| c A, j n: c
j
a
j
} that is a
i
=
F
{c
i
| c A,
c
i
a
i
} (take c
i
= 0 for j
i) tha t is a
i
=
F
{c A | c a
i
} what is true by deﬁnition.
Corollary 41. Let every A
i
is a poset with le ast element and a
i
+
is deﬁned. Then a
+
= λi n: a
i
+
.
Proof. By duality.
5 Deﬁnition o f staroids
Let n be a set. As an example, n may be an ordinal, n may be a natural number, considered as a
set by the fo rmula n = {0,
, n 1}. Let A = A
in
is a family of posets indexed by the set n.
Deﬁnition 42. I will call an anchored relation a pair f = (form f ; GR f) of a family form(f ) o f
sets in de xed by the some index set and a relation GR(f ) P
Q
form(f ). I call GR(f) the graph
of the anchored relation f. I denote Anch(A) the set of small anchored relations of the form A.
Deﬁnition 43. An anchored relation on powersets is an anchored relation f such that every
(form f )
i
is a powerset.
I will denote arity f = dom form f .
Deﬁnition 44 . Every set of anchored relations of the same form constitutes a poset by the formula
f g GR f GR g.
Deﬁnition 45. An anchored relation is an anchored relation between posets when every (form f )
i
is a poset .
Deﬁnition 46. Let f is an anchored relation. For every i arity f and L
Q
((form f)|
(arity f)\{i}
)
(val f )
i
L = {X (form f )
i
| L {(i; X)} GR f }
(“val” is an abbreviation of the word “value”.)
Obvious 47. X (val f )
i
L L {(i; X)} GR f .
Proposition 48. f can be restored knowing form(f ) and (val f)
i
for some i n.
Proof. GR f = {K
Q
form f | K GR f } = {L {(i; X)} | L
Q
(form f)|
(arity f)\{i}
,
X (form f)
i
, L {(i; X)} GR f } = {L {(i; X)} | L
Q
(form f)|
(arity f)\{i}
, X (val f )
i
L}.
Deﬁnition 49. A pre-staroid is an anchored relation f between poset such that (val f )
i
L is a free
star for every i arity f , L
Q
(form f )|
(arity f)\{i}
.
Deﬁnition 50. A staroid is a pre-staroid whose graph is an upper set (on the poset if anchored
relations of the form of this pre-staroid).
Proposition 51. If L
Q
form f and L
i
= 0
(form f)
i
for some i arity f t he n L
f if f is an pre-
staroid.
Proof. Let K = L|
(arity f)\{i}
. We have 0
(val f )
i
K; K {(i; 0)}
f; L
f.
6 Section 5
Deﬁnition 52. Inﬁnitary pre-staroid is such a staroid whose arity is inﬁnite; ﬁnitary pre-staroid
is such a staroid whos e arity is ﬁnite.
Next we will deﬁne com pletary staroids. First goes the general case, next simpler case for the
special case of join-semilattices instead of arb itr ary posets.
Deﬁnition 53. A completary staroid is a poset relation conforming to the formulas:
1. K
Q
form f: (K L
0
K L
1
K GR f ) c {0, 1}
n
: (λi n: L
c(i)
i) GR f for
every L
0
, L
1
Q
form f.
2. If L
Q
form f and L
i
= 0
(form f)
i
for some i arity f then L
f.
Lemma 54. Every completary staroid is an upper set.
Proof. Let f is a completary staroid. Let L
0
L
1
for some L
0
, L
1
Q
form f and L
0
f.
Then taking c = n × {0} we get λi n: L
c(i)
i = λ i n: L
0
i = L
0
f and thus L
1
f because
L
1
L
0
L
1
L
1
.
Proposition 55. A relation between posets whose form is a family of join-semilattices is a com-
pletary staroid iﬀ both:
1. L
0
L
1
GR f c {0, 1}
n
: (λi n: L
c(i)
i) GR f for every L
0
, L
1
Q
form f .
