Filters on posets and generalizations
Victor Porton
78640, Shay Agnon 32-29, Ashkelon, Israel
Abstract
They are studied in details properties of filters on lattices, filters on posets, and
certain generalizations thereof. Also it’s done some more genera l lattice theory
resear ch. There are posed several open problems. Detailed study of filters is
required for my ongoing research which will be published as ” Algebraic General
Topology” series.
Keywords: filters, ideals, lattice of filters, pseudodifference,
pseudocomplement
A.M.S. subject classification: 54A20, 06A06, 06B99
This is a pre print (incorporating errata) of an article [12] published in In-
ternational Journal of Pure and Applied Mathema tics (IJPAM).
Contents
1 Preface 3
2 Notation and basic results 3
2.1 Intersecting and joining elements . . . . . . . . . . . . . . . . . . 4
2.2 Atoms of a poset . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Difference and complement . . . . . . . . . . . . . . . . . . . . . 6
2.4 Center of a la ttice . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Galois connections . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Co-Bro uwerian Lattices . . . . . . . . . . . . . . . . . . . . . . . 11
3 Straight maps and separation subsets 14
3.1 Straight maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Separation subsets and full sta rs . . . . . . . . . . . . . . . . . . 16
3.3 Atomically separable lattices . . . . . . . . . . . . . . . . . . . . 17
Email address: porton@narod.ru (Victor Porton)
URL: http://www.mathematics21.org (Victor Porton)
Preprint submitted to Elsevier January 14, 2014
4 Filtrators 19
4.1 Core part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Filtrators with separable core . . . . . . . . . . . . . . . . . . . . 21
4.3 Intersecting and joining with an element o f the core . . . . . . . 22
5 Filters 22
5.1 Filters on posets . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Filters on meet-semilattice . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Character iz ation of finitely mee t-closed filtrato rs . . . . . . . . . 24
6 Filter objects 24
6.1 Definition of filter objects . . . . . . . . . . . . . . . . . . . . . . 25
6.2 Order of filter objects . . . . . . . . . . . . . . . . . . . . . . . . 25
7 Lattice of filter objects 26
7.1 Minimal and maximal f.o. . . . . . . . . . . . . . . . . . . . . . . 26
7.2 Primary filtrator is filtered . . . . . . . . . . . . . . . . . . . . . . 26
7.3 Formulas for meets and joins of filter objects . . . . . . . . . . . 26
7.4 Distributivity of the la ttice of filter objects . . . . . . . . . . . . 28
7.5 Separability of core for primary filtrators . . . . . . . . . . . . . . 29
7.6 Filters over boolean lattices . . . . . . . . . . . . . . . . . . . . . 30
7.7 Distributivity for an element of boolean co re . . . . . . . . . . . 30
8 Generalized filter base 31
9 Stars 32
9.1 Free stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.2 Stars of elements of filtra tors . . . . . . . . . . . . . . . . . . . . 33
9.3 Stars of filters on boolea n lattices . . . . . . . . . . . . . . . . . . 34
9.4 More about the lattice of filters . . . . . . . . . . . . . . . . . . . 36
10 Atomic filter objects 37
10.1 Prime filtrator elements . . . . . . . . . . . . . . . . . . . . . . . 38
11 Some criteria 39
12 Quasidifference and quasicomp lement 43
13 Complements and core parts 46
13.1 Core part and atomic elements . . . . . . . . . . . . . . . . . . . 48
14 Distributivity of core part over lattice operations 48
15 Fr´echet filter 50
16 Complementive fil ter objects and factoring by a filter 51
17 Number of filters on a s et 53
2
18 Partitioning filter objects 53
19 Open problems 54
19.1 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
19.2 Quasidifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
19.3 Non-formal problems . . . . . . . . . . . . . . . . . . . . . . . . . 56
Appendix A Some counter-examples 56
Appendix A.1 Weak and strong partition . . . . . . . . . . . . . . 57
Appendix B Logic of Generalizations 59
Appendix B.1 The formalistic . . . . . . . . . . . . . . . . . . . . 59
1. Preface
This article is intended to collect in one document the known properties of
filters on posets (and so me generalizations thereof, namely “filtrators” defined
below).
