**Algebraic General Topology. Vol 1**:
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**Axiomatic Theory of Formulas**:
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2000 Mathematics Subject Classiﬁcation. 54J05, 54A05, 54D99, 54E05, 54E15,

54E17, 54E99

Key words and phrases. algebraic general topology, quasi-uniform spaces,

generalizations of proximity spaces, generalizations of nearness spaces,

generalizations of uniform spaces

Abstract. This ﬁle contains future addons for the free e-book “Algebraic

General Topology. Volume 1”, which are yet not enough ripe to be included

into the book.

Contents

Chapter 1. About this document 5

Chapter 2. Applications of algebraic general topology 6

1. “Hybrid” objects 6

2. A way to construct directed topological spaces 6

3. Some inequalities 8

4. Continuity 9

5. A way to construct directed topological spaces 12

6. Integral curves 12

Chapter 3. More on generalized limit 16

1. Hausdorﬀ funcoids 16

2. Restoring functions from limit 16

Chapter 4. Extending Galois connections between funcoids and reloids 18

Chapter 5. Boolean funcoids 20

1. One-element boolean lattice 20

2. Two-element boolean lattice 20

3. Finite boolean lattices 21

4. About inﬁnite case 21

Chapter 6. Interior funcoids 23

Chapter 7. Filterization of pointfree funcoids 25

Chapter 8. Systems of sides 26

1. More on Galois connections 26

2. Deﬁnition 27

3. Concrete examples of sides 28

4. Product 30

5. Negative results 31

6. Dagger systems of sides 31

Chapter 9. Backward Funcoids 32

Chapter 10. Quasi-atoms 33

Chapter 11. Cauchy Filters on Reloids 34

1. Preface 34

2. Low spaces 34

3. Almost sub-join-semilattices 35

4. Cauchy spaces 35

5. Relationships with symmetric reloids 36

6. Lattices of low spaces 37

7. Up-complete low spaces 41

3

CONTENTS 4

8. More on Cauchy ﬁlters 42

9. Maximal Cauchy ﬁlters 43

10. Cauchy continuous functions 44

11. Cauchy-complete reloids 44

12. Totally bounded 44

13. Totally bounded funcoids 45

14. On principal low spaces 45

15. Rest 45

Chapter 12. Funcoidal groups 47

1. On “Each regular paratopological group is completely regular” article 48

Chapter 13. Micronization 53

Chapter 14. More on connectedness 54

1. For topological spaces 54

Chapter 15. Relationships are pointfree funcoids 59

Chapter 16. Manifolds and surfaces 60

1. Sides of a surface 60

2. Special points 60

Bibliography 63

CHAPTER 1

About this document

This ﬁle contains future addons for the free e-book “Algebraic General Topol-

ogy. Volume 1”, which are yet not enough ripe to be included into the book.

Theorem (including propositions, conjectures, etc.) numbers in this document

start from the last theorem number in the book plus one. Theorems references

inside this document are hyperlinked, but references to theorems in the book are

not hyperlinked (because PDF viewer Okular 0.20.2 does not support Backward

button after clicking a cross-document reference, and thus I want to avoid clicking

such links).

5

CHAPTER 2

Applications of algebraic general topology

1. “Hybrid” objects

Algebraic general topology allows to construct “hybrid” objects of “continuous”

(as topological spaces) and discrete (as graphs).

Consider for example D t T where D is a digraph and T is a topological space.

The n-th power (D tT )

n

yields an expression with 2

n

terms. So treating D tT

as one object (what becomes possible using algebraic general topology) rather than

the join of two objects may have an exponential beneﬁt for simplicity of formulas.

2. A way to construct directed topological spaces

2.1. Some notation. I use E and ι notations from volume-2.pdf. FiXme:

Reorder document fragments to describe it before use.

I remind that f|

X

= f ◦ id

X

for binary relations, funcoids, and reloid.

f k

X

= f ◦ (E

X

)

−1

.

fX = id

X

◦f ◦ id

−1

X

.

As proved in volume-2.pdf, the following are bijections and moreover isomor-

phisms (for R being either funcoids or reloids or binary relations):

1

◦

.

n

(f|

X

,fk

X

)

f∈R

o

;

2

◦

.

n

(fX,ι

X

f)

f∈R

o

.

As easily follows from these isomorphisms and theorem 1293:

Proposition 2064. For funcoids, reloids, and binary relations:

1

◦

. f ∈ C(µ, ν) ⇒ f k

A

∈ C(ι

A

µ, ν);

2

◦

. f ∈ C

0

(µ, ν) ⇒ f k

A

∈ C

0

(ι

A

µ, ν);

3

◦

. f ∈ C

00

(µ, ν) ⇒ f k

A

∈ C

00

(ι

A

µ, ν).

2.2. Directed line and directed intervals. Let A be a poset. We will

denote A = A ∪ {−∞, +∞} the poset with two added elements −∞ and +∞, such

that +∞ is strictly greater than every element of A and −∞ is strictly less.

FiXme: Generalize from R to a wider class of posets.

Definition 2065. For an element a of a poset A

1

◦

. J

≥

(a) =

n

x∈A

x≥a

o

;

2

◦

. J

>

(a) =

x∈A

x>a

;

3

◦

. J

≤

(a) =

n

x∈A

x≤a

o

;

4

◦

. J

<

(a) =

x∈A

x<a

;

5

◦

. J

6=

(a) =

n

x∈A

x6=a

o

.

Definition 2066. Let a be an element of a poset A.

1

◦

. ∆(a) =

d

F

n

]x;y[

x,y∈A,x<a∧y>a

o

;

2

◦

. ∆

≥

(a) =

d

F

n

[a;y[

y∈A,y>a

o

;

6

2. A WAY TO CONSTRUCT DIRECTED TOPOLOGICAL SPACES 7

3

◦

. ∆

>

(a) =

d

F

n

]a;y[

y∈A,x<a∧y>a

o

;

4

◦

. ∆

≤

(a) =

d

F

n

]x;a]

x∈A,x<a

o

;

5

◦

. ∆

<

(a) =

d

F

n

]x;a[

x∈A,x<a

o

;

6

◦

. ∆

6=

(a) = ∆(a) \ {a}.

Obvious 2067.

1

◦

. ∆

≥

(a) = ∆(a) u

F

@J

≥

(a);

2

◦

. ∆

>

(a) = ∆(a) u

F

@J

>

(a);

3

◦

. ∆

≤

(a) = ∆(a) u

F

@J

≤

(a);

4

◦

. ∆

<

(a) = ∆(a) u

F

@J

<

(a);

5

◦

. ∆

6=

(a) = ∆(a) u

F

@J

6=

(a).

Definition 2068. Given a partial order A and x ∈ A, the following deﬁnes

complete funcoids:

1

◦

. h|A|i

∗

{x} = ∆(x);

2

◦

. h|A|

≥

i

∗

{x} = ∆

≥

(x);

3

◦

. h|A|

>

i

∗

{x} = ∆

>

(x);

4

◦

. h|A|

≤

i

∗

{x} = ∆

≤

(x);

5

◦

. h|A|

<

i

∗

{x} = ∆

<

(x);

6

◦

. h|A|

6=

i

∗

{x} = ∆

6=

(x).

