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Identity St aroids

by Victor Porton

Email: porton@narod.ru

Web: http://www.mathematics21.org

1 Draft status

This text is a draft.

Read my book [1] before reading this. Well, most p robably I will integrate materials from this

article into my book.

2 Additional propositions

Proposition 1.

hf i

k

X | X ∈ up

Q

i∈n \{k }

A

i

;

Q

i∈n \{k }

Z

i

X

is a ﬁlter base on A

k

for every famil y

(A

i

; Z

i

) of ﬁltrators where i ∈ n for s ome index set n (provided that f is a multifuncoid of the form

A and k ∈ n and every Z

i

for i ∈ n \ {k} is a ﬁlter base and X ∈

Q

i∈n \{k}

A

i

).

Proof. Let K, L ∈ {hf i

k

X | X ∈ up X }. Then there exist X , Y ∈ up X such that K = hf i

k

X,

L = hf i

k

Y . We can take Z ∈ up X such that Z ⊑ X , Y . Then evidently hf i

k

Z ⊑ K and hf i

k

Z ⊑ L

and hf i

k

Z ∈ {hf i

k

X | X ∈ up X }.

Proposition 2. h⇈ f i

k

X =

d

X ∈up X

hf i

k

X for a ﬁltrator

Q

i∈n \{k}

F

i

;

Q

i∈n \{k}

P

i

(i ∈ n

for some index set n) where every Z

i

is a boolean lattice, k ∈ n, and X ∈

Q

i∈n \{k}

F

i

.

Proof. F

k

is separable by obvious 4.136??. (F

k

; P

k

) is with separable core by theorem 4.112??.

Y

h⇈ f i

i

X ⇔ X ∪ {(i; Y) } ∈ GR [⇈ f]⇔X ∪ {(i; Y)} ∈ ⇈ GR [f]⇔up(X ∪ {(i; Y)}) ⊆ GR [f]⇔

∀X ∈ up X , Y ∈ up Y: X ∪ {(i; Y )} ∈ GR [f]⇔∀X ∈ up X , Y ∈ up Y: Y

hf i

i

X ⇔ ∀X ∈ up X ,

Y ∈ up Y: Y ⊓ hf i

i

X

0 ⇔ ∀Y ∈ up Y: 0

{Y ⊓ hf i

i

X | X ∈ up X } ⇔ ∀Y ∈ up Y:

0

hY ⊓ i{hf i

i

X | X ∈ up X } ⇔ (by properties of generalized ﬁlter bases)⇔∀Y ∈ up Y:

d

hY ⊓ i{hf i

i

X | X ∈ up X }

0 ⇔ ∀ Y ∈ up Y: Y ⊓

d

{hf i

i

X | X ∈ up X }

0 ⇔ ∀Y ∈ up Y:

Y

d

X ∈up X

hf i

i

X ⇔ Y

d

X ∈up X

hf i

i

X; so h⇈ f i

i

X =

d

X ∈up X

hf i

i

X.

3 On pseudofuncoids

Deﬁnition 3. Pseudofuncoid from a set A to a set B is a relatio n f between ﬁlters on A and B

such that:

¬(I f 0), I ⊔ J f K ⇔ I f K ∨ J f K (for every I , J ∈ F(A), K ∈ F(B)),

¬(0 f I), K f I ⊔ J ⇔ K f I ∨ K f J (for every I , J ∈ F(B), K ∈ F(A)).

Obvious 4. Pseudofuncoid is just a staroid of the form (F(A); F(B)).

Obvious 5. [f] is a pseudofuncoid for every funcoid f.

Example 6. If A and B are inﬁnite sets, then t he re exist two diﬀerent pseudofuncoids f and g

from A to B such that f ∩ (P × P) = g ∩ (P × P) = [c]∩(P × P) for some funcoid c.

Remark 7. Considering a pseudofuncoid f as a staroid, we get f ∩ (P × P) = f.

1

Proof. Take

f =

(X ; Y) | X ∈ F(A), Y ∈ F(B),

\

X and

\

Y are inﬁnite

and

g = f ∪ {(X ; Y) | X ∈ F(A), Y ∈ F(B), X ⊒ a, Y ⊒ b}

where a and b are nontrivial ultraﬁlters on A and B correspo ndingly, c is the funcoid deﬁned by

the relation

[c]

∗

= δ = {(X; Y ) | X ∈ PA, Y ∈ PB , X and Y are inﬁnite}.

First prove that f is a pseudofuncoid. The formulas ¬(I f 0) and ¬(0 f I) are obvious. We have

I ⊔ J f K ⇔

T

(I ⊔ J ) and

T

Y are inﬁnite⇔

T

I ∪

T

J and

T

Y are inﬁnite⇔(

T

I or

T

J is

inﬁnite) ∧

T

Y is inﬁnite⇔(

T

I and

T

Y are inﬁnite)∨ (

T

J and

T

Y are inﬁnite)⇔ I f K ∨ J f K.

Similarly K f I ⊔ J ⇔ K f I ∨ K f J . So f is a pseudofuncoid.

Let now prove that g is a pseudofuncoid. The formulas ¬(I g 0) and ¬(0 g I) are obvious. Let

I ⊔ J g K. Then either I ⊔ J f K and then I ⊔ J g K or I ⊔ J ⊒ a and then I ⊒ a ∨ J ⊒ a thus

having I g K ∨ J g K. So I ⊔ J g K ⇒ I g K ∨ J g K. The reverse implication is obvious. We have

I ⊔ J g K ⇔ I g K ∨ J g K and similarly K g I ⊔ J ⇔ K g I ∨ K g J . So g is a pseudofuncoid.

Obviously f

g (a g b but not a f b).

It remains to prove f ∩ (P × P) = g ∩ (P × P) = [c]∩(P × P). Really, f ∩ (P × P) = [c]∩(P × P)

is obvious. If (↑

A

X; ↑

B

Y ) ∈ g ∩ (P × P) then either (↑

A

X; ↑

B

Y ) ∈ f ∩ (P × P) or X ∈ up a, Y ∈ up b,

so X and Y are inﬁnite and thus (↑

A

X; ↑

B

Y ) ∈ f ∩ (P × P). So g ∩ (P × P) = f ∩ (P × P).