It seems that until now were published no reference on the theory of filters.
This text is to fill the gap.
This text will also serve as the reference base for my further articles. This
text provides a definitive place to refer as to the collection of theorems about
filters.
Detailed study of filters is required for my ongoing research w hich will be
published as ”Algebraic General Topology” series.
In place of studying filters in this article are instea d researched what the au-
thor calls “filter objects”. Filter objects a re basically the lattice of filters orde red
reverse to set inclusion, with principal filters equated with the poset element
which generates them. (See be low for formal definition of “filter objects”.)
Although our primary interest are properties of filters on a set, in this work
are instead re searched the mor e gene ral theory of “filtra tors” (see below).
This article also contains some original research:
• filtrators ;
• straight maps and separation subsets;
• other minor results, such as the theory of free stars.
2. Notation and basic results
We denote PS the set of all subsets of a set S.
hfi X
def
= {f x | x ∈ X} for any set X and function f.
3
2.1. Intersecting and joining element s
Let A be a poset.
Definition 1 I will call elements a and b of A intersecting and denote a 6≍ b
when exists not least element c such that c ⊆ a ∧ c ⊆ b.
Definition 2 a ≍ b
def
= ¬(a 6≍ b).
Obvious 1 If A is a meet-semilattice then a 6≍ b iff a ∩ b is non-least.
Obvious 2 a
0
6≍ b
0
∧ a
1
⊇ a
0
∧ b
1
⊇ b
0
⇒ a
1
6≍ b
1
.
Definition 3 I will call elements a and b of A joining and denote a ≡ b when
not exists not greatest element c such that c ⊇ a ∧ c ⊇ b.
Definition 4 a 6≡ b
def
= ¬(a ≡ b).
Obvious 3 Intersecting is the dual of non- joining.
Obvious 4 If A is a join-semilattice then a ≡ b iff a ∪ b is its greatest element.
Obvious 5 a
0
≡ b
0
∧ a
1
⊇ a
0
∧ b
1
⊇ b
0
⇒ a
1
≡ b
1
.
2.2. Atoms of a poset
Definition 5 An atom of the poset is an element which has no non-least subele-
ments.
Remark 1 This definition is valid even for posets without lea st element.
I will denote (atoms
A
a) or just (a toms a) the set of atoms contained in
element a of a poset A.
Definition 6 A poset A is called atomic when atoms a 6= ∅ for every non-least
element a ∈ A.
Definition 7 Atomistic poset is such poset that a =
S
atoms a for every
element a of this poset.
Proposition 1 Let A be a poset. If a is an atom of A and B ∈ A then a ⊆
B ⇔ a 6≍ B.
Proof
⇒ a ⊆ B ⇒ a ⊆ a ∧ a ⊆ B, thus a 6≍ B because a is not least.
⇐ a 6≍ B implies existence of non-least element x such that x ⊆ B and x ⊆ a.
Because a is an atom, we have x = a. So a ⊆ B.
4
Theorem 1 If A is a distributive lattice then
atoms(a ∪ b) = atoms a ∪ atoms b
for every a, b ∈ A.
Proof For any atomic element c
c ∈ atoms(a ∪ b) ⇔
c ∩ (a ∪ b) is not least ⇔
(c ∩ a) ∪ (c ∩ b) is not least ⇔
c ∩ a is not least or c ∩ b is not least ⇔
c ∈ atoms a ∨ c ∈ ato ms b.
Theorem 2 a toms
T
S =
T
hatomsi S whenever
T
S is defined for every S ∈
PA where A is a poset.
Proof For any atom c
c ∈ atoms
\
S ⇔
c ⊆
\
S ⇔
∀a ∈ S : c ⊆ a ⇔
∀a ∈ S : c ∈ atoms a ⇔
c ∈
\
hatomsi S.