2. If L
Q
form f and L
i
= 0
(form f)
i
for some i arity f then L
f.
Proof. Let the formulas (1) and (2) hold. Then f is an upper set: Let L
0
L
1
for some L
0
,
L
1
Q
form f a nd L
0
f . Then taking c = n × {0} we get λi n: L
c(i)
i = λi n: L
0
i = L
0
f and
thus L
1
= L
0
L
1
f .
Thus to ﬁnish the proof it is enough to show that
L
0
L
1
GR f K
Y
form f : (K L
0
K L
1
K GR f )
under condition that GR f is an upper set. But this is obvious.
Proposition 56. A completary s taroid is a staroid.
Proof. Let f is a co mpletary s taroid.
Let K
Q
i(arity f)\{i}
(form f)
i
. Let L
0
= K {(i; X
0
)}, L
1
= K {(i; X
1
)} for some X
0
,
X
1
A
i
. Then X
0
X
1
(val f )
i
K L
0
L
1
GR f k {0, 1}: K {(i; X
k
)} GR f K {(i;
X
0
)} f K {(i ; X
1
)} GR f X
0
(val f)
i
K X
1
(val f)
i
K.
So (val f )
i
K is a free star (taken in account that K
i
= 0
(form f)
i
f
K).
f is an upper set by the lemma.
Lemma 57. Every ﬁnitary pre -staroid is completary.
Proof. c {0, 1 }
n
: (λi n: L
c(i)
i) GR f c {0, 1}
n1
: ({(n 1; L
0
(n 1))} (λi n 1:
L
c(i)
i)) GR f ({(n 1; L
1
(n 1))} (λ i n 1: L
c(i)
i)) GR f c {0, 1}
n1
:
L
0
(n 1) (val f )
n1
(λi n 1: L
c(i)
i) L
1
(n 1) (val f )
n1
(λi n 1: L
c(i)
i) c { 0,
1}
n1
K
Q
form f:(K L
0
(n 1) K L
1
(n 1) K (val f)
n1
(λi n 1:L
c(i)
i)) c {0,
1}
n1
K
n1
(form f )
n1
: (K
n1
L
0
(n 1) K
n1
L
1
(n 1) {(n 1; K)} (λi n 1:
L
c(i)
i)) GR f
K
Q
form f : (K L
0
K L
1
K GR f ).
Exercise 1. Prove the simpler special case of the above theorem when the form is a family of join-semilattices.
Theorem 58. For ﬁnite arity the following are the same:
1. pre-staroids;
2. star oids;
Definition of staroids 7
3. completary sta roids.
Proof. f is a ﬁnitary pre- staroid f is a ﬁnitary comple tary staroid.
f is a ﬁnitary completary s taroid f is a ﬁnitary staroid.
f is a ﬁnitary staroid f is a nitary pre-staroid.
Deﬁnition 59. We will denote the set of staroids, pre-staroids, and completary staroids o f a form
A correspondingly as Strd(A), pStrd(A), and cStrd(A).
6 Upgrading and downgrading a set regarding a ltrator
Let x a ﬁltrator (A; Z).
Deﬁnition 60. f = f Z for every f PA (downgrading f ).
Deﬁnition 61. f = {L A | up L f } for every f PZ (upgrading f ).
Obvious 62. a f up a f for every f PZ and a A.
Proposition 63. f = f if f is an upper set.
Proof. f = f Z = {L Z | up L f } = {L Z | up L f } = f PZ = f .
6.1 Upgrading and downgrading staro ids
Let x a family (A; Z) of ﬁltrators.
For a graph f o f a staroid deﬁne f and f taking the ﬁltrator of (
Q
A;
Q
Z).
For a staroid f deﬁne:
form f = Z and GR f = GR f ;
form f = A and GR f = GR f .
Proposition 64. (val f ))
i
L = (val f)
i
L Z
i
for eve ry L
Q
Z|
(arity f)\{i}
.