Remark 8. The above c ounter-example shows that pseudofuncoids (and more generally, any

staroids on ﬁlters) are “second class” objects, they are not full-ﬂedged because they don’t bijectively

correspond to funcoids and the ele gant funcoids theory does not apply to them.

From the above it follows that staroids on ﬁlters do not correspond (by restriction) to staroids

on principal ﬁlters (or staroids on se ts).

4 Complete staroids and multifuncoids

4.1 Complete free stars

Deﬁnition 9. Let A be a poset. Complete free sta rs on A are s uch S ∈ P A that the least element

(if it e xists) is not in S and for every T ∈ PA

∀Z ∈ A: (∀X ∈ T : Z ⊒ X ⇒ Z ∈ S) ⇔ T ∩ S

∅.

Obvious 10. Every complete free star is a free star.

Proposition 11. S ∈ PA where A is a poset is a complete free star iﬀ all the following:

1. The least element (if it exists) is not in S.

2. ∀Z ∈ A: (∀X ∈ T : Z ⊒ X ⇒ Z ∈ S) ⇒ T ∩ S

∅.

3. S is an upper set.

Proof.

⇒. (1) and (2) are obvious. S is an upper s et because S is a free star.

⇐. We need to prove that

∀Z ∈ A: (∀X ∈ T : Z ⊒ X ⇒ Z ∈ S) ⇐ T ∩ S

∅.

Let X

′

∈ T ∩ S. Then ∀X ∈ T : Z ⊒ X ⇒ Z ⊒ X

′

⇒ Z ∈ S because S is an upper set.

Proposition 12. Let S be a complete lattice. S ∈ P A is a complete fre e star iﬀ all the following:

1. The least element (if it exists) is not in S.

2 Section 4

2.

F

T ∈ S ⇒ T ∩ S

∅ for e very T ∈ PS.

3. S is an upper set.

Proof.

⇒. We need to prove only

F

T ∈ S ⇒ T ∩ S

∅. Let

F

T ∈ S. Because S is an upper set, we

have ∀X ∈ T : Z ⊒ X ⇒ Z ⊒

F

T ⇒ Z ∈ S from which we conclude T ∩ S

∅.

⇐. We need to prove only ∀Z ∈ A: ( ∀X ∈ T : Z ⊒ X ⇒ Z ∈ S) ⇒ T ∩ S

∅.

Really, if ∀Z ∈ A: (∀X ∈ T : Z ⊒ X ⇒Z ∈ S) then

F

T ∈S and thus

F

T ∈S ⇒ T ∩S

∅.

Proposition 13. Let A be a complete lattice. S ∈ PA is a complete free star iﬀ the least element

(if it e xists) is not in S and for every T ∈ PA

G

T ∈ S ⇔ T ∩ S

∅.

Proof.

⇒. We need to prove only

F

T ∈ S ⇐ T ∩ S

∅ what follows from that S is an upper set.

⇐. We need to prove only that S is an upper set. To prove this we can use the fact that S is

a free star.

4.1.1 Completely s tarrish posets

Deﬁnition 14. I will call a poset completely starrish when the full star ⋆ a is a free star for every

element a of this poset.

Obvious 15. Every completely starrish poset is starrish.

Proposition 16. Every co mplete join inﬁnite distributive lattice is starrish.

Proof. Let A be a join inﬁnite dis tributive lattice, a ∈ A. Obviously 0

⋆a (if 0 exists); obviously

⋆ a is an upper set. If

F

T ∈ ⋆a, then (

F

T ) ⊓ a is non-least that is

F

ha ⊓ i T is no n-least what is

equivalent to a ⊓ x being non-least for some x ∈ T that is x ∈ ⋆ a.

Theorem 17. If A is a comple tely starrish complete lattice lattice then

atoms

G

T =

[

hatomsiT .

for every T ∈ PA.

Proof. For every ato m c we have: c ∈ atoms

F

T ⇔ c

F

T ⇔

F

T ∈ ⋆ c⇔ ∃X ∈T : X ∈ ⋆ c⇔ ∃X ∈ T :

X

c ⇔ ∃X ∈ T : c ∈ atoms X ⇔ c ∈

S

hatomsiT .

4.2 More on free stars and complete free stars

Obvious 18. ∂ F = ⋆ F for an element F of down-aligned ﬁnitely meet closed ﬁltrator.

Corollary 19. ∂ F = ⋆ F for every ﬁlter F on a poset.

Proposition 20. ⋆ F = ⇈ ∂ F for an e lement F of a ﬁltrator with separable core.

Proof. X ∈ ⇈ ∂ F ⇔ up X ⊆ ∂ F ⇔ ∀X ∈ X : X

F ⇔ X

F ⇔ X ∈ ⋆ F.

Corollary 21. ⋆ F = ⇈ ∂ F for every ﬁlter F on a distributive lattice with least element.

Proposition 22. For a semiﬁltered, star-separable, down-aligned ﬁltrator (A; Z) with ﬁnitely meet

closed and separable core where Z is a complete boolean lattice and both Z and A are atomistic

lattices the following conditions are equivalent for any F ∈ A:

1. F ∈ Z.

Complete staroids and multifuncoids 3

2. ∂ F is a complete free star on Z.

3. ⋆ F is a complete free star on F.

Proof.

(1)⇒(2). That ∂ F does not contain the least element is obvious. That ∂ F is a n upper set is

obvious. So it remains to apply theorem 4.53??.

(2)⇒(3). That ⋆ F does not contain the least element is obvious. That ⋆ F is an upper set is

obvious. So it remains to apply theorem 4.53??.

(3)⇒(1). Apply theorem 4.53??.

Corollary 23. For a ﬁlter F ∈ F on a complete atomic boolean lattice the following conditions

are equivalent:

1. F ∈ P.

2. ∂ F is a complete free star on P.

3. ⋆ F is a complete free star on F.

Theorem 24. Let Z be a boolean lattice. For any set S ∈ P P there exists a principal ﬁlter A

such that ∂ A = S iﬀ S is a complete free star (on P).

Proof.

⇒. From the previous theorem.