Corollary 1 atoms(a ∩ b) = atoms a ∩ atoms b for arbitrary meet-semilattice.
Theorem 3 A complete boolean lattice is atomic iff it is atomistic.
Proof
⇐ Obvious.
⇒ Let A be an atomic boolean lattice. Let a ∈ A. Suppose b =
S
atoms a ⊂ a.
If x ∈ atoms(a \ b) then x ⊆ a \ b and so x ⊆ a and hence x ⊆ b. But we
have x = x ∩ b ⊆ (a \ b) ∩ b = 0 what contradicts to our s upposition.
5
2.3. Difference and complement
Definition 8 Let A be a distributive lattice with least element 0. The differ-
ence (denoted a \ b) of elements a and b is such c ∈ A that b ∩ c = 0 and
a ∪ b = b ∪ c. I will call b substractive from a when a \ b exists.
Theorem 4 If A is a distributive lattice with least element 0, there exists no
more than one difference of elements a, b ∈ A.
Proof Let c and d are both differences a \ b. Then b ∩ c = b ∩ d = 0 and
a ∪ b = b ∪ c = b ∪ d. So
c = c ∩ (b ∪ c) = c ∩ (b ∪ d) = (c ∩ b) ∪ (c ∩ d) = 0 ∪ (c ∩ d) = c ∩ d.
Analogously, d = d ∩ c. Consequently c = c ∩ d = d ∩ c = d.
Definition 9 I will call b complementive to a when t here exist s c ∈ A such
that b ∩ c = 0 and b ∪ c = a.
Proposition 2 b is complementive to a iff b is substractive from a and b ⊆ a.
Proof
⇐ Obvious.
⇒ We deduce b ⊆ a from b ∪ c = a. Thus a ∪ b = a = b ∪ c.
Proposition 3 If b is complementive to a then (a \ b) ∪ b = a .
Proof Because b ⊆ a by the previous proposition.
Definition 10 Let A be a bounded distributive lattice. The complement (de-
noted ¯a) of element a ∈ A is such b ∈ A that a ∩ b = 0 and a ∪ b = 1.
Proposition 4 If A is a bounded distributive lattice then ¯a = 1 \ a.
Proof b = ¯a ⇔ b ∩ a = 0 ∧ b ∪ a = 1 ⇔ b ∩ a = 0 ∧ 1 ∪ a = a ∪ b ⇔ b = 1 \ a.
Corollary 2 If A is a bounded distributive lattice then exists no more than one
complement of an element a ∈ A.
Definition 11 An element of bounded distributive latt ice is called comple-
mented when its complement exists.
Definition 12 A distributive lattice is a complemented lattice iff every its
element is complemented.
6
Proposition 5 For a distributive latt ice (a \ b) \ c = a \ (b ∪ c) if a \ b and
(a \ b) \ c are defined.
Proof ((a \ b) \ c) ∩ c = 0; ((a \ b) \ c) ∪ c = (a \ b) ∪ c; (a \ b) ∩ b = 0;
(a \ b) ∪ b = a ∪ b.
We need to prove ((a\b) \c)∩(b∪c) = 0 and ((a\ b)\c)∪(b∪c) = a∪(b∪c).
In fact,
((a \ b) \ c) ∩ (b ∪ c) =
(((a \ b) \ c) ∩ b) ∪ (((a \ b) \ c) ∩ c) =
(((a \ b) \ c) ∩ b) ∪ 0 =
((a \ b) \ c) ∩ b ⊆
(a \ b) ∩ b = 0,
so ((a \ b) \ c) ∩ (b ∪ c) = 0;
((a \ b) \ c) ∪ (b ∪ c) =
(((a \ b) \ c) ∪ c) ∪ b =
(a \ b) ∪ c ∪ b =
((a \ b) ∪ b) ∪ c =
a ∪ b ∪ c.