Proposition 2069. The complete funcoid corresponding to the order topol-

ogy

1

is equal to |A|.

Proof. Because every open set is a ﬁnite union of open intervals, the com-

plete funcoid f corresponding to the order topology is described by the formula:

hfi

∗

{x} =

d

F

n

]a;b[

a,b∈A,a<x∧b>x

o

= ∆(x) = h|A|i

∗

{x}. Thus f = |A|.

Exercise 2070. Show that |A|

≥

(in general) is not the same as “right order

topology”

2

.

Proposition 2071.

1

◦

.

D

|A|

−1

≥

E

∗

@X = @

n

a∈A

∀y∈A:(y>a⇒X∩[a;y[6=∅)

o

;

2

◦

.

|A|

−1

>

∗

@X = @

n

a∈A

∀y∈A:(y>a⇒X∩]a;y[6=∅)

o

;

3

◦

.

D

|A|

−1

≤

E

∗

@X = @

n

a∈A

∀x∈A:(x<a⇒X∩]x;a]6=∅)

o

;

4

◦

.

|A|

−1

<

∗

@X = @

n

a∈A

∀x∈A:(x<a⇒X∩]x;a[6=∅)

o

.

Proof. a ∈

D

|A|

−1

≥

E

∗

@X ⇔ @{a} 6

D

|A|

−1

≥

E

∗

@X ⇔ h|A|

≥

i

∗

@{a} 6 @X ⇔

∆

≥

(a) 6 @X ⇔ ∀y ∈ A : (y > a ⇒ X ∩ [a; y[6= ∅).

a ∈

|A|

−1

>

∗

@X ⇔ @{a} 6

|A|

−1

>

∗

@X ⇔ h|A|

>

i

∗

@{a} 6 @X ⇔ ∆

>

(a) 6

@X ⇔ ∀y ∈ A : (y > a ⇒ X∩]a; y[6= ∅).

The rest follows from duality.

Remark 2072. On trivial ultraﬁlters these obviously agree:

1

◦

. h|R|

≥

i

∗

{x} = h|R|u ≥i

∗

{x};

2

◦

. h|R|

>

i

∗

{x} = h|R|u >i

∗

{x};

3

◦

. h|R|

≤

i

∗

{x} = h|R|u ≤i

∗

{x};

4

◦

. h|R|

<

i

∗

{x} = h|R|u <i

∗

{x}.

1

See Wikipedia for a deﬁnition of “Order topology”.

2

See Wikipedia

3. SOME INEQUALITIES 8

Corollary 2073.

1

◦

. |R|

≥

= Compl(|R|u ≥);

2

◦

. |R|

>

= Compl(|R|u >);

3

◦

. |R|

≤

= Compl(|R|u ≤);

4

◦

. |R|

<

= Compl(|R|u <).

Obvious 2074. FiXme: also what is the values of \ operation

1

◦

. |R|

≥

= |R|

>

t 1;

2

◦

. |R|

≤

= |R|

<

t 1.

3. Some inequalities

FiXme: Deﬁne the ultraﬁlter “at the left” and “at the right” of a real number.

Also deﬁne “convergent ultraﬁlter”.

Denote ∆

+∞

=

d

x∈R

]x; +∞[ and ∆

−∞

=

d

x∈R

] − ∞; x[.

The following proposition calculates h≥ix and h>ix for all kinds of ultraﬁlters

on R:

Proposition 2075.

1

◦

. h≥i{α} = [α; +∞[ and h>i{α} =]α; +∞[.

2

◦

. h≥ix = h>ix =]α; +∞[ for ultraﬁlter x at the right of a number α.

3

◦

. h≥ix = h>ix = ∆

<

(α) t[α; +∞[= ∆

≤

(α)t]α; +∞[ for ultraﬁlter x at the

left of a number α.

4

◦

. h≥ix = h>ix = ∆

+∞

for ultraﬁlter x at positive inﬁnity.

5

◦

. h≥ix = h>ix = R for ultraﬁlter x at negative inﬁnity.

Proof.

1

◦

. Obvious.

2

◦

.

h≥ix =

F

l

X∈up x

h≥i(Xu]α; +∞[) =

F

l

X∈up x

]α; +∞[=]α; +∞[;

h>ix =

F

l

X∈up x

h>i(Xu]α; +∞[) =

F

l

X∈up x

]α; +∞[=]α; +∞[.

3

◦

. ∆

<

(α) t [α; +∞[= ∆

≤

(α)t]α; +∞[ is obvious.

h>ix =

F

l

X∈up x

h>iX w

F

l

X∈up x

(∆

<

(α)t]α; +∞[) = ∆

<

(α)t]α; +∞[

but h≥ix v ∆

<

(α) t [α; +∞[ is obvious. It remains to take into account that

h>ix v h≥ix.

4

◦

. h≥ix =

d

F

X∈up x

h≥iX =

d

F

X∈up x,inf X∈X

h≥i(Xu]α; +∞[) =

d

F

X∈up x

[inf X; +∞[=

d

F

x>α

[x; +∞[= ∆

+∞

; h>ix =

d

F

X∈up x

h>iX =

d

F

X∈up x,inf X∈X

h>i(Xu]α; +∞[) =

d

F

X∈up x

] inf X; +∞[=

d

F

x>α

[x; +∞[= ∆

+∞

.

5

◦

. h≥ix w h>ix =

d

F

X∈up x

h>iX but h>iX =] − ∞; +∞[ for X ∈ up x

because X has arbitrarily small elements.

Lemma 2076. h|R|ix v h>ix = h≥ix for every nontrivial ultraﬁlter x.

Proof. h>ix = h≥ix follows from the previous proposition.

h|R|ix =

d

X∈up x

h|R|iX =

d

X∈up x

d

y∈X

∆(y).

Consider cases:

4. CONTINUITY 9

x is an ultraﬁlter at the right of some number α.

h|R|ix =

d

X∈up x

d

y∈Xu]α;+∞[

∆(y) v]α; +∞[= h≥ix because

d

y∈Xu]α;+∞[

∆(y) v]α; +∞[.

x is an ultraﬁlter at the left of some number α.

h|R|ix v ∆(α) is obvious. But h≥ix w ∆(α).

x is an ultraﬁlter at positive inﬁnity.

h|R|ix v ∆

+∞

is obvious. But h≥ix = ∆

+∞

.

x is an ultraﬁlter at negative inﬁnity.

Because h≥ix = R.

Corollary 2077. h|R|u ≥ix = h|R|ix for every nontrivial ultraﬁlter x.

Proof. h|R|u ≥ix = h|R|i u h≥ix = h|R|ix.

So h|R|u ≥i and h|R|i agree on all ultraﬁlters except trivial ones.

Proposition 2078. |R|

>

u >= |R|

>

u ≥= |R|

>

.

Proof. |R|

>

v > because h|R|

>

i

∗

x v h>i

∗

x and |R|

>

is a complete funcoid.