⇐. 0

P

S and

F

T ∈ S ⇔ T ∩ S

∅ ⇔ ∃X ∈ T : X ∈ S. Take A = {X | X ∈ P \ S }. We will

prove that A is a prin c ipal ﬁlter. That A is a ﬁlter foll ows from properties of free stars. It

remains to show that A is a principa l ﬁlter. It follows from the following equivalence:

d

P

A ∈ A ⇔

F

P

h¬iA

∈ A ⇔

F

P

h¬iA

S ⇔ ¬∃X ∈ h¬iA: X ∈ S ⇔ ∀X ∈ h¬iA:

X

S ⇔ ∀X ∈ A: X ∈ A ⇔ 1.

Proposition 25.

1. If S is a free star on A then S is a free star on Z, p rovided that Z is a join-s e milattice and

the ﬁltrator (A; Z) is down-aligned and with ﬁnitely join-closed core.

2. If S is a free star on P then ⇈ S is a fr e e star on F, provided that Z is a boolean lattice.

Proof.

1. X ⊔

Z

Y ∈ S ⇔ X ⊔

Z

Y ∈ S ⇔ X ⊔

A

Y ∈ S ⇔ X ∈ S ∨ Y ∈ S ⇔ X ∈ S ∨ Y ∈ S for every

X , Y ∈ Z; 0

S is obvious.

2. There exists a ﬁlter F such that S = ∂ F. For every ﬁlters X , Y ∈ F

X ⊔

A

Y ∈ ⇈S ⇔ up(X ⊔

A

Y) ⊆ S ⇔ ∀K ∈ up(X ⊔

F

Y): K ∈ ∂ F ⇔ ∀K ∈ up(X ⊔

F

Y):

K

F ⇔ X ⊔

F

Y

F ⇔ X ⊔

F

Y ∈ ⋆ F ⇔ X ∈ ⋆ F ∨ Y ∈ ⋆ F ⇔ X

F ∨ Y

F ⇔ ∀X ∈ up X :

X

F ∨∀Y ∈ up Y:Y

F ⇔∀X ∈upX : X ∈ ∂F ∨∀ Y ∈up Y:Y ∈∂ F ⇔up X ⊆S ∨up Y ⊆S ⇔

X ∈ ⇈S ∨ Y ∈ ⇈ S;

0 ∈ ⇈ S ⇔ up 0 ⊆ S ⇔ 0 ∈ S what is false.

Corollary 26. If S is a free star on F then S is a free sta r on P, provided that P is a join-

semilattice.

Proposition 27.

1. If S is a complete free star on A then S is a complete free star on Z, provided that Z is

a complete lattice and the ﬁltrat or (A; Z) is down-aligned and with join-closed core.

2. If S is a complete free star on P then ⇈S is a complete free sta r on F, provided that Z is

a boolean lattice.

4 Section 4

Proof.

1.

F

Z

T ∈ S ⇔

F

Z

T ∈ S ⇔

F

A

T ∈ S ⇔ T ∩ S

∅ ⇔ T ∩ S

∅ for every T ∈ P Z; 0

S is

obvious.

2. There exists a principal ﬁlter F such that S = ∂ F.

F

A

T ∈ ⇈ S ⇔ up

F

A

T ∈ S ⇔ ∀K ∈ up

F

A

T : K ∈ ∂ F ⇔ ∀K ∈ up

F

A

T :

K

F ⇔

F

A

T

F ⇔

F

A

T ∈ ⋆ F ⇔ ∃K ∈ T : K ∈ ⋆ F ⇔ ∃K ∈ T : K

F ⇔ ∃K ∈ T ∀K ∈ up K:

K

F ⇔ ∃K ∈ T ∀K ∈ up K: K ∈ ∂ F ⇔ ∃K ∈ T : up K ⊆ S ⇔ ∃K ∈ T : K ∈ ⇈S ⇔ T ∩ ⇈S

∅.

0 ∈ ⇈ S ⇔ up 0 ⊆ S ⇔ 0 ∈ S what is false.

Corollary 28. If S is a complete free star on F then S is a complete free star on P, provided

that Z is a complete lattice.

5 Complete staroids and multifuncoids

Deﬁnition 29. Consider an indexed family (A

i

; Z

i

) of ﬁltrators. A pre-staroid f of the f orm

Q

Z is complete in argument k ∈ arity f when (val f )

k

L is a comple te free star for every

L ∈

Q

i∈(arity f )\{k}

Z

i

.

Deﬁnition 30. Consider an indexed family (A

i

; Z

i

) of ﬁltrators and pre-multifuncoid f is of

the form

Q

A. Then f is complete in argument k ∈ arity f iﬀ hf i

k

L ∈ Z

k

for every family

L ∈

Q

i∈(arity f )\{k}

Z

i

.

Proposition 31. Consider an indexed family (F

i

; Z

i

) o f primary ﬁltrators over boolean lattices.

Let f be a pre-multifuncoid of the form A and k ∈ arity f. The following are equivalent:

1. Pre-multifuncoid f is complete in argument k.

2. Pre-staroid [f ] is complete in argument k.

Proof. L ∈ GR [f ]⇔L

i

hf i

i

L|

(dom L)\{i}

;

(val [f ])

k

L = ∂ hf i

k

L by the theorem ??17.81.

So (val [f ])

k

L is a complete free star iﬀ hf i

k

L ∈ Z

k

(proposition 22) for every L ∈

Q

i∈(arity f )\{k}

Z

i

.

Example 32. Consider funcoid f = id

FCD(U)

. It is obviously complete in ea ch its two a rguments.

Then [f] is not complete in each of its two arguments because (X ; Y) ∈ [f]⇔X

Y what does not

generate a complete free star if one of the argu ments (say X ) is a ﬁxed nonprincipa l ﬁlter.

Theorem 33. Consider a semiﬁltered, st ar-separable, down-aligned ﬁltrator (A; Z) with ﬁnitely

meet clos e d and separable core where Z is a complete boolean lattice and both Z a nd A are atomistic

lattices.

Let f be a multifuncoid of the aforementioned form. Let k, l ∈ arity f and k

l. The following

are equivalent:

1. f is c omplete in t he argument k.

2. hf i

l

(L ∪ {(k;

F

X)}) =

F

x∈X

hf i

l

(L ∪ {(k; x)}) for every X ∈ PZ

k

, L ∈

Q

i∈(arity f )\{k,l}

Z

i

.

3. hf i

l

(L ∪ {(k;

F

X)}) =

F

x∈X

hf i

l

(L ∪ {(k;x)} ) for every X ∈ P A

k

, L ∈

Q

i∈(arity f)\{k,l}

Z

i

.