Proof. (val f ))
i
L = {X (form f )
i
| L {(i; X)} GR f
Q
Z} = {X Z
i
| L {(i;
X)} GR f } = (val f )
i
L Z
i
.
Proposition 65. Let (A
i
; Z
i
) are ﬁnitely join -closed ﬁltrators with both the base and the core
being join-semilat tices. If f is a staroid of the form A, then f is a staroid of the form Z.
Proof. Let f is a a staroid.
We need to prove that (val f )
i
L is a free star. It follows from the last proposition and the
fact that it is join-closed.
Proposition 66.
Q
Strd
a = ⇈
Q
Strd
a if each a
i
A
i
(for i n where n is some index set) where
A
i
is a separable poset with least element.
Proof. ⇈
Q
Strd
a =
L
Q
A | L
Q
Strd
a
= {L
Q
A | K L: K
a} =
{L
Q
A | L
a} =
Q
Strd
a (taken into acco unt that
Q
A is a sepa rable poset).
6.2 Displacement
Deﬁnition 67. Let f is an indexed family of po intfree funcoids. The displacement of the pre-
staroid
p A = pStrd(λi dom f : FCD(Src f
i
; Src g
i
))
8 Section 6
is de ned as a staroid
q B = pStrd(λi dom f: RLD(Src f
i
; Src g
i
))
such that
q =
(B;C;
B
)
(A;C;
A
)
p
where C = pStrd
Q
idom f
Src f
i
;
Q
idom f
Dst f
i
.
Deﬁnition 68. We w ill deﬁne displaced product of a family f of funcoids by the formula:
Q
(DP)
f = DP
Q
(C)
f
.
Remark 69. The interesting aspect of displaced product of funcoids is that displaced product of
pointfree funcoids is a funcoid (not just a pointfree funcoid).
7 Multifun coids
Deﬁnition 70. I call an pre-multifuncoid sketch f of the form A (where every A
i
is a poset) the
pair (A; α) where for every i dom α
α
i
:
Y
A|
(dom A)\{i}
A
i
.
I denote hf i = α.
Deﬁnition 71. A pre-multifuncoid sketch on powersets is a pre-multifuncoid sketch such that
every A
i
is the set of ﬁlters on a powerset.
Deﬁnition 72. I will call a pre-multifuncoi d a pre-multifuncoid sketch s uch that for every i,
j dom A and L
Q
A
L
i
α
i
L|
(dom L)\{i}
L
j
α
j
L|
(dom L)\{j }
. (4)
Deﬁnition 73. Let A is an indexed fa mily of starrish posets. The pre-staroid correspondi ng to a
pre-multifuncoid f is [f] deﬁned by the formula:
form [f ]=A and L GR [f]L
i
hf i
i
L|
(dom L)\{i}
.
Proposition 74. The pre-staroid corresponding to a pre-multifuncoid is really a pre-staroid.
Proof. By the deﬁnition of starrish posets.
Deﬁnition 75. I will call a multifuncoid a pre-multifuncoid to which corresponds a staroid.
Deﬁnition 76. I will call a completary multifuncoid a pre-multifuncoid to which corresponds a
completary staroid .
Theorem 77. Fix some indexed family A of bo olean lattices. The the set of multifuncoids g
bijectively corresponds to set of pre-staroids f of form A by the formulas:
1. f = [g] for every i dom A, L
Q
A;
2. hgi
i
L = (val f )
i
L.
Proof. Let f is a pre-staroid of the form A. If α is deﬁned by the formula α
i
L = hf i
i
L then
∂α
i
L = (val f )
i
L. Then
L
i
α
i
L|
(dom L)\{i}
L f L
j
α
j
L|
(dom L)\{j }
.
For the sta roid f
deﬁned by the formula L f
L
i
α
i
L|
(dom L)\{i}
we have:
L f
L
i
∂α
i
L|
(dom L)\{i}
L
i
(val f)
i
L|
(dom L)\{i}
L f ;
Multifuncoids 9
thus f
= f.