Lemma 2079. h|R|

>

ix @ h|R|

≥

ix for a nontrivial ultraﬁlter x.

Proof. It enough to prove h|R|

>

ix 6= h|R|

≥

ix.

Take x be an ultraﬁlter with limit point 0 on im z where z is the sequence

n 7→

1

n

.

h|R|

>

ix v h|R|

>

i

∗

im z =

l

n∈im z

∆

>

1

n

v

l

n∈im z

1

n

;

1

n − 1

−

1

n

im z.

Thus h|R|

>

ix im z. But h|R|

≥

ix v h=ix 6 im z.

Corollary 2080. |R|

>

@ |R|

≥

.

Proposition 2081. |R|

>

@ |R|

≥

u >.

Proof. It’s enough to prove |R|

>

6= |R|

≥

u >.

Really, h|R|

≥

u >ix = h|R|

≥

ix 6= h|R|

>

ix (lemma).

Proposition 2082.

1

◦

. |R|

≥

◦ |R|

≥

= |R|

≥

;

2

◦

. |R|

>

◦ |R|

>

= |R|

>

;

3

◦

. |R|

≥

◦ |R|

>

= |R|

>

;

4

◦

. |R|

>

◦ |R|

≥

= |R|

>

.

Proof. ??

Conjecture 2083.

1

◦

. (|R| u ≥) ◦ (|R| u ≥) = |R| u ≥.

2

◦

. (|R| u >) ◦ (|R| u >) = |R| u >.

4. Continuity

I will say that a property holds on a ﬁlter A iﬀ there is A ∈ up A on which the

property holds.

FiXme: f ∈ C(A, B) ∧ f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

) ⇔ (f, f) ∈

C((A, ι

A

|R|

≥

), (B, ι

B

|R|

≥

))

Lemma 2084. Let function f : A → B where A, B ∈ PR and A is connected.

4. CONTINUITY 10

1

◦

. f is monotone and f ∈ C(A, B) iﬀ f ∈ C(A, B) ∩ C(ι

A

|R|

≥

, ι

B

|R|

≥

)

iﬀ f ∈ C(A, B) ∩ C(ι

A

|R|

>

, ι

B

|R|

≥

) iﬀ f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

) ∩

C(ι

A

|R|

≤

, ι

B

|R|

≤

).

2

◦

. f is strictly monotone and ∈ C(A, B) iﬀ f ∈ C(A, B) ∩ C(ι

A

|R|

>

, ι

B

|R|

>

)

iﬀ f ∈ C(ι

A

|R|

>

, ι

B

|R|

>

) ∩ C(ι

A

|R|

<

, ι

B

|R|

<

).

FiXme: Generalize for arbitrary posets. FiXme: Generalize for f being a funcoid.

Proof. Because f is continuous, we have hf ◦ ι

A

|R|i

∗

{x} v hι

B

|R| ◦ fi

∗

{x}

that is hf i

∗

∆(x) v ∆(f (x)) for every x.

If f is monotone, we have hfi

∗

∆

≥

(x) v [f(x); ∞[. Thus

hfi

∗

∆

≥

(x) v ∆

≥

(f(x)), that is hf ◦ ι

A

|R|

≥

i

∗

{x} v hι

B

|R|

≥

◦ fi

∗

{x}, thus

f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

).

If f is strictly monotone, we have hfi

∗

∆

>

(x) v]f(x); ∞[. Thus

hfi

∗

∆

>

(x) v ∆

>

(f(x)), that is hf ◦ ι

A

|R|

>

i

∗

{x} v hι

B

|R|

>

◦ fi

∗

{x}, thus f ∈

C(ι

A

|R|

>

, ι

B

|R|

>

).

Let now f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

).

Take any a ∈ A and let c =

n

b∈B

b≥a,∀x∈[a;b[:f(x)≥f(a)

o

. It’s enough to prove that

c is the right endpoint (ﬁnite or inﬁnite) of A.

Indeed by continuity f (a) ≤ f(c) and if c is not already the right endpoint

of A, then there is b

0

> c such that ∀x ∈ [c; b

0

[: f(x) ≥ f(c). So we have ∀x ∈ [a; b

0

[:

f(x) ≥ f(c) what contradicts to the above.

So f is monotone on the entire A.

f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

) ⇒ f ∈ C(ι

A

|R|

>

, ι

B

|R|

≥

) is obvious. Reversely f ∈

C(ι

A

|R|

>

, ι

B

|R|

≥

) ⇒ f ◦ ι

A

|R|

>

v ι

B

|R|

≥

◦ f ⇔ ∀x ∈ R : hfihι

A

|R|

>

i

∗

{x} v

hι

B

|R|

≥

i

∗

hfi

∗

{x} ⇔ ∀x ∈ R : hfi∆

>

(x) v ∆

≥

f(x) ⇔ ∀x ∈ R : hfi∆

>

(x) t

{f(x)} v ∆

≥

f(x) ⇔ ∀x ∈ R : hf i∆

>

(x) t {x} v ∆

≥

f(x) ⇔ ∀x ∈ R : hf i∆

≥

(x) v

∆

≥

f(x) ⇔ ∀x ∈ R : hfihι

A

|R|

≥

i

∗

{x} v hι

B

|R|

≥

i

∗

hfi

∗

{x} ⇔ ∀x ∈ R : f ◦ ι

A

|R|

≥

v

ι

B

|R|

≥

◦ f ⇔ f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

).

Let f ∈ C(ι

A

|R|

>

, ι

B

|R|

>

). Then f ∈ C(ι

A

|R|

>

, ι

B

|R|

≥

) and thus it is mono-

tone. We need to prove that f is strictly monotone. Suppose the contrary. Then

there is a nonempty interval [p; q] ⊆ A such that f is constant on this interval. But

this is impossible because f ∈ C(ι

A

|R|

>

, ι

B

|R|

>

).

Prove that f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

) ∩ C(ι

A

|R|

≤

, ι

B

|R|

≤

) implies f ∈ C(A, B).

Really, it implies hf i∆

≤

(x) v ∆

≤

(fx) and hfi∆

≥

(x) v ∆

≥

(fx) thus hfi∆(x) =

hfi(∆

≤

(x) t {x} t ∆

≥

(x)) ⊆ ∆

≤

f(x) t {f(x)} t ∆

≥

f(x) = ∆(f(x)).

Prove that f ∈ C(ι

A

|R|

>

, ι

B

|R|

>

) ∩ C(ι

A

|R|

<

, ι

B

|R|

<

) f ∈ C(A, B). Really, it

implies hfi∆

<

(x) v ∆

<

(fx) and hfi∆

>

(x) v ∆

>

(fx) thus hfi∆(x) = hfi(∆

<

(x)t

{x} t ∆

>

(x)) ⊆ ∆

<

f(x) t {f(x)} t ∆

>

f(x) = ∆(f(x)).

Theorem 2085. Let function f : A → B where A, B ∈ PR.