Proof.

(3)⇒(2). Obvious.

(2)⇒(1). Let Y ∈ Z.

F

X

hf i

k

(L ∪ {(l; Y )}) ⇔ Y

hf i

l

(L ∪ {(k;

F

X)}) ⇔ Y

F

x∈X

hf i

l

(L ∪ {(k;

x)}) ⇔ (proposition 4.144??)⇔∃x ∈ X: Y

hf i

l

(L ∪ {(k; x)}) ⇔ ∃x ∈ X: x

hf i

k

(L ∪ (l; Y )).

Complete staroids and multifuncoids 5

It is equivalent (proposition 22 and the fact that [f ] is an upper se t) to hf i

k

(L ∪ {(l ; Y )})

being a principal ﬁlter and thus (val [f])

l

L being a complete free star.

(1)⇒(3). Y

hf i

l

(L ∪ {(k;

F

X)}) ⇔

F

X

hf i

k

(L ∪ {(l; Y )}) ⇔ ∃x ∈ X: x

hf i

k

(L ∪ {(l;

Y )}) ⇔ ∃x ∈ X: Y

hf i

l

(L ∪ {(k; x)}) ⇔ Y

F

x∈X

hf i

l

(L ∪ {(k; x)}) for every principal

Y .

6 Identity staroids and multifuncoids

6.1 Identity relations

Denote id

A[n]

= {(λi ∈ n: x) | x ∈ A} = {n × {x} | x ∈ A} the n-ary identity relation on a set A (for

each index set n).

Proposition 34.

Q

X

id

A[n]

⇔

T

i∈n

X

i

∩ A

∅.

Proof.

Q

X

id

A[n]

⇔ ∃t ∈ A: n × {t} ∈

Q

X ⇔ ∃t ∈ A∀i ∈ n: t ∈ X

i

⇔

T

i∈n

X

i

∩ A

∅.

6.2 Universal deﬁnitions of identity st aroids

Consider a ﬁltrator (A; Z) and A ∈ A.

I will deﬁne below small identity staroids id

A[n]

Strd

and big identity staroids ID

A[n]

Strd

. That they are

really staroids and even completary staroids (under certain conditions) is proved below.

Deﬁnition 3 5. Consider a ﬁltrator (A; Z). Let Z be a complete lattice. Let A ∈ A, let n be an

index set.

form id

A[n]

Strd

= Z

n

; L ∈ GR id

A[n]

Strd

⇔

d

i∈n

Z

L

i

∈ ∂ A .

Obvious 36. X ∈ GR id

A[n]

Strd

⇔ ∀A ∈ up A:

d

i∈n

Z

X

i

⊓ A

0 if our ﬁltrator is with separable core.

Deﬁnition 37. The subset X of a poset A has a nontrivial lower bound (I denote this predicate

as MEET(X)) iﬀ there is nonleast a ∈ A such that ∀x ∈ X: a ⊑ x.

Deﬁnition 38. Staroid ID

A[n]

Strd

(for any A ∈ A whe re A is a poset) is deﬁned by the formulas:

form ID

A[n]

Strd

= A

n

; L ∈ GR ID

A[n]

Strd

⇔ MEET({L

i

| i ∈ n} ∪ {A} ).

Obvious 39. If A is complete latt ic e , then L ∈ GR ID

A[n]

Strd

⇔

d

L

A.

Obvious 40. If A is complete latt ic e and a is an atom, the n L ∈ GR ID

a[n]

Strd

⇔

d

L ⊒ a.

Obvious 41. If A is a complete lattice then ther e exists a multifuncoid Λ ID

A[n]

Strd

such that

ΛID

A[n]

Strd

k

L =

d

i∈n

L

i

⊓ A for every k ∈ n, L ∈ A

n \{k}

.

Proposition 42. If (A; Z) is a meet-closed ﬁltrator and Z is a complete lattice a nd A is a meet-

semilattice. There exists a multifuncoid Λid

A[n]

Strd

such that

Λid

A[n]

Strd

k

L =

d

i∈n

Z

L

i

⊓

A

A for e very

k ∈ n, L ∈ Z

n \{k}

.

Proof. We need to prove that L ∪ {(k; X)} ∈ id

A[n]

Strd

⇔

d

i∈n

Z

L

i

⊓

A

A

A

X. But

l

i∈n

Z

L

i

⊓

A

A

A

X ⇔

l

i∈n

Z

L

i

⊓

A

X

A

A ⇔

l

i∈n

Z

(L ∪ {(k ; X)})

i

A

A ⇔ L ∪ {(k; X)} ∈ id

A[n]

Strd

.

6.3 Identities are staroids

Proposition 43. Let A be a complete distributive lattice and A ∈ A. Then ID

A[n]

Strd

is a staroid.

6 Section 6

Proof. That L

GR ID

A[n]

Strd

if L

k

= 0 for some k ∈ n is obvious. It remains to pr ove

L ∪ {(k; X ⊔ Y )} ∈ GR ID

A[n]

Strd

⇔ L ∪ {(k; X)} ∈ GR ID

A[n]

Strd

∨ L ∪ {(k; Y )} ∈ GR ID

A[n]

Strd

.

It is eq uivalent to

l

i∈n\{k}

L

i

⊓ (X ⊔ Y )

A ⇔

l

i∈n \{k}

L

i

⊓ X

A ∨

l

i∈n \{k}

L

i

⊓ Y

A.

Really,

d

i∈n \{k}

L

i

⊓ (X ⊔ Y )

A ⇔

d

i∈n \{k}

L

i

⊓ X

⊔

d

i∈n \{k}

L

i

⊓ Y

A ⇔

d

i∈n \{k}

L

i

⊓ X

A ∨

d

i∈n \{k}

L

i

⊓ Y

A.

Proposition 44. Let (A; Z) be a starrish ﬁltrator over a complete meet inﬁnite distributive lattice

and A ∈ A. Then id

A[n]

Strd

is a staroid.

Proof. That L

GR id

A[n]

Strd

if L

k

= 0 for some k ∈ n is obvious. It remains to prove

L ∪ {(k; X ⊔ Y )} ∈ GR id

A[n]

Strd

⇔ L ∪ {(k; X)} ∈ GR id

A[n]

Strd

∨ L ∪ {(k; Y )} ∈ GR id

A[n]

Strd

.

