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

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Funcoids and Reloids:
a Generalization of Proximities and Uniformities
Victor Porton
September 11, 2013
Abstract
It is a part of my Algebraic General Topology research.
In this article, I introduce the concepts of funcoids, which general-
ize proximity spaces and reloids, which generalize uniform spaces. The
concept of a funcoid is a generalized concept of proximity, the concept of
a reloid is the concept of u niformity cleared (generalized) from superflu-
ous details. Also funcoids generalize pretopologies and preclosures. Also
funcoids and reloids are generalizations of binary relations whose domains
and ranges are filters (instead of sets).
Also funcoids and reloids can be considered as a generalization of (di-
rected) graphs, this provides us a common generalization of analysis and
discrete mathematics.
The concept of continuity is defined by an algebraic formula (instead
of t he old messy epsilon-delta notation) for arbitrary morphisms (includ-
ing funcoids and reloids) of a partially ordered category. In one formula
continuity, proximity continuity, and uniform continuity are generalized.
Contents
1 Common 3
1.1 Earlier works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Used concepts, notation and sta tements . . . . . . . . . . . . . . 3
1.2.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Partially ordered dagger categories 5
2.1 Partially ordered categories . . . . . . . . . . . . . . . . . . . . . 5
2.2 Dagger categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Some special classes of morphisms . . . . . . . . . . . . . 7
Keywords: algebraic general topology, quasi-uniform spaces, generalizations of proximity
spaces, generalizations of nearness spaces, generalizations of uniform spaces; A.M.S. subj ect
classification: 54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99
1
3 Funcoids 8
3.1 Informal introduction into funcoids . . . . . . . . . . . . . . . . . 8
3.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.1 Composition of funcoids . . . . . . . . . . . . . . . . . . . 11
3.3 Funcoid as continuatio n . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Lattices of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 More on composition of funcoids . . . . . . . . . . . . . . . . . . 17
3.6 Domain and range of a funco id . . . . . . . . . . . . . . . . . . . 1 9
3.7 Categorie s of funcoids . . . . . . . . . . . . . . . . . . . . . . . . 20
3.8 Spec ifying funcoids by functions o r rela tions on atomic filter objects 21
3.9 Direct product of filter objects . . . . . . . . . . . . . . . . . . . 24
3.10 Atomic funco ids . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.11 Complete funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.12 Completion of funcoids . . . . . . . . . . . . . . . . . . . . . . . . 34
3.13 Monovalued and injective funcoids . . . . . . . . . . . . . . . . . 36
3.14 T
0
-, T
1
- and T
2
-separable funcoids . . . . . . . . . . . . . . . . . 38
3.15 Filter objects closed regarding a funcoid . . . . . . . . . . . . . . 38
4 Reloids 39
4.1 Composition of reloids . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Direct product of filter objects . . . . . . . . . . . . . . . . . . . 42
4.3 Restricting re loid to a filter object. Do main a nd image . . . . . . 43
4.4 Categorie s of reloids . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Monovalued and injective reloids . . . . . . . . . . . . . . . . . . 46
4.6 Complete reloids and completion of reloids . . . . . . . . . . . . . 47
5 Relationships between funcoids and reloids 51
5.1 Funcoid induced by a reloid . . . . . . . . . . . . . . . . . . . . . 51
5.2 Reloids induced by funcoid . . . . . . . . . . . . . . . . . . . . . 56
5.3 Galois connections of funcoids and reloids . . . . . . . . . . . . . 58
6 Continuous morphisms 59
6.1 Tra ditional definitions of continuity . . . . . . . . . . . . . . . . . 59
6.1.1 Pre-topology . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1.2 Proximity spaces . . . . . . . . . . . . . . . . . . . . . . . 60
6.1.3 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Our three definitions of continuity . . . . . . . . . . . . . . . . . 61
6.3 Continuity of a restricted morphism . . . . . . . . . . . . . . . . 62
7 Connectedness regarding funcoids and reloids 63
7.1 Some lemma s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2 Endomorphism series . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.3 Connectedness r egarding binary relations . . . . . . . . . . . . . 65
7.4 Connectedness r egarding funcoids and reloids . . . . . . . . . . . 67
7.5 Algebraic properties of S a nd S
. . . . . . . . . . . . . . . . . . 69
2
8 Postface 70
8.1 Misc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
A Some counter-examples 70
A.1 Sec ond product. Oblique product . . . . . . . . . . . . . . . . . . 75
1 Common
1.1 Earlier works
Some mathematicians researched generalizations of proximities and uniformities
before me but they have failed to reach the right degre e of generalization which
is presented in this work allowing to represent properties of spaces with algebraic
(or categorical) formulas.
Proximity structures were introduced by Smirnov in [5].
Some references to predecessors:
In [6], [7], [12], [2], [19] are s tudied generalized uniformities and proximi-
ties.
Proximities and uniformities are also studied in [10], [11], [18], [20], [21].
[8] and [9] contains recent progress in quasi-unifor m spaces. [9] has a very
long list of related literature.
Some works ([17]) about proximity spaces consider relations hips of proximities
and compact topological spaces. In this work the attempt to define or research
their g eneralization, compactness of funcoids or reloids , is not done. It seems
potentially productive to attempt to borrow the definitions and procedures from
the above mentioned works. I hope to do this study in a separate article.
[4] studies mappings between proximity structures. (In this work no at-
tempt to research mappings between funcoids is done.) [13] researches relation-
ships of quasi-uniform spaces and topological spaces. [1] studies how proximity
structures can be treated as uniform structures and compactification regarding
proximity and uniform spaces.