2.4. Center of a lattice
Definition 13 The center Z(A) of a bounded distributive lattice A is the set
of its complemented elements.
Remark 2 For definition of center of non-distributive lattices see [3].
Remark 3 In [9] the word center and the notation Z(A) is used in a different
sense.
Definition 14 A complete lattice A is join infinite distributive when x ∩
S
S =
S
hx∩i S; complete lattice is meet infinite distributive when x∪
T
S =
T
hx∪i S for all x ∈ A and S ∈ PA.
Definition 15 Infinitely distributive complete lattice is a complete lattice
which is both join infinite distributive and meet infinite distributive.
Definition 16 A sublattice K of a complete lattice L is a closed sublattice of
L if K contains the meet and the join of any its nonempty subset.
Theorem 5 Center of a infinitely distributive lattice is its closed sublattice.
7
Proof See [6].
Remark 4 See [7] for a more strong result.
Theorem 6 The center of a bounded distributive lattice constitutes its sublat-
tice.
Proof Let A be a bounded distributive lattice and Z(A) is its center. Let
a, b ∈ Z(A). Conseq ue ntly ¯a,
¯
b ∈ Z(A). Then ¯a ∪
¯
b is the complement of a ∩ b
because
(a ∩ b) ∩ (¯a ∪
¯
b) = (a ∩ b ∩ ¯a) ∪ (a ∩ b ∩
¯
b) = 0 ∪ 0 = 0 and
(a ∩ b) ∪ (¯a ∪
¯
b) = (a ∪ ¯a ∪
¯
b) ∩ (b ∪ ¯a ∪
¯
b) = 1 ∩ 1 = 1.
So a ∩ b is complemented, analogously a ∪ b is complemented.
Theorem 7 The center of a bounded distributive lattice constitutes a boolean
lattice.
Proof Because it is a distributive complemented lattice.
2.5. Galois connections
See [1] and [5] for more detailed treatment of Galois connections.
Definition 17 Let A and B be two posets. A Galois connection between A
and B is a pair of functions f = (f
∗
; f
∗
) with f
∗
: A → B and f
∗
: B → A such
that:
∀x ∈ A, y ∈ B : (f
∗
x ⊆
B
y ⇔ x ⊆
A
f
∗
y).
f
∗
is called upper adjoint of f
∗
and f
∗
is called lower adjoint of f
∗
.
Theorem 8 A pair (f
∗
; f
∗
) of functions f
∗
: A → B and f
∗
: B → A is a
Galois connection iff both of the following:
1. f
∗
and f
∗
are monotone.
2. x ⊆
A
f
∗
f
∗
x and f
∗
f
∗
y ⊆
B
y for every x ∈ A and y ∈ B.
Proof
⇒ 2. x ⊆
A
f
∗
f
∗
x since f
∗
x ⊆
B
f
∗
x; f
∗
f
∗
y ⊆
B
y since f
∗
y ⊆
A
f
∗
y.
1. Let a, b ∈ A and a ⊆
A
b. Then a ⊆
A
b ⊆
A
f
∗
f
∗
b. So by definition
f
∗
a ⊆ f
∗
b that is f
∗
is monotone. Analogously f
∗
is monotone.
⇐ f
∗
x ⊆
B
y ⇒ f
∗
f
∗
x ⊆
A
f
∗
y ⇒ x ⊆
A
f
∗
y. The other direction is analogous.
8
Theorem 9
1. f
∗
◦ f
∗
◦ f
∗
= f
∗
.
2. f
∗
◦ f
∗
◦ f
∗
= f
∗
.
Proof
1. Let x ∈ A. We have x ⊆
A
f
∗
f
∗
x; consequently f
∗
x ⊆
B
f
∗
f
∗
f
∗
x. On the
other hand, f
∗
f
∗
f
∗
x ⊆
B
f
∗
x. So f
∗
f
∗
f
∗
x = f
∗
x.
2. Analogous ly.
Proposition 6 f
∗
◦ f
∗
and f
∗
◦ f
∗
are idempotent.