It is eq uivalent to

l

i∈n\{k}

Z

L

i

⊓ (X ⊔ Y )

A ⇔

l

i∈n \{k}

Z

L

i

⊓ X

A ∨

l

i∈n \{k}

Z

L

i

⊓ Y

A.

Really,

d

i∈n \{k}

Z

L

i

⊓ (X ⊔ Y )

A ⇔

d

i∈n \{k}

Z

L

i

⊓ X

⊔

d

i∈n \{k}

Z

L

i

⊓ Y

A ⇔

d

i∈n \{k}

Z

L

i

⊓ X

A ∨

d

i∈n \{k}

Z

L

i

⊓ Y

A.

Proposition 45. Let (A; Z) be a distributive la ttice ﬁltrator with least element and ﬁnitely join-

closed core which is a join semilattice. ID

A[n]

Strd

is a completary staroid for every A ∈ A.

Proof. ∂ A is a free star by theorem ??4 .47.

L

0

⊔ L

i

∈ GR ID

A[n]

Strd

⇔ ∀i ∈ n: (L

0

⊔ L

i

) i ∈ ∂ A ⇔ ∀i ∈ n: L

0

i ⊔ L

1

i ∈ ∂ A ⇔ ∀i ∈ n:

(L

0

i ∈∂ A ∨ L

1

i ∈∂ A ) ⇔ ∃c ∈ {0, 1}

n

∀i ∈n:L

c(i)

i ∈∂ A⇔ ∃c ∈ {0, 1}

n

:(λi ∈ n: L

c(i)

i)∈ GR ID

A[n]

Strd

.

Lemma 46. X ∈ GR id

A[n]

Strd

⇔ Cor

′

d

i∈n

A

X

i

A for a join-clo sed ﬁltrator (A; Z) such that both A

and Z are complete lattices, provided that A ∈ A.

Proof. X ∈ GR id

A[n]

Strd

⇔

d

i∈n

Z

X

i

A ⇔ Cor

′

d

i∈n

A

X

i

A.

Conjecture 4 7. id

A[n]

Strd

is a completary staroid for every set-th eoretic ﬁlter A.

Proposition 48. Let each (A

i

;Z

i

) for i ∈ n (where n is an index set) is a ﬁnitely join-closed ﬁltrator,

such that each A

i

and each Z

i

are join-semilattices. If f is a completary staroid of the form A then

f is a completary staroid of the form Z.

[TODO: Move this proposition (and note its corollary).]

Proof. L

0

⊔

Z

L

1

∈ GR f ⇔ L

0

⊔

Z

L

1

∈ GR f ⇔ L

0

⊔

A

L

1

∈ GR f ⇔ ∃c ∈ {0, 1}

n

:

(λi ∈ n: L

c(i)

i) ∈ GR f ⇔ ∃c ∈ {0, 1}

n

: (λi ∈ n: L

c(i)

i) ∈ GR f for every L

0

, L

1

∈

Q

Z.

Conjecture 4 9. ⇈id

A[n]

Strd

is a completary staroid if A is a ﬁlter o n a set and n is an index set.

6.4 Special case of sets and ﬁlters

Proposition 50. ↑

Z

n

X ∈ GR id

a[n]

Strd

⇔ ∀A ∈ a:

Q

X

id

A[n]

for every ﬁlter a on a powerset and

index set n.

Proof. ∀A ∈ a:

Q

X

id

A[n]

⇔ ∀A ∈ a:

T

i∈n

X

i

∩ A

∅ ⇔ ∀A ∈ a:

d

i∈n

P

↑

Z

X

i

↑A ⇔

d

i∈n

P

(↑

Z

n

X

i

)

a ⇔

d

i∈n

P

(↑

Z

n

X)

i

a ⇔ ↑

Z

n

X ∈ GR id

a[n]

.

Identity staroids and multifuncoids 7

Proposition 51. Y ∈ GR id

A[n]

Strd

⇔ ∀A ∈ A: Y ∈ GR ↑

Strd

id

A[n]

for every ﬁlter A on a powerset and

Y ∈ P

n

.

Proof. Take Y = ↑

Z

n

X.

∀A ∈ A: Y ∈ GR ↑

Strd

id

A[n]

⇔ ∀A ∈ A: ↑

Z

n

X ∈ GR ↑

Strd

id

A[n]

⇔ ∀A ∈ A:

Q

X

id

A[n]

⇔

↑

Z

n

X ∈ GR id

A[n]

Strd

⇔ Y ∈ GR id

A[n]

Strd

.

Proposition 52. ↑

Z

n

X ∈ GR id

a[n]

Strd

⇔ ∀A ∈ a∃t ∈ A∀i ∈ n: t ∈ X

i

.

Proof. ↑

Z

n

X ∈ GR id

a[n]

Strd

⇔ ∃A ∈ a∃t ∈ A: n × {t} ∈

Q

X ⇔ ∀A ∈ a∃t ∈ A∀i ∈ n: t ∈ X

i

.

6.5 Relationships between big and small identity staroids

Deﬁnition 53. a

Strd

n

=

Q

i∈n

Strd

a f or every ele ment a of a poset and an index set n.

Proposition 54. ⇈id

a[n]

Strd

⊑ID

a[n]

Strd

⊑ a

Strd

n

for every ﬁlter a (on any distributive lattice) and an index

set n.

Proof.

GR ⇈ id

a[n]

Strd

⊆ GR ID

a[n]

Strd

. L ∈ GR ⇈id

a[n]

Strd

⇔ up L ⊆ GR id

a[n]

Strd

⇔ ∀L ∈ up L: L ∈ GR i d

a[n]

Strd

⇔

(proposition 4.112??)⇔∀L ∈ up L∀A ∈ up a:

d

i∈n

Z

L

i

A ⇔ ∀L ∈ up L∀A ∈ up a:

d

i∈n

Z

L

i

⊓ A

0 ⇒

S

i∈n

L

i

∪ a has ﬁnite intersection property⇔L ∈ GR ID

a[n]

Strd

.

GR ID

a[n]

Strd

⊆ GR a

Strd

n

. L ∈ GR ID

a[n]

Strd

⇔ MEET({L

i

| i ∈ n} ∪ {a}) ⇒ ∀i ∈ a: L

i

a ⇔

L ∈ GR a

Strd

a

.