1.2 Used concepts, notation and statements
The set of functions from a set A to a set B is denoted as B
A
.
I will often skip pare ntheses and write f x instead of f (x) to denote the result
of a function f acting on the argument x.
I will call small sets members of some Grothendieck universe. (Let us assume
the axiom of existence of a Grothendieck universe.)
Let f is a sma ll binary relation.
I will denote hf i X = {f α | α X} and X [f ] Y x X, y Y : x f y
for small sets X, Y .
3
By just hfi and [f ] I will denote the correspo nding function and relation on
small sets.
λx D : f (x) = {(x; f (x)) | x D} for every formula f (x) depended
on a variable x and set D.
I will deno te the least and the greatest element of a poset A as 0
A
and 1
A
respectively.
For elements a and b of a lattice with a minimal element I will denote a b
when a b is the minimal element of the lattice and a 6≍ b otherwise. See [15]
for a more gene ral notion.
Propositi on 1 Let f , g, h be binary relations. Then g f 6≍ h g 6≍ h f
1
.
Proof
g f 6≍ h
a, c : a ((g f ) h) c
a, c : (a (g f) c a h c)
a, b, c : (a f b b g c a h c)
b, c :
b g c b
h f
1
c
b, c : b
g
h f
1

c
g 6≍ h f
1
.
1.2.1 Filters
In this work the word filter will refer to a filter on a set (in contrast to [15]
where filters are considered on arbitrary posets). Note that I do not require
filters to be proper.
I will call the set of filters on a set A (base set) ordere d reverse to set-
theoretic inclusion of filters the se t of filter objects on A and denote it F (A)
or just F when the base set is implied a nd call its element filter objects (f.o.
for short). I will denote up F the filter corr esponding to a filter object F. So
we have A B up A up B for every filter objects A and B on the same set.
In this particular manuscript, we will not equate principal filter objects with
corres ponding sets as it is done in [15]. Instead we will have Base (A) equal to
the unique base of a f.o. A. I will denote
A
X (or just X when A is implied)
the principal filter object on A cor responding to the set X.
Filters are studied in the work [15].
Every set F (A) is a c omplete lattice and we will apply lattice operations to
subsets of such sets without ex plic itly mentioning F (A).
Prior r e ading of [15] is needed to fully understand this work.
Filter objects corresponding to ultrafilter s are atoms of the lattice F (A) and
will be called atom ic filter objects (on A).
4
Also we will need to introduce the concept of generalized filter base.
Definition 1 Generalized filter base is a set S PF \
0
F
such that
∀A, B S∃C S : C A B.
Propositi on 2 Let S is a generalized lter base. If A
1
, . . . , A
n
S (n
N
),
then
∃C S : C A
1
. . . A
n
.
Proof C an be easily proved by induction.
Theorem 1 If S is a generalized filter base, then up
T
S =
S
hupi S.
Proof Obviously up
T
S
S
hupi S. Reverse ly, let K up
T
S; then K =
A
1
. . . A
n
where A
i
up A
i
where A
i
S, i = 1, . . . , n, n
N
; so exists
C S such that C A
1
. . . A
n
⊆↑ (A
1
. . . A
n
) = K, K up C,
K
S
hupi S.
Corollary 1 If S is a generalized lter base, then
T
S = 0
F
0
F
S.
Proof
T
S = 0
F
up
T
S
S
hupi S ∃X S : up X 0
F
S.
Obvious 1. If S is a filter base on a set A then
A
S is a generalized filter
base.
Definition 2 I will call shifted filtrator a triple (A; Z; ) where A and Z are
posets and is an order embedding from Z to A.
Some concepts and notation can be defined for shifted filtrators through
similar co nc e pts for filtrators : h↑i up a = up
(A;h↑iZ)
a; h↑i Cor a = Cor
(A;h↑iZ)
a,
etc.
For a set A and the set of f.o. F on this set we will consider the shifted
filtrator (F; A; ).
2 Partially ordered dagger categories
2.1 Partially ordered categories
Definition 3 I will call a partially ordered (pre)category a (pre)category
together with partial order on each of its Hom-sets with the additional require-
ment that
f
1
f
2
g
1
g
2
g
1
f
1
g
2
f
2
for every morphisms f
1
, g
1
, f
2
, g
2
such that Src f
1
= Src f
2
Dst f
1
= Dst f
2
=
Src g
1
= Src g
2
Dst g
1
= Dst g
2
.
5
2.2 Dagger categories
Definition 4 I will call a dagger precategory a precategory together with an
involutive contravariant identity-on-objects prefunctor x 7→ x
.
In other words, a dagger precategory is a precategory equipped with a
function x 7→ x
on its set of morphisms which reverses the sou rce and the
destination and is subject to the following identities for every morphisms f and
g:
1. f
††
= f;
2. (g f )
= f
g
.
Definition 5 I will call a dagger category a category together with an invo-
lutive contravariant identity-on-objects functor x 7→ x
.
In other words, a dagger category is a category equipped with a function
x 7→ x
on its set of morphisms which reverses the sou rce and the destination
and is subject to the following identities for every morphisms f and g and object
A:
1. f
††
= f;
2. (g f )
= f
g
;
3. (1
A
)
= 1
A
.
Theorem 2 If a category is a dagger precategory then it is a dagger category.
Proof We nee d to prove only that (1
A
)
= 1
A
. Really
(1
A
)
= (1
A
)
1
A
= (1
A
)
(1
A
)
††
= ((1
A
)
1
A
)
= (1
A
)
††
= 1
A
.