1

◦

. f is locally monotone and f ∈ C(A, B) iﬀ f ∈ C(A, B)∩C(ι

A

|R|

≥

, ι

B

|R|

≥

)

iﬀ f ∈ C(A, B) ∩ C(ι

A

|R|

>

, ι

B

|R|

≥

) iﬀ f ∈ C(ι

A

|R|

≥

, ι

B

|R|

≥

) ∩

C(ι

A

|R|

≤

, ι

B

|R|

≤

).

2

◦

. f is locally strictly monotone and ∈ C(A, B) iﬀ f ∈ C(A, B) ∩

C(ι

A

|R|

>

, ι

B

|R|

>

) iﬀ f ∈ C(ι

A

|R|

>

, ι

B

|R|

>

) ∩ C(ι

A

|R|

<

, ι

B

|R|

<

).

Proof. By the lemma it is (strictly) monotone on each connected component.

See also related math.SE questions:

1

◦

. http://math.stackexchange.com/q/1473668/4876

2

◦

. http://math.stackexchange.com/a/1872906/4876

3

◦

. http://math.stackexchange.com/q/1875975/4876

4. CONTINUITY 11

4.1. Directed topological spaces. Directed topological spaces are deﬁned

at

http://ncatlab.org/nlab/show/directed+topological+space

Definition 2086. A directed topological space (or d-space for short) is a pair

(X, d) of a topological space X and a set d ⊆ C([0; 1], X) (called directed paths or

d-paths) of paths in X such that

1

◦

. (constant paths) every constant map [0; 1] → X is directed;

2

◦

. (reparameterization) d is closed under composition with increasing con-

tinuous maps [0; 1] → [0; 1];

3

◦

. (concatenation) d is closed under path-concatenation: if the d-paths a,

b are consecutive in X (a(1) = b(0)), then their ordinary concatenation

a + b is also a d-path

(a + b)(t) = a(2t), if 0 ≤ t ≤

1

2

,

(a + b)(t) = b(2t − 1), if

1

2

≤ t ≤ 1.

I propose a new way to construct a directed topological space. My way is more

geometric/topological as it does not involve dealing with particular paths.

Definition 2087. Let T be the complete endofuncoid corresponding to a topo-

logical space and ν v T be its “subfuncoid”. The d-space (dir)(T, ν) induced by

the pair (T, ν) consists of T and paths f ∈ C([0; 1], T ) ∩ C(|[0; 1]|

≥

, ν) such that

f(0) = f(1).

Proposition 2088. It is really a d-space.

Proof. Every d-path is continuous.

Constant path are d-paths because ν is reﬂexive.

Every reparameterization is a d-path because they are C(|[0; 1]|

≥

, ν) and we

can apply the theorem about composition of continuous functions.

Every concatenation is a d-path. Denote f

0

= λt ∈ [0;

1

2

] : a(2t) and f

1

= λt ∈

[

1

2

; 1] : b(2t − 1). Obviously f

0

, f

1

∈ C([0; 1], µ) ∩ C(|[0; 1]|

≥

, ν). Then we conclude

that a + b = f

1

t f

1

is in f

0

, f

1

∈ C([0; 1], µ) ∩ C(|[0; 1]|

≥

, ν) using the fact that the

operation ◦ is distributive over t.

Below we show that not every d-space is induced by a pair of an endofuncoid

and its subfuncoid. But are d-spaces not represented this way good anything except

counterexamples?

Let now we have a d-space (X, d). Deﬁne funcoid ν corresponding to the d-

space by the formula ν =

d

a∈d

(a ◦ |R|

≥

◦ a

−1

).

Example 2089. The two directed topological spaces, constructed from a ﬁxed

topological space and two diﬀerent reﬂexive funcoids, are the same.

Proof. Consider the indiscrete topology T on R and the funcoids 1

FCD(R,R)

and 1

FCD(R,R)

t({0}×

FCD

∆

≥

). The only d-paths in both these settings are constant

functions.

Example 2090. A d-space is not determined by the induced funcoid.

Proof. The following a d-space induces the same funcoid as the d-space of all

paths on the plane.

Consider a plane R

2

with the usual topology. Let d-paths be paths lying inside

a polygonal chain (in the plane).

6. INTEGRAL CURVES 12

Conjecture 2091. A d-path a is determined by the funcoids (where x spans

[0; 1])

(λt ∈ R : a(x + t))|

∆(0)

.

5. A way to construct directed topological spaces

I propose a new way to construct a directed topological space. My way is more

geometric/topological as it does not involve dealing with particular paths.

Conjecture 2092. Every directed topological space can be constructed in the

below described way.

Consider topological space T and its subfuncoid F (that is F is a funcoid

which is less that T in the order of funcoids). Note that in our consideration F is

an endofuncoid (its source and destination are the same).

Then a directed path from point A to point B is deﬁned as a continuous function

f from [0; 1] to F such that f(0) = A and f(1) = B. FiXme: Specify whether the

interval [0; 1] is treated as a proximity, pretopology, or preclosure.

Because F is less that T , we have that every directed path is a path.

Conjecture 2093. The two directed topological spaces, constructed from a

ﬁxed topological space and two diﬀerent funcoids, are diﬀerent.

For a counter-example of (which of the two?) the conjecture consider funcoid

T u (Q ×

FCD

Q) where T is the usual topology on real line.We need to consider

stability of existence and uniqueness of a path under transformations of our funcoid

and under transformations of the vector ﬁeld. Can this be a step to solve Navier-

Stokes existence and smoothness problems?

6. Integral curves

We will consider paths in a normed vector space V .

Definition 2094. Let D be a connected subset of R. A path is a function

D → V .

Let d be a vector ﬁeld in a normed vector space V .

Definition 2095. Integral curve of a vector ﬁeld d is a diﬀerentiable function

f : D → V such that f

0

(t) = d(f(t)) for every t ∈ D.

Definition 2096. The deﬁnition of right side integral curve is the above deﬁ-

nition with right derivative of f instead of derivative f

0

. Left side integral curve is

deﬁned similarly.

6.1. Path reparameterization. C

1

is a function which has continuous de-

rivative on every point of the domain.

By D

1

I will denote a C

1

function whose derivative is either nonzero at every

point or is zero everywhere.

Definition 2097. A reparameterization of a C

1

path is a bijective C

1

function

φ : D → D such that φ

0

(t) > 0. A curve f

2

is called a reparametrized curve f

1

if

there is a reparameterization φ such that f

2

= f

1

◦ φ.

It is well known that this deﬁnes an equivalence relation of functions.

Proposition 2098. Reparameterization of D

1

function is D

1

.

Proof. If the function has zero derivative, it is obvious.

Let f

1

has everywhere nonzero derivative. Then f

0

2

(t) = f

0

1

(φ(t))φ

0

(t) what is

trivially nonzero.

6. INTEGRAL CURVES 13

Definition 2099. Vectors p and q have the same direction (p q) iﬀ there

exists a strictly positive real c such that p = cq.

Obvious 2100. Being same direction is an equivalence relation.

Obvious 2101. The only vector with the same direction as the zero vector is

zero vector.