Proposition 55. ⇈id

a[a]

Strd

⊏ ID

a[a]

Strd

= a

Strd

a

for every nontrivial ultraﬁlter a on a se t.

Proof.

GR ⇈ id

a[a]

Strd

GR ID

a[a]

Strd

. Let L

i

= ↑

Base(a)

i. Then trivially L ∈ GR ID

a[a]

Strd

. But to disprove

L ∈ GR ⇈ id

a[a]

Strd

it’s enough to show L

GR id

a[a]

Strd

for some L ∈ up L. Really, take

L

i

= L

i

= ↑

Base(a)

i. Then L ∈ GR id

a[a]

Strd

⇔ ∀A ∈ a∃t ∈ A∀i ∈ a: t ∈ i what is clearly false (we

can always take i ∈ a such that t

i for any point t).

GR ID

a[a]

Strd

= GR a

Strd

a

. L ∈ GR ID

a[a]

Strd

⇔ ∀i ∈ n: L

i

⊒ a ⇔ ∀i ∈ a: L

i

a ⇔ L ∈ GR a

Strd

a

.

Corollary 56. a

Strd

a

isn’t an atom when a is a nontrivial ultr aﬁlter.

Corollary 57. Staroidal product of an inﬁnite indexed family of ultraﬁlters may be non-atomic.

Proposition 58. id

a[n]

Strd

is determined by the value of ⇈id

a[n]

Strd

. Mo reover id

a[n]

Strd

= ⇈id

a[n]

Strd

.

Proof. Use general properties of u pgrading and downgrading (proposition 17.63??).

Lemma 59. L ∈ GR ID

a[n]

Strd

iﬀ

S

i∈n

L

i

∪ a has ﬁnite intersection property (for primary ﬁltrators).

Proof. L∈GR ID

a[n]

Strd

⇔

d

i∈n

L ⊓ a

0

F

⇔ ∀X ∈

d

i∈n

L ⊓ a:X

∅ what is equivalent of

S

i∈n

L

i

∪a

having ﬁnite intersection property.

Proposition 60. ID

a[n]

Strd

is determined by the value of ID

a[n]

Strd

, moreover ID

a[n]

Strd

= ⇈ID

a[n]

Strd

(for

primary ﬁltrators).

Proof. L ∈ ⇈ID

a[n]

Strd

⇔ up L ⊆ ID

a[n]

Strd

⇔ up L ⊆ ID

a[n]

Strd

⇔ ∀L ∈ up L: L ∈ ID

a[n]

Strd

⇔ ∀L ∈ up L:

d

i∈n

L

i

⊓ a

0

F

⇔

S

i∈n

L

i

∪ a has ﬁnite intersection property⇔(lemma)⇔L ∈ GR ID

a[n]

Strd

.

8 Section 6

Proposition 61. id

a[n]

Strd

⊑ ID

a[n]

Strd

for every ﬁlter a and an index set n.

Proof. id

a[n]

Strd

= ⇈id

a[n]

Strd

⊑ ID

a[n]

Strd

.

Proposition 62. id

a[a]

Strd

⊏ ID

a[a]

Strd

for every nontrivial ultraﬁlter a.

Proof. Suppose id

a[a]

Strd

= ID

a[a]

Strd

. Then ID

a[a]

Strd

= ⇈ ID

a[a]

Strd

= ⇈ id

a[a]

Strd

what contradicts to the

above .

Obvious 63. L ∈ GR ID

a[n]

Strd

⇔ a ⊓

d

i∈n

L

i

0

F

if a is an elem e nt of a complete lattice.

Obvious 64. L ∈ GR ID

a[n]

Strd

⇔ ∀i ∈ n: L

i

⊒ a ⇔ ∀i ∈ n: L

i

a if a is an ultra ﬁlter on A.

6.6 Identity staroids on principal ﬁlters

For principal ﬁlter ↑A (where A is a set) the above deﬁnitions coincide with n-ary identity relation,

as formulated in the below propositions:

Proposition 65. ↑

Strd

id

A[n]

= id

↑A[n]

Strd

.

Proof. L ∈ GR ↑

Strd

id

A[n]

⇔

Q

L

id

A[n]

⇔ ∃t ∈ A∀i ∈ n: t ∈ L

i

⇔

T

i∈n

L

i

∩ A

∅⇔ L ∈ GR id

↑A[n]

Strd

.

Thus ↑

Strd

id

A[n]

= id

↑A[n]

Strd

.

Corollary 66. id

↑A[n]

Strd

is a principal staroid.

Problem 67. Is ID

A[n]

Strd

principal for every principal ﬁlt er A on a set and index set n?

Proposition 68. ↑

Strd

id

A[n]

⊑ ID

↑A[n]

Strd

for every set A.

Proof. L ∈ GR ↑

Strd

id

A[n]

⇔ L ∈ GR id

↑A[n]

Strd

⇔ ↑A

d

i∈n

A

L

i

⇐ ↑A

d

i∈n

Z

L

i

⇔ L ∈GR ID

↑A[n]

Strd

.

Proposition 69. ↑

Strd

id

A[n]

⊏ ID

↑A[n]

Strd

for some set A and index set n.

Proof. L ∈ GR ↑

Strd

id

A[n]

⇔

d

i∈n

Z

L

i

↑A what is not implied by

d

i∈n

A

L

i

↑A that is

L ∈ GR ID

↑A[n]

Strd

. (For a counter example take n = N, L

i

= (0; 1/i), A = R.)

Proposition 70. ⇈↑

Strd

id

A[n]

= ⇈id

↑A[n]

Strd

.

Proof. ⇈↑

Strd

id

A[n]

= ⇈id

↑A[n]

Strd

is obvious from the above.

Proposition 71. ⇈↑

Strd

id

A[n]

⊑ ID

↑A[n]

Strd

.

Proof. X ∈ GR ⇈↑

Strd

id

A[n]

⇔ up X ⊆GR ↑

Strd

id

A[n]

⇔ ∀Y ∈ up X : Y ∈ GR ↑

Strd

id

A[n]

⇔ ∀Y ∈ up X :

Y ∈ id

↑A[n]

Strd

⇔ ∀Y ∈ up X :

d

i∈n

Z

Y

i

⊓ ↑A

0 ⇒

d

i∈n

A

X

i

⊓ ↑A

0 ⇔ X ∈ GR ID

↑A[n]

Strd

.