For a partially ordered dagger (pre)category I will additionally require (for
every morphisms f and g)
f
g
f g.
An example of da gger c ategory is the category Rel whose objects are sets and
whose morphisms are binary relations betwe en these sets with usual compositio n
of binary relations and with f
= f
1
.
Definition 6 A morphism f of a dagger category is called unitary when it is
an isomorphism and f
= f
1
.
Definition 7 Sy mmetric (endo)morphism of a dagger precategory is such a
morphism f that f = f
.
6
Definition 8 Transitive (endo)morphism of a precategory is such a morphism
f that f = f f.
Theorem 3 The following conditions are equivalent for a morphism f of a
dagger precategory:
1. f is symmetric and transitive.
2. f = f
f.
Proof
(1)(2) If f is symmetric and transitive then f
f = f f = f.
(2)(1) f
= (f
f)
= f
f
††
= f
f = f, so f is symmetric. f = f
f =
f f , s o f is transitive.
2.2.1 Some special classes of morphis ms
Definition 9 For a partially ordered dagger category I will call monovalued
morphism such a morphism f that f f
1
Dst f
.
Definition 10 For a partially ordered dagger category I will call entirely de-
fined morphism such a morphism f that f
f 1
Src f
.
Definition 11 For a partially ordered dagger category I will call injective mor-
phism such a morphism f that f
f 1
Src f
.
Definition 12 For a partially ordered dagger category I will call surjective
morphism such a morphism f that f f
1
Dst f
.
Remark 1 It’s easy to show that this is a generalization of monovalued, entirely
defined, injective, and surjective binary relations as morphisms of the category
Rel.
Obvious 2. Injective morphism” is a dual of monovalued morphism” and
“surjective morphism” is a dual of “entirely defined morphism”.
Definition 13 For a given partially ordered dagger category C the category
of monovalued (entirel y defined, injective, surjective) morphisms of
C is the category with the same set of objects as of C and the set of morphisms
being the set of monovalued (entirely defined, injective, surjective) morphisms
of C with the composition of morphisms the same as in C.
7
We need to prove that these are really categories , that is that composition
of monovalued (entirely defined, injective, surjective) morphisms is monovalued
(entirely defined, injective, surjective) and that identity mor phisms are mo no-
valued, entirely defined, injective, and surjective.
Proof We will prove only for monovalued morphisms and entirely defined
morphisms, a s injective and surjective morphisms are their duals.
Monovalued Let f and g a re monovalued morphisms, Dst f = Src g. (g f)
(g f)
= g f f
g
g 1
Dst f
g
= g 1
Src g
g
= g g
1
Dst g
=
1
Dst(gf )
. So g f is monovalued.
That identity morphisms are monova lue d follows from the following: 1
A
(1
A
)
= 1
A
1
A
= 1
A
= 1
Dst 1
A
1
Dst 1
A
.
Entirely define d Let f and g are entirely defined morphisms, Dst f = Src g.
(g f)
(g f) = f
g
g f f
1
Src g
f = f
1
Dst f
f = f
f
1
Src f
= 1
Src(gf )
. So g f is entirely defined.
That identity morphisms are entirely defined follows from the following:
(1
A
)
1
A
= 1
A
1
A
= 1
A
= 1
Src 1
A
1
Src 1
A
.
Definition 14 I will call a bijective morphism a morphism which is entirely
defined, monovalued, injective, and surjective.
Obvious 3 . Bijective morphisms form a full subcategory.
Propositi on 3 If a morphism is bijective then it is an isomorphism.
Proof Let f is bijective. Then f f
1
Dst f
, f
f 1
Src f
, f
f 1
Src f
,
f f
1
Dst f
. Thus f f
= 1
Dst f
and f
f = 1
Src f
that is f
is an inverse
of f.
3 Funcoids
3.1 Informal intro duct ion into funcoids
Funcoids are a generalization of proximity spaces and a generalization of pre-
topologica l spaces. Also funcoids are a generalization of binary relations.
That funcoids are a common generalization of spaces” (proximity spaces,
(pre)topological spaces) and binary relations (including monovalued functions)
makes them smart for describing properties of functions in regard of spa ces. For
example the statement f is a continuous function from a space µ to a space
ν can be described in terms of funcoids as the for mula f µ ν f (see below
for details).
Most naturally funcoids appear as a generalization of proximity spaces.
8
Let δ be a proximity that is certain binary relation so that A δ B is defined
for every sets A and B. We will extend it from sets to filter objects by the
formula:
A δ
B A up A, B up B : A δ B.
Then (as it will b e proved below) there exist two functions α, β F
F
such that
A δ
B B
F
αA 6= 0
F
A
F
βB 6= 0
F
.
The pair (α; β) is called funcoid when B
F
αA 6= 0
F
A
F
βB 6= 0
F
. So
funcoids are a generalization of proximity spaces.
Funcoids consist of two components the first α and the s e cond β. The first
component of a funcoid f is denoted as hfi and the second component is denoted
as
f
1
. (The similarity of this notation with the notatio n for the image of a
set under a function is not a coincidence, we will see that in the case of principal
funcoids (see below) these coincide.)
One of the most important properties of a funcoid is that it is uniquely
determined by just one of its components. That is a funcoid f is unique ly
determined by the function hf i. Moreover a funcoid f is uniquely determined
by hf i |
P
S
domhfi
that is by values of function hfi on sets (if we equate principal
filters with sets).
Next we will consider some examples of funcoids determined by specified
values of the first component o n sets.