Theorem 2102. A D

1

function y is some reparameterization of a D

1

integral

curve x of a continuous vector ﬁeld d iﬀ y

0

(t) d(y(t)) that is vectors y

0

(t) and

d(y(t)) have the same direction (for every t).

Proof. If y is a reparameterization of x, then y(t) = x(φ(t)). Thus y

0

(t) =

x

0

(φ(t))φ

0

(t) = d(x(φ(t)))φ

0

(t) = d(y(t))φ

0

(t). So y

0

(t) d(y(t)) because φ

0

(t) > 0.

Let now x

0

(t) d(x(t)) that is that is there is a strictly positive function c(t)

such that x

0

(t) = c(t)d(x(t)).

If x

0

(t) is zero everywhere, then d(x(t)) = 0 and thus x

0

(t) = d(x(t)) that is x

is an Integral curve. Note that y is a reparameterization of itself.

We can assume that x

0

(t) 6= 0 everywhere. Then F (x(t)) 6= 0. We have that

c(t) =

||x

0

(t)||

||d(x(t))||

is a continuous function. FiXme: Does it work for non-normed

spaces?

Let y(u(t)) = x(t), where

u(t) =

Z

t

0

c(s)ds,

which is deﬁned and ﬁnite because c is continuous and monotone (thus having

inverse deﬁned on its image) because c is positive.

Then

y

0

(u(t))u

0

(t) = x

0

(t)

= c(t)d(x(t)), so

y

0

(u(t))c(t) = c(t)d(y(u(t)))

y

0

(u(t)) = d(y(u(t)))

and letting s = u(t) we have y

0

(s) = d(y(s)) for a reparameterization y of x.

6.2. Vector space with additional coordinate. Consider the normed vec-

tor space with additional coordinate t.

Our vector ﬁeld d(t) induces vector ﬁeld

ˆ

d(t, v) = (1, d(v)) in this space. Also

ˆ

f(t) = (t, f (t)).

Proposition 2103. Let f be a D

1

function. f is an integral curve of d iﬀ

ˆ

f is

a reparametrized integral curve of

ˆ

d.

Proof. First note that

ˆ

f always has a nonzero derivative.

ˆ

f

0

(t)

ˆ

d(

ˆ

f(t)) ⇔

(1, f

0

(t)) (1, d(f(t))) ⇔ f

0

(t) = d(f(t)).

Thus we have reduced (for D

1

paths) being an integral curve to being a

reparametrized integral curve. We will also describe being a reparametrized in-

tegral curve topologically (through funcoids).

6.3. Topological description of C

1

curves. Explicitly construct this fun-

coid as follows:

R(d, φ) =

n

v∈V

b

vd<φ,v6=0

o

for d 6= 0 and R(0, φ) = {0}, where

b

ab is the angle

between the vectors a and b, for a direction d and an angle φ.

6. INTEGRAL CURVES 14

Definition 2104. W (d) =

d

RLD

n

R(d,φ)

φ∈R,φ>0

o

u

d

RLD

r>0

B

r

(0). FiXme: This is

deﬁned for inﬁnite dimensional case. FiXme: Consider also FCD instead of RLD.

Proposition 2105. For ﬁnite dimensional case R

n

we have W (d) =

d

RLD

n

R(d,φ)

φ∈R,φ>0

o

u ∆

(RLD)n

where

∆

(RLD)n

= ∆ ×

RLD

· · · ×

RLD

∆

| {z }

n times

.

Proof. ??

Finally our funcoids are the complete funcoids Q

+

and Q

−

described by the

formulas

hQ

+

i

∗

@{p} = hp+iW (d(p)) and hQ

−

i

∗

@{p} = hp+iW (−d(p)).

Here ∆ is taken from the “counter-examples” section.

In other words,

Q

+

=

l

p∈R

(@{p} ×

FCD

hp+iW (d(p))); Q

−

=

l

p∈R

(@{p} ×

FCD

hp+iW (−d(p))).

That is hQ

+

i

∗

@{p} and hQ

−

i

∗

@{p} are something like inﬁnitely small spherical

sectors (with inﬁnitely small aperture and inﬁnitely small radius).

FiXme: Describe the co-complete funcoids reverse to these complete funcoids.

Theorem 2106. A D

1

curve f is an reparametrized integral curve for a direc-

tion ﬁeld d iﬀ f ∈ C(ι

D

|R|

>

, Q

+

) ∩ C(ι

D

|R|

<

, Q

−

).

Proof. Equivalently transform f ∈ C(ι

D

|R|, Q

+

); f ◦ ι

D

|R| v Q

+

◦ f;

hf ◦ ι

D

|R|i

∗

@{t} v hQ

+

◦ fi

∗

@{t}; hfi

∗

∆

>

(t) u D v hQ

+

i

∗

f(t); hf i

∗

∆

>

(t) v

hQ

+

i

∗

f(t); hf i

∗

∆

>

(t) v f (t) + W(D(f(t))); hf i

∗

∆

>

(t) − f(t) v W (D(f(t)));

∀r > 0, φ > 0∃δ > 0 : hf i

∗

(]t; t + δ[) − f (t) ⊆ R(d(f (t)), φ) ∩ B

r

(f(t));

∀r > 0, φ > 0∃δ > 0∀0 < γ < δ : hf i

∗

(]t; t + γ[) − f(t) ⊆ R(d(f(t)), φ) ∩ B

r

(f(t));

∀r > 0, φ > 0∃δ > 0∀0 < γ < δ :

hfi

∗

(]t; t + γ[) − f(t)

γ

⊆ R(d(f(t)), φ)∩B

r/δ

(f(t));

∀r > 0, φ > 0∃δ > 0 : ∂

+

f(t) ⊆ R(d(f(t)), φ) ∩ B

r/δ

(f(t));

∀r > 0, φ > 0 : ∂

+

f(t) ⊆ R(d(f(t)), φ);

∂

+

f(t) d(f(t))

where ∂

+

is the right derivative.

In the same way we derive that C(|R|

<

, Q

−

) ⇔ ∂

−

f(t) d(f(t)).

Thus f

0

(t) d(f (t)) iﬀ f ∈ C(|R|

>

, Q

+

) ∩ C(|R|

<

, Q

−

).

6.4. C

n

curves. We consider the diﬀerential equation f

0

(t) = d(f(t)).

We can consider this equation in any topological vector space V (https://en.

wikipedia.org/wiki/Frechet_derivative), see also https://math.stackexchange.com/

q/2977274/4876. Note that I am not an expert in topological vector spaces and

thus my naive generalizations may be wrong in details.

n-th derivative f

(n)

(t) = d

n

(f(t)); f

(n+1)

(t) = d

0

n

(f(t)) ◦ f

0

(t) = d

0

n

(f(t)) ◦

d(f(t)). So d

n+1

(y) = d

0

n

(y) ◦ d(y).

Given a point y ∈ V deﬁne

R

n

(y) =

v ∈ V

\

vd

0

(y) <

d

1

1!