Proposition 72. ⇈↑

Strd

id

A[n]

⊏ ID

↑A[n]

Strd

for some set A.

Proof. We need to prove ⇈↑

Strd

id

A[n]

ID

↑A[n]

Strd

that is it’s enough to prove (se e the above proof)

that ∀Y ∈ up X :

d

i∈n

Z

Y

i

⊓ ↑A

0 :

d

i∈n

A

X

i

⊓ ↑A

0. A co unter-examp le follows:

∀Y ∈ up X :

d

i∈n

Z

Y

i

⊓ ↑A

0 does not hold f or n = N, X

i

= ↑(−1/i; 0) for i ∈ n, A = (−∞; 0).

To show this, it’s enough to prove

d

i∈n

Z

Y

i

⊓ ↑A

0 for Y

i

= ↑(−1/i; 0) but this is obvious since

d

i∈n

Z

Y

i

= 0.

On the other hand,

d

i∈n

A

X

i

⊓ ↑A

0 for the same X and A.

The above theorems are summarized in the following diagram:

Identity staroids and multifuncoids 9

↑

Strd

id

A[n]

= id

↑A[n]

Strd

⇈↑

Strd

id

A[n]

= ⇈id

↑A[n]

Strd

⇈

⊒

ID

↑A[n]

Strd

⊒

⇈

ID

↑A[n]

Strd

Remark 73. ⊑ on the diag ram means inequality which can become strict for so me A and n.

6.7 Identity s taroids represented as mee ts and joins

Proposition 74. id

a[n]

Strd

=

d

{↑

Strd

id

A[n]

| A ∈ a} for every set- theoretic ﬁlter a where the meet

may be taken on every of the following posets: anchored relations, staroid s.

Proof. That id

a[n]

Strd

⊑ ↑

Strd

id

A[n]

for every A ∈ a is obvious.

Let f ⊑ ↑

Strd

id

A[n]

for every A ∈ a. L ∈ GR f ⇒ L ∈ GR ↑

Strd

id

A[n]

⇒ ∀A ∈ a:

d

i∈n

A

L

i

A ⇒

d

i∈n

A

L

i

a ⇒ L ∈ GR id

a[n]

Strd

. Thus f ⊑ id

a[n]

Strd

.

Proposition 75. ID

A[n]

Strd

=

F

ID

a[n]

Strd

| a ∈ atoms A

=

F

{a

Strd

n

| a ∈ atoms A} where the meet

may be taken on e very of the following posets: anchored relations, staroids, completary staroids,

provided that A is a ﬁlter on a set.

Proof. ID

A[n]

Strd

⊒ ID

a[n]

Strd

for every a ∈ atoms A is obvious.

Let f ⊒ ID

a[n]

Strd

for every a ∈ atoms A. Then ∀L ∈ GR ID

a[n]

Strd

: L ∈ GR f that is

∀L ∈ form f : (MEET({L

i

| i ∈ n} ∪ {a}) ⇒ L ∈ GR f).

But ∃a ∈ atoms A: MEET({L

i

| i ∈ n} ∪ {a}) ⇔ ∃a ∈ atoms A:

d

i∈n

A

L

i

a ⇐

d

i∈n

A

L

i

A ⇔

L ∈ ID

A[n]

Strd

.

So L ∈ ID

A[n]

Strd

⇒ L ∈ GR f . Thus f ⊒ ID

A[n]

Strd

.

Then use the fact that ID

a[n]

Strd

= a

Strd

n

.

Proposition 76. id

A[n]

Strd

=

F

id

a[n]

Strd

| a ∈ atoms A

where the meet may be taken on every of the

following posets: anchored relations, staroids, provided that A is a ﬁlter on a set.

Proof. id

A[n]

Strd

⊒ id

a[n]

Strd

for every a ∈ atoms A is obvious.

Let f ⊒ id

a[n]

Strd

for every a ∈ atoms A. Then ∀L ∈ GR id

a[n]

Strd

: L ∈ GR f that is

∀L ∈ form f :

l

i∈n

Z

L

i

a ⇒ L ∈ GR f

!

.

But ∃a ∈ atoms A:

d

i∈n

Z

L

i

a ⇐

d

i∈n

Z

L

i

A ⇔ L ∈ id

A[n]

Strd

.

So L ∈ id

A[n]

Strd

⇒ L ∈ GR f. Thus f ⊒ id

A[n]

Strd

.

7 Finite case

Theorem 77. Let n be a ﬁnite set.

1. id

A[n]

Strd

= ID

A[n]

Strd

if A and Z are meet-semilatt ices and (A; Z) is a ﬁnitely meet-closed ﬁltrator.

10 Section 7

2. ID

A[n]

Strd

= ⇈id

A[n]

Strd

if (A; Z) is a primary ﬁltrator over a distributive la ttice.

Proof.

1. L ∈ GR ID

A[n]

Strd

⇔ L ∈ GR ID

A[n]

Strd

⇔ MEET({L

i

| i ∈ n} ∪ {A}) ⇔

d

i∈n

A

L

i

⊓ A

0 ⇔ (by

ﬁniteness)⇔

d

i∈n

Z

L

i

⊓ A

0 ⇔ L ∈ id

A[n]

Strd

for every L ∈

Q

Z.

2. L ∈ GR ⇈ id

A[n]

Strd

⇔ up L ⊆ GR id

A[n]

Strd

⇔ ∀K ∈ up L: K ∈ GR id

A[n]

Strd

⇔ ∀K ∈ up L:

d

i∈n

Z

K

i

∈ ∂ A⇔ ∀K ∈ up L:

d

i∈n

Z

K

i

A⇔ (by ﬁniteness and theorem 4.44??)⇔∀K ∈ up L:

d

i∈n

A

K

i

A ⇔ A ∈

T

h⋆i

d

i∈n

A

K

i

| K ∈ up L

⇔ (by the formula for ﬁnite meet of ﬁlters,

theorem 4.111??)⇔A ∈

T

h⋆i

d

i∈n

A

L

i

⇔∀K ∈

d

i∈n

A

L

i

:A ∈ ⋆ K ⇔ ∀K ∈

d

i∈n

A

L

i

:A

K ⇔(by

separability of core, theorem 4.112??)⇔

d

i∈n

A

L

i

A ⇔ L ∈ ID

A[n]

Strd

.