Funcoids as a gener alization of pretopological spaces: Let α be a pretopo-
logical space that is a map α F
for some set . Then we define α
X
def
=
S
F
{αx | x X} fo r every set X P . We will prove that there exists a
unique funcoid f such that α
= hfi |
P
. So funcoids are a generalization of
pretopological spaces. Funcoids ar e also a generalizatio n of preclosure opera-
tors: For every preclosure operator p on a set it exists a unique funcoid f
such that hf i |
P
= p.
For every binary relation p on a set it exists unique funcoid f such that
X P : hfi X = hpi X (where hpi is de fined in the introduction), recall
that a funcoid is uniquely determined by the values of its first component on
sets. I will call such funcoids principal. So funcoids are a generalization of
binary r elations.
Composition of binary re lations (i.e. of principal funcoids) complies with
the formulas:
hg fi = hgi hfi and
(g f )
1
=
f
1
g
1
.
By the same fo rmulas we can define comp osition of every two funcoids. Funcoids
with this composition form a category (the category of funcoids).
Also funcoids can be reversed (like reversal of X and Y in a binary relation)
by the formula (α; β)
1
= (β; α). In particular case if µ is a proximity we have
µ
1
= µ because proximities are symmetric.
Funcoids behave similarly to (multivalued) functions but acting on filter
objects instead of acting on sets. Below these will be defined domain and image
of a funcoid (the domain and the image of a funcoid ar e filter objects).
9
3.2 Basic definitions
Definition 15 Let’s call a funcoid from a set A t o a set B a quadruple
(A; B; α; β) where α F (B)
F(A)
, β F (A)
F(B)
such that
∀X F (A) , Y F (B) : (Y 6≍ αX X 6≍ βY).
Further we will assume that all funcoids in consideration are small without
mentioning it explicitly.
Definition 16 Source and destination of every funcoid (A; B; α; β) are de-
fined as
Src (A; B; α; β) = A and Dst (A; B; α; β) = B.
I will denote FCD (A; B) the set of funcoids from A to B.
I will denote FCD the set of all funcoids (for s mall sets).
Definition 17 h(A; B; α; β)i
def
= α for a funcoid (A; B; α; β).
Definition 18 (A; B; α; β)
1
= (B; A; β; α) for a funcoid (A; B; α; β).
Propositi on 4 If f is a funcoid then f
1
is also a funcoid.
Proof I t follows from sy mmetry in the definition of funcoid.
Obvious 4 . (f
1
)
1
= f for a funcoid f.
Definition 19 The relation [f] P (F (Src f) × F (Dst f)) is defined (for every
funcoid f and X F (Src f), Y F (Dst f)) by the formula X [f] Y
def
= Y 6≍
hfi X .
Obvious 5. X [f ] Y Y 6≍ hfi X X 6≍
f
1
Y for every funcoid f and
X F (Src f), Y F (Dst f).
Obvious 6 .
f
1
=[f]
1
for a funcoid f .
Theorem 4 Let A, B are small sets.
1. For given value of hfi exists no more than one funcoid f FCD (A; B).
2. For given value of [f] exists no m ore than one funcoid f FCD (A; B).
Proof Let f, g FCD (A; B) .
Obviously hfi = hgi [f ]=[g] and
f
1
=
g
1
[f]=[g]. So it’s enough
to prove that [f]=[g] hf i = hgi.
Provided that [f ]=[g] we have Y 6≍ hf i X X [f ] Y X [g] Y Y 6≍
hgi X and conseq ue ntly h f i X = hgi X for every X F (A) and Y F (B)
because a set of filter objects is separable [15], thus hfi = hgi.
10
Propositi on 5 hf i 0
F(Src f )
= 0
F(Dst f)
for every funcoid f.
Proof Y 6≍ hf i 0
F(Src f )
0
F(Src f )
6≍
f
1
Y 0 Y 6≍ 0
F(Dst f)
. Thus
hfi 0
F(Src f)
= 0
F(Dst f )
by separability of filter objects.
Propositi on 6 hf i (I J ) = hf i I hfi J for every funcoid f and I, J
F (Src f).
Proof
hfi (I J ) =
{Y F | Y 6≍ hf i (I J )} =
Y F | I J 6≍
f
1
Y
= (by corollary 10 in [15])
Y F | I 6≍
f
1
Y J 6≍
f
1
Y
=
{Y F | Y 6≍ hfi I Y 6≍ hf i J } =
{Y F | Y 6≍ hfi I hfi J } =
(hfi I hfi J ).
Thus hf i (I J ) = hfi I hfi J because F (Dst f) is separable.
Propositi on 7 For every f FCD (A; B) for every small sets A and B we
have:
1. K [f ] I J K [f ] I K [f ] J for every I, J F (B), K F (A).
2. I J [f] K I [f ] K J [f] K for every I, J F (A), K F (B).
Proof 1. K [f ] I J (I J )hfi K 6= 0
F(B)
(I hfi K)(J hfi K) 6=
0
F(B)
I hfi K 6= 0
F(B)
J h f i K 6= 0
F(B)
K [f ] I K [f] J .
2. Simila r.
3.2.1 Composition of funcoids
Definition 20 Fun coids f and g are composable when Dst f = Src g.
Definition 21 Composition of composable funcoids is defined by the formula
(B; C; α
2
; β
2
) (A; B; α
1
; β
1
) = (A; C; α
2
α
1
; β
1
β
2
).
Propositi on 8 If f, g are composable fu ncoids then g f is a funcoid.
