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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 deﬁne 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This article is outdated.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 ﬁlters 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 iﬀ each A

i

has a least element. In this case

Least(A) =

Y

i∈dom 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

A∀i ∈ d om A: (c

i

⊑ a

i

∧ c

i

⊑ b

i

) ⇔

∀i ∈ dom A∃x ∈

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

i∈dom x

atoms x

i

=

G

i∈dom 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

i∈n

; Z

i∈n

) 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

i∈n

; Z

i∈n

) 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 ﬁltrator (

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 ﬁltrator 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

A∀i ∈ dom A:

(x

i

⊓ y

i

= 0

A

i

∧ x

i

⊔ y

i

= 1

A

i

) ⇔ ∀i ∈ dom A∃y ∈ 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

i∈dom a

up a

i

;

4 Section 4

2. down a =

Q

i∈dom 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

i∈dom 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 A∃X ∈ up x

i

: X ≍

A

i

y

i

⇔ ∃X ∈ up x∀i ∈ 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 ﬁltrator, then (

Q

A;

Q

Z) is a down-aligned ﬁltrator.

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

i∈n

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}

n−1

: ({(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}

n−1

:

L

0

(n − 1) ∈ (val f )

n−1

(λi ∈ n − 1: L

c(i)

i) ∨ L

1

(n − 1) ∈ (val f )

n−1

(λi ∈ n − 1: L

c(i)

i) ⇔ ∃c ∈ { 0,

1}

n−1

∀K ∈

Q

form f:(K ⊒L

0

(n − 1) ∨ K ⊒ L

1

(n − 1)⇒ K ∈ (val f)

n−1

(λi ∈ n − 1:L

c(i)

i)) ⇔∃c ∈ {0,

1}

n−1

∀K

n−1

∈ (form f )

n−1

: (K

n−1

⊒ L

0

(n − 1) ∨ K

n−1

⊒ 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 ﬁnitary 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 ﬁltrator

Let ﬁx 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 ﬁx 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 ﬁned 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

i∈dom f

Src f

i

;

Q

i∈dom 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

k∈L \{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

i∈arity f

F((form f )

i

);

Q

i∈arity f

(form f )

i

.

Theorem 79. Let (A

i

; Z

i

) is a family of join-closed down-aligned ﬁltrators ﬁltrators 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

i∈dom 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