Proposition 78. Let (A; Z) be a ﬁnitely meet closed ﬁltrator. ID

A[n]

Strd

and id

A[n]

Strd

are the same

for ﬁnite n.

Proof. Because

d

i∈dom L

Z

L

i

=

d

i∈dom L

A

L

i

for ﬁnitary L.

8 Counter-examples a nd conjectures

The following example shows tha t the theo rem 33 can’t be stre nghtened:

Example 79. For some multifuncoid f on powersets complete in argument k the following formula

is false:

hf i

l

(L ∪ {(k;

F

X)}) =

F

x∈X

hf i

l

(L ∪ {(k ; x)}) for every X ∈ P P

k

, L ∈

Q

i∈(arity f )\{k,l}

F

i

.

Proof. Consider multifuncoid f = Λid

↑U [3]

Strd

where U is an inﬁnite set ( of the fo rm P

3

) and L = (Y )

where Y is a nonprincipal ﬁlter on U .

hf i

0

(L ∪ {(k;

F

X)}) = Y ⊓

F

X;

F

x∈X

hf i

0

(L ∪ {(k; x)}) =

F

x∈X

(Y ⊓ x).

It can be Y ⊓

F

X =

F

x∈X

(Y ⊓ x) only if Y is principal: Really: Y ⊓

F

X =

F

x∈X

(Y ⊓ x)

implies Y

F

X ⇒

F

x∈X

(Y ⊓ x)

0 ⇒ ∃x ∈ X: Y

x and thus Y is principal. But we claimed

above that it is nonprincipal.

Example 80. There ex ists a staroid f and an indexed family X of princ ipal ﬁlters (with arity f =

dom X and (form f)

i

= Base(X

i

) for e very i ∈ arity f ), such that f ⊑

Q

Strd

X and Y ⊓ X

GR f

for some Y ∈ GR f .

Remark 8 1. Such examples obviously do not exist if both f is a principal staroid and X and Y are

indexed fa milies of principal ﬁlters (be cause for powerset algebras staroidal produ c t is equivale nt

to Cartesian product). This makes the above example inspired.

Proof. (Monroe Eskew) Let a be any (trivial or nontrivial) ultraﬁlter on an inﬁ nite s e t U . Let

A, B ∈ a be such that A ∩ B ⊂ A, B. In other words, A, B are arbitrary nonempty sets such that

∅

A ∩ B ⊂ A, B and a be an ultraﬁlter on A ∩ B.

Let f be the staroid whose graph consists of functions p: U → a such that eit her p(n) ⊇ A for

all but ﬁnitely many n or p(n) ⊇ B for all but ﬁnitely many n. Let’s prove f is really a staroid.

It’s obvious px

∅ for every x ∈ U . Let k ∈ U, L ∈ a

U \{k}

. It is enough (taking symmetry into

account) to prove that

L ∪ {(k; x ⊔ y)} ∈ GR f ⇔ L ∪ {(k; x)} ∈ GR f ∨ L ∪ {(k; y)} ∈ GR f. (1)

Really, L ∪ {(k; x ⊔ y)} ∈ GR f iﬀ x ⊔ y ∈ a and L(n) ⊇ A for all but ﬁnitely many n or L(n) ⊇ B

for all but ﬁnitely many n ; L ∪ {(k; x)} ∈ GR f iﬀ x ∈ a and L(n) ⊇ A for all but ﬁnitely many n

or L(n) ⊇ B; and simila rly for y.

Counter-examples and conjectures 11

But x ⊔ y ∈ a ⇔ x ∈ a ∨ y ∈ a because a is an ultraﬁlter. So, the formula (1) holds, and we have

proved that f is really a staroid.

Take X be the c onstant function with value A and Y be the constant function with value B.

∀p ∈ GR f : p

X be c ause p

i

∩ X

i

∈ a; so GR f ⊆ GR

Q

Strd

X that is f ⊑

Q

Strd

X.

Finally, Y ⊓ X

GR f because X ⊓ Y = λi ∈ U: A ∩ B.

Some conjectures similar to t he above ex ample:

Conjecture 82. There exists a completa ry staroid f and an indexed family X of principal ﬁl ters

(with arity f = dom X and (form f)

i

= Base(X

i

) for every i ∈ arity f), such that f ⊑

Q

Strd

X and

Y ⊓ X

GR f for som e Y ∈ GR f .

Conjecture 83. There exists a staroid f and an indexed family x of ultraﬁlters (with arity f =

dom x a nd (form f)

i

= Base(x

i

) for every i ∈ arity f), such that f ⊑

Q

Strd

x and Y ⊓ x

GR f for

some Y ∈ GR f.

Other conjecture s:

Conjecture 84. If staroid 0

f ⊑ a

Strd

n

for an ultraﬁlter a and an index set n, then n × {a}∈ G R f .

(Can it be generalized for arbitrary staroidal products?)

Conjecture 8 5. The following posets are atomic:

1. anchored relations on powersets;

2. staroids on powersets;

3. completary star oids on powersets.

Conjecture 8 6. The following posets are atomistic:

1. anchored relations on powersets;

2. staroids on powersets;

3. completary star oids on powersets.

The above conjectures seem diﬃcult, bec ause we know almost not hing about structure of atomic

staroids.

Conjecture 8 7. A staroid on powersets is pri ncipal iﬀ it is com plete in every argument.

Conjecture 8 8. If a is an ultraﬁlter, then id

a[n]

Strd

is an atom of the lattice of:

1. anchored relations of the form ( P Base(a))

n

;

2. staroids of the form (P Base(a))

n

;

3. completary star oids of the form (P Base(a))

n

.

Conjecture 8 9. If a is an ultraﬁlter, then ⇈id

a[n]

Strd

is an atom of the lattice of:

1. anchored relations of the form F(Base(a))

n

;

2. staroids of the form F(Base(a))

n

;

3. completary star oids of the form F(Base(a))

n

.

Informal problem: Formulate and prove associativity of staroidal product.

Bib liography

[1] Victor Porton. Algebraic General Topology. Volume 1. 2014.

12 Section