Let now α is an indexed fa mily of functions α
i
A
i
(dom A)\{i}
conforming to the formula (4).
Let relation f b e tween posets is deﬁned by the formula L f L
i
α
i
L|
(dom L)\{i}
. T he n
(val f )
i
L = {K A
i
| K
α
i
L|
(dom L)\{i}
} = K = ∂α
i
L|
(dom L)\{i}
and thus (val f)
i
L is a c ore star that is f is a pre-staroid. For the indexed family α
deﬁned by
the formula α
i
L = hf i
i
L we have
∂α
i
L = hf i
i
L = {K A
i
| K
α
i
L} = ∂α
i
L;
thus α
= α.
We have shown that these are bijections.
Theorem 78. hf i
j
(L {(i; X Y )}) = hf i
j
(L {(i; X)}) hf i
j
(L {(i; Y )}) for every staroid f
if (form f )
j
is a boolean lattice and i, j arity f.
Proof. Let i arity f and L
Q
kL \{i,j }
A
k
. Let Z A
i
.
Z
hf i
j
(L {(i; X Y )}) L {(i; X Y ), (j; Z)} f X Y (va l f )
i
(L {(j;
Z)}) X (val f)
i
(L {(j; Z)} Y (va l f)
i
(L {(j; Z)}) L {(i; X), (j; Z)} f L {(i; Y ),
(j; Z)} f
A
i
Z
hf i
j
(L {(i; X)}) Z
hf i
j
(L {(i; Y )})
Thus hf i
j
(L {(i; X Y )}) = h f i
j
(L {(i; X)}) hf i
j
(L {(i; Y )}).
Let us cons ider the ﬁltrator
Q
iarity f
F((form f )
i
);
Q
iarity f
(form f )
i
.
Theorem 79. Let (A
i
; Z
i
) is a family of join-closed down-aligned ﬁltrators ltrators whose both
base and core are join-semilattices. Let f is a pre-staroid of the form Z. Then f is a staroid o f
the form A.
Proof. First prove that GR f is a pre-staroid. We need to prove that 0
(GR f )
i
(that
is up 0
(GR f)
i
what is true by the theorem conditions) and that for every X , Y A
i
and
L
Q
i(arity f)\{i}
A
i
where i arity f
L {(i; X Y)} GR f L {(i; X )} GR f L {(i; Y)} GR f.
The reverse implication is obvious. Let L {(i; X Y)} GR f. The n for every L L and X X ,
Y Y we have and X
Z
i
Y X
A
i
Y thus L {(i; X
Z
i
Y )} GR f and thus
L {(i; X)} GR f L {(i; Y )} GR f
consequently L {(i; X )} GR f L {(i; Y)} GR f .
It is left to prove that f is an upper set, but this is obvious.
There is a conjecture similar to the above th e ore ms:
Conjecture 80. L [f ][f ]
Q
idom A
atoms L
i
for every multifuncoid f of the form whose
elements are atomic posets. (Do e s this conjecture hold for the special case of form whose elements
are posets on ﬁlters on a set?)
Conjecture 81. Let be a se t, F be the set of f.o. on , P be the set of principal f.o. on , let
n be an index set. Consider the ﬁltrator (F
n
; P
n
). Then if f is a completary s taroid of the form
P
n
, then f is a completary staroid of the form F
n
.
8 Join of multi funcoids
Pre-multifuncoid sketches are ordered by the formula f g hf i hg i where in the rig ht part
of this formula is the product order . I will denote , ,
d
,
F
(without an inde x) the or de r poset
operations on the poset of pre-multifuncoid sketchs.
10 Section 8
Remark 82. To describe this, the deﬁnition of order poset is u sed twice. Let f and g are posets
of the same f orm A
hf i hgi i dom A: hf i
i
hgi
i
and hf i
i
hgi
i
L
Y
A|
(dom A)\{i}
: hf i
i
L hgi
i
L.
Theorem 83. f
pFCD(A)
g = f g for every pre-multifuncoids f and g of the same form A of
distributive lattices .