Proof f
∗
◦ f
∗
is idempotent bec ause f
∗
f
∗
f
∗
f
∗
y = f
∗
f
∗
y. f
∗
◦ f
∗
is similar.
Theorem 10 Each of two adjoints is uniquely determined by the other.
Proof Let p and q be both upper adjoints of f. We have for all x ∈ A and
y ∈ B:
x ⊆ p(y) ⇔ f(x) ⊆ y ⇔ x ⊆ q(y).
For x = p(y) we obtain p(y) ⊆ q(y) and for x = q(y) we obtain q(y) ⊆ p(y). So
p(y) = q(y).
Theorem 11 Let f be a function from a poset A t o a poset B.
1. Both:
1. If f is monotone and g(b) = max {x ∈ A | fx ⊆ b} is defined for every
b ∈ B then g is the upper adjoint of f .
2. If g : B → A is the upper adjoint of f then g(b) = max {x ∈ A | fx ⊆ b}
for every b ∈ B.
2. Both:
1. If f is monotone and g(b) = min {x ∈ A | f x ⊇ b} is defined for every
b ∈ B then g is the lower adjoint of f.
2. If g : B → A is the lower adjoint of f then g(b) = min {x ∈ A | fx ⊇ b}
for every b ∈ B.
Proof We will prove only the first as the second is its dual.
9
1. Let g(b) = max {x ∈ A | fx ⊆ b} for every b ∈ B. Then
x ⊆ gy ⇔ x ⊆ max { x ∈ A | fx ⊆ y} ⇒ fx ⊆ y
(because f is monotone) and
x ⊆ gy ⇔ x ⊆ max {x ∈ A | fx ⊆ y} ⇐ fx ⊆ y.
So fx ⊆ y ⇔ x ⊆ gy that is f is the lower adjoint o f g.
2. We have
g(b) = max {x ∈ A | fx ⊆ b} ⇔
fgb ⊆ b ∧ ∀x ∈ A : (fx ⊆ b ⇒ x ⊆ gb)
what is true by properties of adjoints.
Theorem 12 Let f be a function from a poset A t o a poset B.
1. If f is an upper adjoint, f preserves all existing infima in A.
2. If A is a complete lattice and f preserves all infima, then f is an upper
adjoint of a function B → A.
3. If f is a lower adjoint, f preserves all existing suprema in A.
4. If A is a complete lattice and f preserves all suprema, then f is a lower
adjoint of a function B → A.
Proof We will prove only first two items be c ause the rest items are similar.
1. Let S ∈ PA and
T
S exists . f
T
S is a lower bound for h f i S because f is
order-preserving. If a is a lower bound for hfi S then ∀x ∈ S : a ⊆ fx that is
∀x ∈ S : x ⊆ ga where g is the lower adjoint of f . Thus
T
S ⊆ ga and hence
f
T
S ⊆ a. So f
T
S is the greatest lower bound fo r hfi S.
2. Let A be a complete lattice and f preser ves all infima. Let g(a) =
T
{x ∈ A | fx ⊇ a}.
Since f prese rves infima, we have
f(g(a)) =
\
{f(x) | x ∈ A, f(x) ⊇ a} ⊇ a.
g(f(b)) =
T
{x ∈ A | f x ⊇ fb} ⊆ b.
Obviously f is monotone and thus g is also mo notone.
So f is the upper adjoint of g.
Corollary 3 Let f be a function from a complete lattice A to a poset B. Then:
1. f is an upper adjoint of a function B → A iff f preserves all infima in A.
2. f is a lower adjoint of a function B → A iff f preserves all suprema in A.
10
2.6. Co-Brouwerian Lattices
Definition 18 Let A be a poset. Let a ∈ A. Pseudocomplement of a is
max {c ∈ A | c ≍ a} .
If z is pseudocomplement of a we will denote z = a
∗
.
Definition 19 Let A be a poset. Let a ∈ A. Dual pseudocomplement of a
is
min {c ∈ A | c ≡ a} .