(y)|v| +

d

2

(y)

2!

|v|

2

+ · · · +

d

n−1

(y)

(n−1)!

|v|

n−1

+ O(|v|

n

), v 6= 0

for d

0

(y) 6= 0 and R

n

= {0} if d

0

(y) = 0.

6. INTEGRAL CURVES 15

Definition 2107. R

∞

(y) = R

0

(y) u R

1

(y) u R

2

(y) u . . . .

FiXme: It does not work: https://math.stackexchange.com/a/2978532/4876.

Definition 2108. W

n

(y) = R

n

(y) u

d

RLD

r>0

B

r

(0); W

∞

(y) = R

∞

(y) u

d

RLD

r>0

B

r

(0).

Finally our funcoids are the complete funcoids Q

n

+

and Q

n

−

described by the

formulas

Q

n

+

∗

@{p} = hp+iW

n

(p) and

Q

n

−

∗

@{p} = hp+iW

−

n

(p)

where W

−

is W for the reverse vector ﬁeld −d(y).

FiXme: Related questions: http://math.stackexchange.com/q/1884856/4876

http://math.stackexchange.com/q/107460/4876 http://mathoverﬂow.net/q/88501

Lemma 2109. Let for every x in the domain of the path for small t > 0 we

have f(x + t) ∈ R

n

(F (f(x))) and f(x − t) ∈ R

n

(−F (f(x))). Then f is C

n

smooth.

Proof. FiXme: Not yet proved!

See also http://math.stackexchange.com/q/1884930/4876.

Conjecture 2110. A path f is conforming to the above diﬀerentiable equa-

tion and C

n

(where n is natural or inﬁnite) smooth iﬀ f ∈ C(ι

D

|R|

>

, Q

n

+

) ∩

C(ι

D

|R|

<

, Q

n

−

).

Proof. Reverse implication follows from the lemma.

Let now a path f is C

n

. Then

f(x + t) =

n

X

i=0

f

(i)

(x)

t

i

i!

+ o(t

i

) = f(x) + f

0

(x)t +

n

X

i=2

f

(i)

(x)

t

i

i!

+ o(t

i

)

6.5. Plural funcoids. Take I

+

and Q

+

as described above in forward direc-

tion and I

−

and Q

−

in backward direction. Then

f ∈ C(I

+

, Q

+

) ∧ f ∈ C(I

−

, Q

−

) ⇔ f × f ∈ C(I

+

×

(A)

I

−

, Q

+

×

(A)

Q

−

)?

To describe the above we can introduce new term plural funcoids. This is

simply a map from an index set to funcoids. Composition is deﬁned component-

wise. Order is deﬁned as product order. Well, do we need this? Isn’t it the same

as inﬁmum product of funcoids?

6.6. Multiple allowed directions per point.

hQi

∗

@{p} =

l

d∈d(p)

hp+iW (d).

It seems (check!) that solutions not only of diﬀerential equations but also of

diﬀerence equations can be expressed as paths in funcoids.

CHAPTER 3

More on generalized limit

Definition 2111. I will call a permutation group ﬁxed point free when every

element of it except of identity has no ﬁxed points.

Definition 2112. A funcoid f is Kolmogorov when hfi

∗

{x} 6= hf i

∗

{y} for

every distinct points x, y ∈ dom f.

1. Hausdorﬀ funcoids

Definition 2113. Limit lim F = x of a ﬁlter F regarding funcoid f is such a

point that hfi

∗

{x} w F.

Definition 2114. Hausdorﬀ funcoid is such a funcoid that every proper ﬁlter

on its image has at most one limit.

Proposition 2115. The following are pairwise equivalent for every funcoid f:

1

◦

. f is Hausdorﬀ.

2

◦

. x 6= y ⇒ hfi

∗

{x} hfi

∗

{y}.

Proof.

1

◦

⇒2

◦

. If 2

◦

does not hold, then there exist distinct points x and y such that

hfi

∗

{x} 6 hfi

∗

{y}. So x and y are both limit points of hf i

∗

{x}u hfi

∗

{y},

and thus f is not Hausdorﬀ.

2

◦

⇒1

◦

. Suppose F is proper.

hfi

∗

{x} w F ∧ hfi

∗

{y} w F ⇒ hfi

∗

{x} 6 hfi

∗

{y} ⇒ x = y.

Corollary 2116. Every entirely deﬁned Hausdorﬀ funcoid is Kolmogorov.

Remark 2117. It is enough to be “almost entirely deﬁned” (having nonempty

value everywhere except of one point).

Obvious 2118. For a complete funcoid induced by a topological space this

coincides with the traditional deﬁnition of a Hausdorﬀ topological space.

2. Restoring functions from limit

Consider alternative deﬁnition of generalized limit:

xlim f = λr ∈ G : ν ◦ f◦ ↑ r.

Or:

xlim

a

f =

(

(

r

−1

∗

a, ν ◦ f◦ ↑ r)

r ∈ G

)

(note this requires explicit ﬁlter in the deﬁnition of generalized limit).

Operations on the set of generalized limits can be deﬁned (twice) pointwise.

FiXme: First deﬁne operations on funcoids.

Proposition 2119. The above deﬁned xlim

hµi

∗

{x}

f is a monovalued function

if µ is Kolmogorov and G is ﬁxed point free.

16

2. RESTORING FUNCTIONS FROM LIMIT 17

Proof. We need to prove

r

−1

hµi

∗

{x} 6=

s

−1

hµi

∗

{x} for r, s ∈ G, r 6= s.

Really, by deﬁnition of generalized limit, they commute, so our formula is equivalent

to hµi

∗

r

−1

∗

{x} 6= hµi

∗

s

−1

∗

{x}; hµi

∗

r

−1

◦ s

∗

s

−1

∗

{x} 6= hµi

∗

s

−1

∗

{x}.

But r

−1

◦ s 6= e, so because it is ﬁxed point free,

r

−1

◦ s

∗

s

−1

∗

{x} 6=

s

−1

∗

{x}

and thus by kolmogorovness, we have the thesis.

Lemma 2120. Let µ and ν be Hausdorﬀ funcoids. If function f is deﬁned at

point x, then

fx = lim

(xlim

hµi

∗

{x}

f)hµi

∗

{x}

∗

{x}

Remark 2121. The right part is correctly deﬁned because xlim

a

f is monoval-

ued.

Proof. lim

(xlim

hµi

∗

{x}

f)hµi

∗

{x}

∗

{x} = limhν ◦ fi

∗

{x} = limhνi

∗

fx = fx.

Corollary 2122. Let µ and ν be Hausdorﬀ funcoids. Then function f can

be restored from values of xlim

hµi

∗

{x}

f.

CHAPTER 4

Extending Galois connections between funcoids

and reloids

Definition 2123.

1

◦

. Φ

∗

f = λb ∈ B :

d

n

x∈A

fxvb

o

;

2

◦

. Φ

∗

f = λb ∈ A :

d

n

x∈B

fxwb

o

.

Proposition 2124.

1

◦

. If f has upper adjoint then Φ

∗

f is the upper adjoint of f.

2

◦

. If f has lower adjoint then Φ

∗

f is the lower adjoint of f .