Proof Let f = (A; B; α
1
; β
1
), g = (B; C; α
2
; β
2
). For every X F (A), Y
F (C) we have
Y 6≍ (α
2
α
1
)X Y 6≍ α
2
α
1
X α
1
X 6≍ β
2
Y X 6≍ β
1
β
2
Y X 6≍ (β
1
β
2
)Y.
11
So (A; C; α
2
α
1
; β
1
β
2
) is a funcoid.
Obvious 7 . hg f i = hgi hfi for every composable funcoids f and g.
Propositi on 9 (h g) f = h (g f) for every composable funcoids f , g, h.
Proof
h(h g) f i = hh gihfi = (hhihgi)hfi = hhi(hgihfi) = hhihg f i =
hh (g f)i.
Theorem 5 (g f)
1
= f
1
g
1
for every composable fu ncoids f and g.
Proof
(g f)
1
=
f
1
g
1
=
f
1
g
1
.
3.3 Funcoid as c ontinuation
Let f is a funcoid.
Definition 22 hf i
is the function P (Src f) F (Dst f) defined by the for-
mula
hfi
X = hfi
Src f
X.
Definition 23 [f ]
is the relation between P (Src f ) and P (Dst f) defined by
the formula
X [f]
Y =
Src f
X [f]
Dst f
Y.
Obvious 8 .
1. hfi
= hfi
Src f
;
2. [f]
=
Dst f
1
[f]
Src f
.
Theorem 6 For every funcoid f and X F (Src f) and Y F (Dst f)
1. hfi X =
T
hfi
up X ;
2. X [f] Y X up X , Y up Y : X [f]
Y .
Proof 2. X [f] Y Y hfi X 6= 0
F(Dst f)
Y up Y :
Dst f
Y hfi X 6=
0
F(Dst f )
Y up Y : X [f ]
Dst f
Y .
Analogously X [f ] Y X up X :
Src f
X [f ] Y. C ombining these two
equivalences we get
X [f ] Y X up X , Y up Y :
Src f
X [f ]
Dst f
Y X up X , Y up Y : X [f]
Y.
1. Y hfi X 6= 0
F(Dst f )
X [f ] Y X up X :
Src f
X [f ] Y X
up X : Y hfi
X 6= 0
F(Dst f )
.
12
Let’s denote W =
Y hfi
X | X up X
. We will prove that W
is a generalized filter base. To prove this it is enough to show that V =
hfi
X | X up X
is a generalized filter base.
Let P, Q V . Then P = hf i
A, Q = hfi
B where A, B up X ; A B
up X and R P Q for R = hfi
(A B) V . So V is a genera lized filter
base and thus W is a generalized filter ba se.
0
F(Dst f )
/ W
T
W 6= 0
F(Dst f )
by the corollary 1 of the theorem 1. That
is
X up X : Y hfi
X 6= 0
F(Dst f )
Y
\
hfi
up X 6= 0
F(Dst f )
.
Comparing with the a bove, Y hf i X 6= 0
F(Dst f )
Y
T
hfi
up X 6=
0
F(Dst f )
. So hfi X =
T
hfi
up X bec ause the lattice of filter objects is sepa -
rable.
Propositi on 10 For every f FCD (A; B) we have (for every I, J PA)
hfi
= 0
F(B)
, hf i
(I J) = hf i
I hfi
J
and
¬(I [f ]
), I J [f]
K I [f ]
K J [f ]
K (for every I, J PA, K P B),
¬( [f]
I), K [f]
I J K [f ]
I K [f ]
J (for every I, J PB, K P A).
Proof hf i
= hfi
A
= hf i 0
F(A)
= 0
F(B)
; hf i
(I J) = hfi
A
(I J) =
hfi
A
I
A
J
= hfi
A
I hfi
A
J = hfi
I hfi
J.
I [f ]
0
F(B)
6≍ hfi
A
I 0; I J [f ]
K ⇔↑
A
(I J) [f ]
B
K ⇔↑
B
K 6≍ hfi
(I J) ⇔↑
B
K 6≍ hfi
I hfi
J ⇔↑
B
K 6≍ hfi
I
B
K 6≍ hfi
J
I [f]
K J [f]
K.
The r est follows from symmetry.
Theorem 7 Fix small sets A and B. Let L
F
= λf FCD (A; B) : hf i
and
L
R
= λf FCD (A; B) :[f]
.
1. L
F
is a bijection from the set FCD (A; B) to the set of functions α F (B)
PA
that obey the conditions (for every I, J PA)
α = 0
F(B)
, α(I J) = αI αJ. (1)
For such α it holds (for every X F (A))
L
1
F
α
X =
\
hαi up X . (2)
2. L
R
is a bijection from the set FCD (A; B) to the set of binary relations δ
P (PA × P B) t hat obey the conditions
¬(I δ ), I J δ K I δ K J δ K (for every I, J PA, K P B),
¬( δ I), K δ I J K δ I K δ J (for every I, J PB, K P A).
(3)
13
For such δ it holds (for every X F (A), Y F (B))
X
L
1
R
δ
Y X up X, Y up Y : X δ Y. (4)
Proof Injectivity of L
F
and L
R
, formulas (2) (for α im L
F
) and (4) (for
δ im L
R
), formulas (1) and (3) follow from two previous theorems. T he only
thing remained to prove is that for every α and δ that obey the above c onditions
a cor responding funcoid f exists.
2. Let define α F (B)
PA
by the formula (αX) = {Y P B | X δ Y }
for every X P A. (It is obvious that {Y P B | X δ Y } is a free star.)
Analogously it can be defined β F (A)
PB
by the formula (βY ) = {X PA | X δ Y }.