Proof. α
i
x =
def
f
i
x g
i
x. It is enough to prove that α is a multifuncoid.
We need to prove:
L
i
α
i
L|
(dom L)\{i}
L
j
α
j
L|
(dom L)\{j }
.
Really, L
i
α
i
L|
(dom L)\{i}
L
i
f
i
L|
(dom L)\{i}
g
i
L|
(dom L)\{i}
L
i
f
i
L|
(dom L)\{i}
L
i
g
i
L|
(dom L)\{i}
L
j
f
j
L|
(dom L)\{j }
L
j
g
j
L|
(dom L)\{j }
L
j
f
j
L|
(dom L)\{j }
g
j
L|
(dom L)\{j}
L
j
α
j
L|
(dom L)\{j }
.
Theorem 84.
F
pFCD(A)
F =
F
F for every set F of pre-multifuncoids of the same f orm A of join
inﬁnite distributive complete lattices.
Proof. α
i
x =
def
F
f F
f
i
x. It is enough to prove that α is a multifuncoid.
We need to prove:
L
i
α
i
L|
(dom L)\{i}
L
j
α
j
L|
(dom L)\{j }
.
Really, L
i
α
i
L|
(dom L)\{i}
L
i
F
f F
f
i
L|
(dom L)\{i}
⇔∃f F : L
i
f
i
L|
(dom L)\{i}
⇔∃f F :
L
j
f
j
L|
(dom L)\{j}
L
j
F
f F
f
j
L|
(dom L)\{j }
L
j
α
j
L|
(dom L)\{j}
.
Proposition 85. The mapping f
[f ] is an order embedding, f or multifuncoids of the form A of
separable starrish posets.
Proof. The mapping f
[f] is deﬁned because A are starrish poset. The mapping is injective
because A are separable posets. That f
[f ] is a monot one function is obvious.
Remark 86. This order embedding is useful to describe propert ie s of posets of pre-staroids.
Theorem 87. If f, g are multifuncoids of the same form A of distributive la ttices, then
f
pFCD(A)
g FCD(A).
Proof. Let A
f
pFCD(A)
g
and B A. Then for every k dom A
A
k
f
pFCD(A)
g
A|
(dom A)\{k}
=(f g)A|
(dom A)\{k}
=f (A|
(dom A)\{k}
) g(A|
(dom A)\{k}
).
Thus A
k
f(A|
(dom A)\{k }
) A
k
g(A|
(dom A)\{k}
); A [f ]A [g]; B [f ]B [g];
B
k
f(B |
(dom A)\{k}
) B
k
g(B |
(dom A)\{k}
); f(B |
(dom A)\{k }
) g(B |
(dom A)\{k}
) = (f
g)B |
(dom A)\{k}
=
f
pFCD(A)
g
B|
(dom A)\{k}
B
k
. Thus B
f
pFCD(A)
g
.
Theorem 88. If F is a set multifuncoids of the same form A of join iniﬁ nite distributive c omplete
lattices, then
F
pFCD(A)
f FCD(A).
Proof. Let A
h
F
pFCD(A)
f
i
and B A. Then for every k dom A.
A
k
F
pFCD(A)
F
A|
(dom A)\{k }
=(
F
F )A|
(dom A)\{k }
=
F
f F
f(A|
(dom A)\{k}
).
Thus f F : A
k
f(A|
(dom A)\{k}
); f F : A [f ]; B [f ]B [g]; f F : B
k
f(B |
(dom A)\{k}
);
F
f F
f(B|
(dom A)\{k }
) = (f g)B |
(dom A)\{k }
=
F
pFCD(A)
F
B|
(dom A)\{k}
B
k
.
Thus B
h
F
pFCD(A)
F
i
.
Conjecture 89. The formula f
FCD(A)
g cFCD(A) is not true in general for completary
multifuncoids (even for multifuncoids on powersets) f and g of the same form A.
Join of multifuncoids 11