Proof. By theorem 131.

Lemma 2125. Φ

∗

(RLD)

out

= (FCD).

Proof. (Φ

∗

(RLD)

out

)f =

d

n

g∈FCD

(RLD)

out

gwf

o

=

d

FCD

n

g∈Rel

(RLD)

out

gwf

o

=

d

FCD

n

g∈Rel

gwf

o

= (FCD)f.

Lemma 2126. Φ

∗

(RLD)

out

6= (FCD).

Proof. (Φ

∗

(RLD)

out

)f =

d

n

g∈FCD

(RLD)

out

gvf

o

(Φ

∗

(RLD)

out

)⊥ 6= ⊥.

Lemma 2127. Φ

∗

(FCD) = (RLD)

out

.

Proof. (Φ

∗

(FCD))f =

d

n

g∈RLD

(FCD)gwf

o

=

d

RLD

n

g∈Rel

(FCD)gwf

o

=

d

RLD

n

g∈Rel

gwf

o

=

(RLD)

out

f.

Lemma 2128. Φ

∗

(RLD)

in

= (FCD).

Proof. (Φ

∗

(RLD)

in

)f =

d

n

g∈FCD

(RLD)

in

gvf

o

=

d

n

g∈FCD

gv(FCD)f

o

= (FCD)f.

Theorem 2129. The picture at ﬁgure 1 describes values of functions Φ

∗

and Φ

∗

.

All nodes of this diagram are distinct.

Proof. Follows from the above lemmas.

Figure 1

(FCD) (RLD)

in

(RLD)

out

other

Φ

∗

Φ

∗

Φ

∗

Φ

∗

18

4. EXTENDING GALOIS CONNECTIONS BETWEEN FUNCOIDS AND RELOIDS 19

Question 2130. What is at the node “other”?

Trying to answer this question:

Lemma 2131. (Φ

∗

(RLD)

out

)⊥ = Ω

FCD

.

Proof. We have (RLD)

out

Ω

FCD

= ⊥. x 6v Ω

FCD

⇒ (RLD)

out

x w Cor x A ⊥.

Thus max

n

x∈FCD

(RLD)

out

x=⊥

o

= Ω

FCD

.

So (Φ

∗

(RLD)

out

)⊥ = Ω

FCD

.

Conjecture 2132. (Φ

∗

(RLD)

out

)f = Ω

FCD

t (FCD)f.

The above conjecture looks not natural, but I do not see a better alternative

formula.

Question 2133. What happens if we keep applying Φ

∗

and Φ

∗

to the node

“other”? Will we this way get a ﬁnite or inﬁnite set?

CHAPTER 5

Boolean funcoids

1. One-element boolean lattice

Let A be a boolean lattice and B = P 0. It’s sole element is ⊥.

f ∈ pFCD(A; B) ⇔ ∀X ∈ A : (hfiX 6 ⊥ ⇔ hf

−1

i⊥ 6 X) ⇔ ∀X ∈ A : (0 ⇔

f

−1

⊥ 6 X) ⇔ ∀X ∈ A :

f

−1

⊥ X ⇔ ∀X ∈ A :

f

−1

⊥ = ⊥

A

⇔

f

−1

⊥ =

⊥

A

⇔

f

−1

= {(⊥; ⊥

A

)}.

Thus card pFCD(A; P0) = 1.

2. Two-element boolean lattice

Consider the two-element boolean lattice B = P 1.

Let f be a pointfree protofuncoid from A to B (that is (A; B; α; β) where

α ∈ B

A

, β ∈ A

B

).

f ∈ pFCD(A; B) ⇔ ∀X ∈ A, Y ∈ B : (hfiX 6 Y ⇔

f

−1

Y 6 X) ⇔ ∀X ∈

A, Y ∈ B : ((0 ∈ hfiX ∧ 0 ∈ Y ) ∨ (1 ∈ hfiX ∧ 1 ∈ Y ) ⇔ hf

−1

iY 6 X).

T =

n

X∈A

0∈hfiX

o

is an ideal. Really: That it’s an upper set is obvious. Let

P ∪ Q ∈

n

X∈A

0∈hfiX

o

. Then 0 ∈ hfi(P ∪ Q) = hfiP ∪ hfiQ; 0 ∈ hfiP ∨ 0 ∈ hfiQ.

Similarly S =

n

X∈A

1∈hfiX

o

is an ideal.

Let now T, S ∈ PA be ideals. Can we restore hfi? Yes, because we know

0 ∈ hf iX and 1 ∈ hfiX for every X ∈ A.

So it is equivalent to ∀X ∈ A, Y ∈ B : ((X ∈ T ∧ 0 ∈ Y ) ∨ (X ∈ S ∧ 1 ∈ Y ) ⇔

hf

−1

iY 6 X).

f ∈ pFCD(A; B) is equivalent to conjunction of all rows of this table:

Y equality

∅

f

−1

∅ = ∅

{0} X ∈ T ⇔

f

−1

{0} 6 X

{1} X ∈ S ⇔

f

−1

{1} 6 X

{0,1} X ∈ T ∨ X ∈ S ⇔

f

−1

{0, 1} 6 X

Simpliﬁed:

Y equality

∅

f

−1

∅ = ∅

{0} T = ∂

f

−1

{0}

{1} S = ∂

f

−1

{1}

{0,1} T ∪ S = ∂

f

−1

{0, 1}

From the last table it follows that T and S are principal ideals.

So we can take arbitrary either

f

−1

{0}, hf

−1

i{1} or principal ideals T and

S.

In other words, we take

f

−1

{0}, hf

−1

i{1} arbitrary and independently. So

we have pFCD(A; B) equivalent to product of two instances of A. So it a boolean

lattice. FiXme: I messed product with disjoint union below.)

20

4. ABOUT INFINITE CASE 21

3. Finite boolean lattices

We can assume B = PB for a set B, card B = n. Then

f ∈ pFCD(A; B) ⇔ ∀X ∈ A, Y ∈ B : (hfiX 6 Y ⇔

f

−1

Y 6 X) ⇔ ∀X ∈

A, Y ∈ B : (∃i ∈ Y : i ∈ hfiX ⇔

f

−1

Y 6 X).

Having values of

f

−1

{i} we can restore all hf

−1

iY . [need this paragraph?]

Let T

i

=

n

X∈A

i∈hfiX

o

.

Let now T

i

∈ PA be ideals. Can we restore hfi? Yes, because we know

i ∈ hfiX for every X ∈ A.

So, it is equivalent to:

∀X ∈ A, Y ∈ B : (∃i ∈ Y : X ∈ T

i

⇔

f

−1

Y 6 X). (1)

Lemma 2134. The formula (1) is equivalent to:

∀X ∈ A, i ∈ B : (X ∈ T

i

⇔

f

−1

{i} 6 X). (2)

Proof. (1)⇒(2). Just take Y = {i}.

(2)⇒(1). Let (2) holds. Let also X ∈ A, Y ∈ B. Then hf

−1

iY 6 X ⇔

S

i∈Y

hf

−1

i{i} 6 X ⇔ ∃i ∈ Y : hf

−1

i{i} 6 X ⇔ ∃i ∈ Y : X ∈ T

i

.