Let’s continue α and β to α
F (B)
F(A)
and β
F (A)
F(B)
by the formulas
α
X =
\
hαi up X and β
Y =
\
hβi up Y
and δ to δ
P (F (A) × F (B)) by the formula
X δ
Y X up X , Y up Y : X δ Y.
Y α
X 6= 0
F(B)
Y
T
hαi up X 6= 0
F(B)
T
hY∩i hαi up X 6= 0
F(B)
. Let’s
prove that
W = hY∩i hαi up X
is a generalized filter base: To prove it is enough to show that hαi up X is a
generalized filter base. If A, B hαi up X then ex ist X
1
, X
2
up X such that
A = αX
1
and B = αX
2
.
Then α(X
1
X
2
) hαi up X . So hαi up X is a generalized filter base and
thus W is a generalized filter base.
Accordingly to the corolla ry 1 of the theorem 1,
T
hY∩i hαi up X 6= 0
F(B)
is
equivalent to
X up X : Y αX 6= 0
F(B)
,
what is equivalent to X up X , Y up Y :
B
Y αX 6= 0
F(B)
X
up X , Y up Y : Y (αX) X up X , Y up Y : X δ Y . Combining
the equivalencies we get Y α
X 6= 0
F(B)
X δ
Y. Analogously X β
Y 6=
0
F(A)
X δ
Y. So Y α
X 6= 0
F(B)
X β
Y 6= 0
F(A)
, that is (A; B; α
; β
)
is a funcoid. Fr om the fo rmula Y α
X 6= 0
F(B)
X δ
Y it follows that
X [(A; B; α
; β
)]
Y ⇔↑
B
Y α
A
X 6= 0
F(B)
⇔↑
A
X δ
B
Y X δ Y.
1. Let define the relation δ P (P A × PB) by the fo rmula X δ Y ⇔↑
B
Y αX 6= 0
F(B)
.
That ¬( δ I) and ¬(I δ ) is obvious. We have IJ δ K ⇔↑
B
Kα(IJ) 6=
0
F(B)
⇔↑
B
K (αI αJ) 6= 0
F(B)
⇔↑
B
K αI 6= 0
F(B)
B
K αI 6= 0
F(B)
I δ K J δ K and
K δ I J ⇔↑
B
(I J) αK 6= 0
F(B)
B
I
B
J
αK 6= 0
F(B)
B
I αK
B
J αK
6= 0
F(B)
⇔↑
B
I αK 6= 0
F(B)
B
J αK 6=
0
F(B)
K δ I K δ J.
14
That is the formulas (3) are true.
Accordingly the above there exists a funcoid f such that
X [f] Y X up X , Y up Y : X δ Y.
X P A, Y PB :
B
Y hf i
A
X 6= 0
F(B)
⇔↑
A
X [f ]
B
Y X δ Y ⇔↑
B
Y αX 6= 0
F(B)
,
consequently X P A : αX = hf i
A
X = hfi
X.
Note that by the last theorem to every proximity δ corresponds a unique
funcoid. So funcoids are a generaliz ation of (quasi-)proximity structures.
Reverse funcoids c an be cons ide red as a generalization of conjugate quasi-
proximity.
Definition 24 Any small (multivalued) function F : A B corresponds to
a funcoid
FCD(A;B)
F FCD (A; B), where by definition
FCD(A;B)
F
X =
T
B
hhF ii up X for every X F (A).
Using the last theorem it is easy to show that this definition is monovalued
and does not contradict to former stuff. (Take α =
B
hF i.)
Definition 25 Fun coids corresponding to a binary relation (= multivalued func-
tion) are called principal funcoids.
We may eq uate principal funcoids with cor responding binar y r elations by
the method of appendix B in [15]. This is useful for de scribing r elationships o f
funcoids and binary relations, such as for the formulas of continuous functions
and continuous funcoids (see below).
Theorem 8 If S is a generalized lter base on Sr c f then hfi
T
S =
T
hhfii S
for every funcoid f.
Proof hf i
T
S hf i X for e very X S and thus hfi
T
S
T
hhfii S.
By properties of generaliz ed filter bases:
hfi
T
S =
T
hfi
up
T
S =
T
hfi
{X | ∃P S : X up P} =
T
hfi
X | ∃P S : X up P
T
{hfi P | P S} =
T
hhfii S.
3.4 Lattices of funcoids
Definition 26 f g
def
= [f][g] for f, g FCD.
Thus every FCD (A; B) is a po set. (It’s taken into account that [f ]6=[g] if
f 6= g.)
Definition 27 I will call a shifted filtrator of funcoids the s hifted filtrator
(FCD (A; B) ; P (A × B) ;
FCD(A;B)
)
for some small sets A, B.
up f
def
= up
FCD (A; B) ; P (A × B) ;
FCD(A;B)
f for every funcoid f FCD (A; B).
15
Lemma 1 hf i
X =
T
Dst f
hF i X | F up f
for every funcoid f and
set X P (Src f).
Proof Obviously hfi
X
T
Dst f
hF i X | F up f
.
Let B up hfi
X. Let F
B
= X × B ((Src f) \ X) × (Dst f).
hF
B
i X = B.
We have 6= P X hF
B
i P = B hfi
P and 6= P * X hF
B
i P =
Dst f hfi
P . Thus hF
B
i P hf i
P for every set P P (Src f) and so
FCD(Src f ;Dst f)
F
B
f that is F
B
up f.
Thus B up hfi
X : B up
T
Dst f
hF i X | F up f
because B
up
Dst f
hF
B
i X.