If z is dual pseudocomplement of a we will denote z = a
+
.
Definition 20 Let A be a join-semilattice. Let a, b ∈ A. Pseudodifference of
a and b is
min {z ∈ A | a ⊆ b ∪ z} .
If z is a pseudodifference of a and b we will denote z = a \
∗
b.
Remark 5 I do not require that a
∗
is undefined if there are no pseudocom-
plement of a and likewise for dual pseudocomplement and pseudodifference. In
fact below I will define quasicomplement, dual quasicomplement, and quasidif-
ference which will generalize pseudo-* counterparts. I will denote a
∗
the more
general case of quasicomplement than of pseudocomplement, and likewise for
other notation.
Obvious 6 Dual pseudocomplement is the dual of pseudocomplement.
Definition 21 Co-brouwerian lattice is a lattice for which is defined pseu-
dodifference of any two its elements.
Proposition 7 Every non-empty co-brouwerian lattice A has least element.
Proof Let a be an arbitrary lattice element. Then a\
∗
a = min {z ∈ A | a ⊆ a ∪ z} =
min A. So min A exists.
Definition 22 Co-Heyting lattice is co-brouwerian latt ice with greatest ele-
ment.
Theorem 13 For a co-brouwerian lattice a ∪ − is an upper adjoint of − \
∗
a
for every a ∈ A.
Proof g(b) = min {x ∈ A | a ∪ x ⊇ b} = b \
∗
a exists for every b ∈ A and thus
is the lower adjoint of a ∪ −.
Corollary 4 ∀a, x, y ∈ A : (x\
∗
a ⊆ y ⇔ x ⊆ a ∪ y) for a co-brouwerian lattice.
11
Definition 23 Let a, b ∈ A where A is a complete lattice. Quasidifference a\
∗
b
is defin ed by the formula
a \
∗
b =
\
{z ∈ A | a ⊆ b ∪ z} .
Remark 6 The more detailed theory of quasidifference (as well as quasicom-
plement and dual quasicomplement) will be considered below.
Lemma 1 (a \
∗
b) ∪ b = a ∪ b for elements a, b of a meet infinite distributive
complete lattice.
Proof
(a \
∗
b) ∪ b =
\
{z ∈ A | a ⊆ b ∪ z} ∪ b =
\
{z ∪ b | z ∈ A, a ⊆ b ∪ z} =
\
{t ∈ A | t ⊇ b, a ⊆ t} =
a ∪ b.
Theorem 14 The following are equivalent for a complete latt ice A:
1. A is meet infinite distributive.
2. A is a co-brouwerian lattice.
3. A is a co-Heyting lattice.
4. a ∪ − has lower adjoint for every a ∈ A.
Proof
(2)⇔(3) Obvious (taking in acc ount completeness of A).
(4)⇒(1) Let −\
∗
a be the lower adjoint of a∪−. Let S ∈ PA. For every y ∈ S
we have y ⊇ (a ∪ y) \
∗
a by properties o f Galois connections; consequently
y ⊇ (
T
ha∪i S) \
∗
a;
T
S ⊇ (
T
ha∪i S) \
∗
a. So
a ∪
\
S ⊇ ((
\
ha∪i S) \
∗
a) ∪ a ⊇
\
ha∪i S.
But a ∪
T
S ⊆
T
ha∪i S is obvious.
(1)⇒(2) Let a\
∗
b =
T
{z ∈ A | a ⊆ b ∪ z}. To prove that A is a co- brouwerian
lattice is enough to prove that a ⊆ b ∪ (a \
∗
b). But it follows from the
lemma.
12
(2)⇒(4) a \
∗
b = min {z ∈ A | a ⊆ b ∪ z}. So a ∪ − is an upper adjoint of
− \
∗
a.
(1)⇒(4) Because a ∪ − preserves all meets.
Corollary 5 Co-brouwerian lattices are distributive.