Further transforming: ∀i ∈ B : T

i

= ∂hf

−1

i{i}.

So

f

−1

{i} are arbitary elements of B and T

i

are corresponding arbitrary

principal ideals.

In other words, pFCD(A; B)

∼

=

AΠ . . . ΠA (card B times). Thus pFCD(A; B) is

a boolean lattice.

4. About inﬁnite case

Let A be a complete boolean lattice, B be an atomistic boolean lattice.

f ∈ pFCD(A; B) ⇔ ∀X ∈ A, Y ∈ B : (hfiX 6 Y ⇔

f

−1

Y 6 X) ⇔ ∀X ∈

A, Y ∈ B : (∃i ∈ atoms Y : i ∈ atomshf iX ⇔

f

−1

Y 6 X).

Let T

i

=

n

X∈A

i∈atomshfiX

o

.

T

i

is an ideal: Really: That it’s an upper set is obvious. Let P ∪ Q ∈

n

X∈A

i∈atomshfiX

o

. Then i ∈ atomshfi(P ∪Q) = atomshfiP ∪atomshfiQ; i ∈ hfiP ∨i ∈

hfiQ.

Let now T

i

∈ PA be ideals. Can we restore hfi? Yes, because we know

i ∈ atomshfiX for every X ∈ A and B is atomistic.

So, it is equivalent to:

∀X ∈ A, Y ∈ B : (∃i ∈ atoms Y : X ∈ T

i

⇔

f

−1

Y 6 X). (3)

Lemma 2135. The formula (3) is equivalent to:

∀X ∈ A, i ∈ atoms

B

: (X ∈ T

i

⇔ hf

−1

ii 6 X). (4)

Proof. (3)⇒(4). Let (3) holds. Take Y = i. Then atoms Y = {i} and thus

X ∈ T

i

⇔ ∃i ∈ atoms Y : X ∈ T

i

⇔

f

−1

Y 6 X ⇔

f

−1

i 6 X.

(4)⇒(3). Let (2) holds. Let also X ∈ A, Y ∈ B. Then

f

−1

Y 6 X ⇔

hf

−1

i

d

atoms Y 6 X ⇔

d

i∈atoms Y

f

−1

i 6 X ⇔ ∃i ∈ atoms Y :

f

−1

i 6 X ⇔ ∃i ∈ atoms Y : X ∈ T

i

.

Further equivalently transforming: ∀i ∈ atoms

B

: T

i

= ∂

f

−1

i.

So

f

−1

i are arbitary elements of B and T

i

are corresponding arbitrary prin-

cipal ideals.

4. ABOUT INFINITE CASE 22

In other words, pFCD(A; B)

∼

=

Q

i∈card atoms

B

A. Thus pFCD(A; B) is a boolean

lattice.

So ﬁnally we have a very weird theorem, which is a partial solution for the

above open problem (The weirdness is in its partiality and asymmetry):

Theorem 2136. If A is a complete boolean lattice and B is an atomistic

boolean lattice (or vice versa), then pFCD(A; B) is a boolean lattice.

[4] proves “THEOREM 4.6. Let A, B be bounded posets. A ⊗ B is a com-

pletely distributive complete Boolean lattice iﬀ A and B are completely distributive

Boolean lattices.” (where A ⊗ B is equivalent to the set of Galois connections be-

tween A and B) and other interesting results.

CHAPTER 6

Interior funcoids

Having a funcoid f let deﬁne interior funcoid f

◦

.

Definition 2137. Let f ∈ FCD(A, B) = pFCD(T A, T B) be a co-complete

funcoid. Then f

◦

∈ pFCD(dual T A, dual T B) is deﬁned by the formula hf

◦

i

∗

X =

hfiX.

Proposition 2138. Pointfree funcoid f

◦

exists and is unique.

Proof. X 7→ hfiX is a component of pointfree funcoid dual T A → dual T B

iﬀ hfi is a component of the corresponding pointfree funcoid T A → T B that is

essentially component of the corresponding funcoid FCD(A, B) what holds for a

unique funcoid.

It can be also deﬁned for arbitrary funcoids by the formula f

◦

= (CoCompl f)

◦

.

Obvious 2139. f

◦

is co-complete.

Theorem 2140. The following values are pairwise equal for a co-complete

funcoid f and X ∈ T Src f:

1

◦

. hf

◦

i

∗

X;

2

◦

.

n

y∈Dst f

hf

−1

i

∗

{y}vX

o

3

◦

.

d

n

Y ∈T Dst f

hf

−1

i

∗

Y vX

o

4

◦

.

d

n

Y∈F Dst f

hf

−1

iYvX

o

Proof.

1

◦

=2

◦

.

n

y∈Dst f

hf

−1

i

∗

{y}vX

o

=

n

x∈Dst f

hf

−1

i

∗

{x}X

o

=

n

x∈Dst f

{x}hfiX

o

= hfiX = hf

◦

i

∗

X.

2

◦

=3

◦

. If

f

−1

∗

Y v X then (by completeness of f

−1

) Y =

n

y∈Y

hf

−1

i

∗

{y}vX

o

and

thus

l

Y ∈ T Dst f

hf

−1

i

∗

Y v X

v

y ∈ Dst f

hf

−1

i

∗

{y} v X

.

The reverse inequality is obvious.

3

◦

=4

◦

. It’s enough to prove that if

f

−1

Y v X for Y ∈ F Dst f then exists

Y ∈ up Y such that hf

−1

i

∗

Y v X. Really let

f

−1

Y v X. Then

d

hhf

−1

i

∗

i

∗

up Y v X and thus exists Y ∈ up Y such that hf

−1

i

∗

Y v X

by properties of generalized ﬁlter bases.

This coincides with the customary deﬁnition of interior in topological spaces.

Proposition 2141. f

◦◦

= f for every funcoid f.

Proof. hf

◦◦

i

∗

X = ¬¬hfi¬¬X = hfiX.

Proposition 2142. Let g ∈ FCD(A, B), f ∈ FCD(B, C), h ∈ FCD(A, C) for

some sets A. B, C.

g v f

◦

◦ h ⇔ f

−1

◦ g v h, provided f and h are co-complete.

23

6. INTERIOR FUNCOIDS 24

Proof. g v f

◦

◦ h ⇔ ∀X ∈ A : hgi

∗

X v hf

◦

◦ hi

∗

X ⇔ ∀X ∈ A :

hgi

∗

X v hf

◦

i

∗

hhi

∗

X ⇔ ∀X ∈ A : hgi

∗

X v ¬hfi

∗

¬hhi

∗

X ⇔ ∀X ∈ A : hgi

∗

X

hfi

∗

¬hhi

∗

X ⇔ ∀X ∈ A :

f

−1

∗

hgi

∗

X ¬hhi

∗

X ⇔ ∀X ∈ A :

f

−1

∗

hgi

∗

X v

hhi

∗

X ⇔ ∀X ∈ A :