So
T
Dst f
hF i X | F up f
hfi
X.
Theorem 9 hf i X =
T

FCD(Src f ;Dst f)
F
X | F up f
for every fun-
coid f and X F (Src f).
Proof
T

FCD(Src f;Dst f )
F
X | F up f
=
T
T
Dst f
hhF ii up X | F up f
=
T
T
Dst f
hF i X | X up X
| F up f
=
T
T
Dst f
hF i X | F up f
| X up X
=
T
Dst f
hfi
X | X up X
= hfi X (the lemma used).
Conjecture 1 Every filtrator of fu ncoids is:
1. with separable core;
2. with co-separable core.
Below it is shown tha t FCD (A; B) are complete lattices for every small sets
A and B. We will apply lattice operatio ns to subsets of such sets without
explicitly mentioning FCD (A; B).
Theorem 10 FCD (A; B) is a complete lattice (for every small sets A and B).
For every R P FCD (A; B) and X P A, Y P B
1. X [
S
R]
Y f R : X [f]
Y ;
2. h
S
Ri
X =
S
hfi
X | f R
.
Proof Accordingly [14] to prove that it is a complete lattice it’s enough to
prove existence of all joins.
2 αX
def
=
S
hfi
X | f R
. We have α = 0
F(Dst f )
;
α(I J) =
[
hfi
(I J) | f R
=
[
hfi
I hfi
J | f R
=
[
hfi
I | f R
[
hfi
J | f R
= αI αJ.
16
So hhi
A
= α for some funcoid h. Obviously
f R : h f. (5)
And h is the least funcoid for which holds the condition (5). So h =
S
R.
1 X [
S
R]
Y ⇔↑
Dst f
Y ∩h
S
Ri
X 6= 0
F(Dst f)
⇔↑
Dst f
Y
S
hfi
X | f R
6=
0
F(Dst f )
f R :
Dst f
Y hfi
X 6= 0
F(Dst f )
f R : X [f]
Y
(used the theorem 40 in [15]).
In the next theorem, compared to the previous one, the class of infinite
unions is replaced w ith lesser class of finite unions and simultaneously class of
sets is changed to more wide class of filter objects.
Theorem 11 For every f, g FCD (A; B) and X F (A) (for every small sets
A, B)
1. hf gi X = hf i X hgi X ;
2. [f g]=[f] [g].
Proof
1. Let αX
def
= hfi X hgi X ; βY
def
=
f
1
Y
g
1
Y for every X F (A),
Y F (B). Then
Y αX 6= 0
F(B)
Y hf i X 6= 0
F(B)
Y hgi X 6= 0
F(B)
X
f
1
Y 6= 0
F(A)
X
g
1
Y 6= 0
F(A)
X βY 6= 0
F(A)
.
So h = (A; B; α; β) is a funcoid. Obviously h f and h g. If p f and
p g for some funcoid p then hpi X hfi X hgi X = hhi X that is p h.
So f g = h.
2. X [f g] Y Y hf gi X 6= 0
F(B)
Y (hfi X hgi X ) 6= 0
F(B)
Y hfi X 6= 0
F(B)
Y hgi X 6= 0
F(B)
X [f] Y X [g] Y for every
X F (A), Y F (B).
3.5 More on composition of funcoids
Propositi on 11 [g f]=[g] hfi =
g
1
1
[f ] for every composable fun-
coids f and g.
17
Proof X [g f] Y Y hg fi X 6= 0
F(Dst g)
Y hgi hfi X 6= 0
F(Dst g)
hfi X [g] Y X ([g] hfi)Y for every X F (Src f), Y F (Dst g). [g f]=
(f
1
g
1
)
1
=
f
1
g
1
1
=
(
f
1
g
1
)
1
=
g
1
1
[f].
The following theorem is a variant for funcoids o f the statement (which
defines compositions of relations) that x(g f)z y(x f y y g z) for every
x and z and every binary relations f and g.
Theorem 12 For every small sets A, B, C and f FCD (A; B) , g FCD (B; C)
and X F (A), Z F (C)
X [g f] Z y atoms 1
F(B)
: (X [f] y y [g] Z).
Proof
y atoms 1
F(B)
: (X [f ] y y [g] Z) y atoms 1
F(B)
:
Z hgi y 6= 0
F(C)
y hfi X 6= 0
F(B)
y atoms 1
F(B)
: (Z hgi y 6= 0
F(C)
y hfi X )
Z hgi hfi X 6= 0
F(C)
X [g f] Z.
Reversely, if X [g f] Z then hfi X [g] Z, consequently exists y atoms hfi X
such that y [g] Z; we have X [f] y.
Theorem 13 For every small sets A, B, C
1. f (g h) = f g f h for g, h FCD (A; B) and f FCD (B; C);
2. (g h ) f = g f h f for g, h FCD (B; C) and f FCD (A; B).
Proof I will prove only the first equality because the other is analo gous.
For every X F(A), Z F(C)
X [f (g h)] Z y a toms 1
F(B)
: (X [g h ] y y [f] Z)
y atoms 1
F(B)
: ((X [g] y X [h] y) y [f] Z)
y atoms 1
F(B)
: (X [g] y y [f] Z X [h] y y [f] Z)
y atoms 1
F(B)
: (X [g] y y [f] Z) y atoms 1
F(B)
: (X [h] y y [f ] Z)
X [f g] Z X [f h] Z
X [f g f h] Z.
Conjecture 2 g f =
T
FCD(Src f;Dst g)
(G F ) | F up f, G up g
for
every composable funcoids f and g.