The following theorem is essentially borrowed from [8 ]:
Theorem 15 A lattice A with least element 0 is co-brouwerian with pseudodif-
ference \
∗
iff \
∗
is a binary operation on A sat isfying the following identities:
1. a \
∗
a = 0;
2. a ∪ (b \
∗
a) = a ∪ b;
3. b ∪ (b \
∗
a) = b;
4. (b ∪ c) \
∗
a = (b \
∗
a) ∪ (c \
∗
a).
Proof
⇐ We have
c ⊇ b \
∗
a ⇒ c ∪ a ⊇ a ∪ (b \
∗
a) = a ∪ b ⊇ b;
c ∪ a ⊇ b ⇒ c = c ∪ (c \
∗
a) ⊇ (a \
∗
a) ∪ (c \
∗
a) = (a ∪ c) \
∗
a ⊇ b \
∗
a.
So c ⊇ b \
∗
a ⇔ c ∪ a ⊇ b that is a ∪ − is an upper adjoint of − \
∗
a. By
a theorem above our lattice is co-brouwerian. By an other theorem above
\
∗
is a ps eudodifference.
⇒ 1. Obvious.
2.
a ∪ (b \
∗
a) =
a ∪
\
{z ∈ A | b ⊆ a ∪ z} =
\
{a ∪ z | z ∈ A, b ⊆ a ∪ z} =
a ∪ b.
3. b∪(b\
∗
a) = b∪
T
{z ∈ A | b ⊆ a ∪ z} =
T
{b ∪ z | z ∈ A, b ⊆ a ∪ z} =
b.
13
4. Obviously (b∪c)\
∗
a ⊇ b\
∗
a and (b∪c)\
∗
a ⊇ c\
∗
a, thus (b∪c )\
∗
a ⊇
(b \
∗
a) ∪ (c \
∗
a). We have
(b \
∗
a) ∪ (c \
∗
a) ∪ a =
((b \
∗
a) ∪ a) ∪ ((c \
∗
a) ∪ a) =
(b ∪ a) ∪ (c ∪ a) =
a ∪ b ∪ c ⊇
b ∪ c.
From this by the definition of adjoints: (b \
∗
a)∪(c\
∗
a) ⊇ (b ∪c)\
∗
a.
Theorem 16 (
S
S) \
∗
a =
S
{x \
∗
a | x ∈ S} for a ∈ A and S ∈ PA where A
is a complete co-brouwerian lattice.
Proof Because lower adjoint preserves all suprema.
Theorem 17 (a \
∗
b) \
∗
c = a \
∗
(b ∪ c) for elements a, b, c of a complete
co-brou werian lattice.
Proof a \
∗
b =
T
{z ∈ A | a ⊆ b ∪ z}.
(a \
∗
b) \
∗
c =
T
{z ∈ A | a \
∗
b ⊆ c ∪ z}.
a \
∗
(b ∪ c) =
T
{z ∈ A | a ⊆ b ∪ c ∪ z}.
It’s left to prove a \
∗
b ⊆ c ∪ z ⇔ a ⊆ b ∪ c ∪ z.
Let a \
∗
b ⊆ c ∪ z. Then a ∪ b ⊆ b ∪ c ∪ z by the lemma and consequently
a ⊆ b ∪ c ∪ z.
Let a ⊆ b ∪ c ∪ z. Then a \
∗
b ⊆ (b ∪ c ∪ z) \
∗
b ⊆ c ∪ z by a theorem above.
3. Straight maps and separation subsets
3.1. Straight maps
Definition 24 Let f be a monotone map from a meet-semilattice A to some
poset B. I call f a straight m ap when
∀a, b ∈ A : (f a ⊆ fb ⇒ fa = f (a ∩ b)).
Proposition 8 The following statements are equivalent for a monotone map
f:
1. f is a straight map.
2. ∀a, b ∈ A : (f a ⊆ fb ⇒ fa ⊆ f(a ∩ b)).