18
3.6 Domain and range of a funcoid
Definition 28 Let A be a small set. The identity funcoid I
FCD( A )
=
A; A; (=)|
F(A)
; (=)|
F(A)
.
Obvious 9 . The identity funcoid is a funcoid.
Definition 29 Let A be a small set, A F (A). The restricted identity
funcoid
I
FCD
A
= (A; A; A∩; A∩).
Propositi on 12 The restricted identity fun coid is a funcoid.
Proof We need to prove that (A X ) Y 6= 0
F(A)
(A Y) X 6= 0
F(A)
what is obvious.
Obvious 1 0.
1.
I
FCD(A)
1
= I
FCD( A )
;
2.
I
FCD
A
1
= I
FCD
A
.
Obvious 1 1. For every X , Y F (A)
1. X
I
FCD(A)
Y X Y 6= 0
F(A)
.
2. X
I
FCD
A
Y A X Y 6= 0
F(A)
.
Definition 30 I will define restricting of a funcoid f to a filter object A
F (Src f) by the formula
f|
A
def
= f I
FCD
A
.
Definition 31 Image of a funcoid f wil l be defined by the formula im f =
hfi 1
F(Src f)
.
Domain of a funcoid f is defined by the formula dom f = im f
1
.
Propositi on 13 hf i X = hf i (X dom f) for every f FCD, X F (Src f).
Proof For every Y F (Dst f) we have Y hfi (X dom f) 6= 0
F(Dst f )
X dom f
f
1
Y 6= 0
F(Src f )
X im f
1
f
1
Y 6= 0
F(Src f)
X
f
1
Y 6= 0
F(Src f )
Y hfi X 6= 0
F(Dst f )
. Thus hf i X = hf i (X dom f)
because the lattice of filter objects is separable.
Propositi on 14 X dom f 6= 0
F(Src f )
hfi X 6= 0
F(Dst f )
for every f FCD,
X F (Src f).
19
Proof X dom f 6= 0
F(Src f )
X
f
1
1
F(Dst f )
6= 0
F(Src f )
1
F(Dst f )
hfi X 6= 0
F(Dst f )
hfi X 6= 0
F(Dst f )
.
Corollary 2 dom f =
S
a atoms 1
F(Src f)
| hf i a 6= 0
F(Dst f )
.
Proof T his follows from the fact that F (Src f) is an atomistic lattice.
Propositi on 15 dom f|
A
= A dom f for every funcoid f and A F (Src f).
Proof dom f |
A
= im
I
FCD
A
f
1
=
I
FCD
A
f
1
1
(Dst f )
= A∩
f
1
1
(Dst f )
=
A dom f.
Theorem 14 im f =
T
Dst f
himi up f and dom f =
T
Src f
hdomi up f
for every funcoid f.
Proof im f = hf i 1
F(Src f)
=
T

FCD(Src f;Dst f )
F
1
F(Src f )
| F up f
=
T
Dst f
im F | F up f
=
T
Dst f
himi up f (used the theorem 9).
The sec ond formula follows from symmetry.
Propositi on 16 For every composable funcoids f , g:
1. If im f dom g then im (g f) = im g.
2. If im f dom g then dom (g f) = dom f.
Proof
1. im (g f ) = hg f i 1
F(Src f)
= hgi hfi 1
F(Src f )
= hgi im f = hgi (im f dom g) =
hgi dom g = hgi 1
F(Src g)
= im g.
2. dom (g f ) = im
f
1
g
1
what by proved above is equal to im f
1
that
is dom f.
3.7 Categories of funcoids
I will define two categories, the category of funcoids and the category of
funcoid triples.
The category of funcoids is defined as follows:
Objects a re small sets.
The s e t of mo rphisms from a set A to a set B is FCD (A; B).
The composition is the composition of funcoids.
Identity morphism for a set is the identity funcoid for that set.
20
To show it is really a category is trivial.
The category of funcoid triples is defined as follows:
Objects a re filter objects on small sets.
The mor phisms from a f.o . A to a f.o. B are triples (f ; A; B) where
f FCD (Base (A) ; Base (B)) and dom f A im f B.
The co mpo sition is defined by the formula (g; B; C)(f; A; B) = (g f; A; C).
Identity morphism for an f.o. A is I
FCD
A
.
To prove that it is really a c ategory is trivial.
3.8 Specifying funcoids by functions or relations on atomic
filter objects
Theorem 15 For every funcoid f and X F (Src f), Y F (Dst f)
1. hfi X =
S
hhfii atoms X ;
2. X [f] Y x atoms X , y atoms Y : x [f] y.
Proof 1.
Y hfi X 6= 0
F(Dst f )
X
f
1
Y 6= 0
F(Src f )
x atoms X : x
f
1
Y 6= 0
F(Src f )
x atoms X : Y hfi x 6= 0
F(Dst f )
.
hfi X =
S
hi hhfii atoms X =
S
hhfii atoms X .
2. If X [f ] Y, then Y hf i X 6= 0
F(Dst f )
, consequently ex ists y atoms Y
such that y hfi X 6= 0
F(Dst f)
, X [f] y. Repeating this second time we get that
there exists x atoms X such that x [f] y. From this follows
x atoms X , y atoms Y : x [f] y.
The reverse is obvious.
Theorem 16 Let A and B be small sets.
1. A function α F (B)
atoms 1
F(A)
such that (for every a atoms 1
F(A)
)
αa
\
D
[
hαi atoms
A
E
up a (6)
can be continued to the function hf i for a unique f FCD (A; B);
hfi X =
[
hαi atoms X (7)
for every X F