3. ∀a, b ∈ A : (f a ⊆ fb ⇒ fa 6 ⊃f(a ∩ b)).
14
4. ∀a, b ∈ A : (f a ⊃ f(a ∩ b) ⇒ f a * f b).
Proof
(1)⇔(2)⇔(3) Due fa ⊇ f(a ∩ b).
(3)⇔(4) Obvious.
Remark 7 The definition of str aight map can be generalized for any poset A
by the formula
∀a, b ∈ A : (fa ⊆ fb ⇒ ∃c ∈ A : (c ⊆ a ∧ c ⊆ b ∧ f a = fc)).
This generalization is not yet researched however.
Proposition 9 Let f be a monotone map from a meet-semilattice A to some
poset B. If
∀a, b ∈ A : (f (a ∩ b) = fa ∩ fb)
then f is a straight map.
Proof Let fa ⊆ fb. Then f (a ∩ b) = f a ∩ fb = fa.
Proposition 10 Let f be a monotone map from a meet-semilattice A to some
poset B. If
∀a, b ∈ A : (f a ⊆ fb ⇒ a ⊆ b)
then f is a straight map.
Proof fa ⊆ fb ⇒ a ⊆ b ⇒ a = a ∩ b ⇒ f a = f(a ∩ b).
Theorem 18 If f is a straight monotone map from a meet-semilattice A then
the following statements are equivalent:
1. f is an injection.
2. ∀a, b ∈ A : (f a ⊆ fb ⇒ a ⊆ b).
3. ∀a, b ∈ A : (a ⊂ b ⇒ f a ⊂ fb).
4. ∀a, b ∈ A : (a ⊂ b ⇒ f a 6= fb).
5. ∀a, b ∈ A : (a ⊂ b ⇒ f a + fb).
6. ∀a, b ∈ A : (f a ⊆ fb ⇒ a 6 ⊃b).
Proof
15
(1)⇒(3) Let a, b ∈ A. Let fa = fb ⇒ a = b. Let a ⊂ b. f a 6= fb because
a 6= b. f a ⊆ fb because a ⊆ b. So f a ⊂ fb.
(2)⇒(1) Let a, b ∈ A. Let fa ⊆ fb ⇒ a ⊆ b. Let fa = fb. Then a ⊆ b ∧ b ⊆ a
and consequently a = b.
(3)⇒(2) Let ∀a, b ∈ A : (a ⊂ b ⇒ fa ⊂ f b). Let a * b. Then a ⊃ a ∩ b. So
fa ⊃ f(a ∩ b). If f a ⊆ f b then fa ⊆ f(a ∩ b) what is a contradiction.
(3)⇒(5)⇒(4) Obvious.
(4)⇒(3) Because a ⊂ b ⇒ a ⊆ b ⇒ fa ⊆ fb.
(5)⇔(6) Obvious.
3.2. Separation subset s and full stars
Definition 25 ∂
Y
a = {x ∈ Y | x 6≍ a} for an element a of a poset A and
Y ∈ PA.
Definition 26 Full star of a is ⋆a = ∂
A
a.
Proposition 11 If A is a meet-semilattice, then ⋆ is a straight monotone map.
Proof Monotonicity is obvious. Let ⋆a * ⋆(a ∩ b). Then it exists x ∈ ⋆a such
that x /∈ ⋆(a ∩ b). So x ∩ a /∈ ⋆b but x ∩ a ∈ ⋆a and conseq ue ntly ⋆a * ⋆b.
Definition 27 A separation subset of a poset A is such its subset Y that
∀a, b ∈ A : (∂
Y
a = ∂
Y
b ⇒ a = b).
Definition 28 I call separable such poset that ⋆ is an injection.
Obvious 7 A poset is separable iff it has separation subset.
Definition 29 A poset A has disjunction property of Wallman iff for any
a, b ∈ A either b ⊆ a or there exists a non-least element c ⊆ b such that a ≍ c.
Theorem 19 For a meet-semilattice with least element the following statements
are equivalent:
1. A is separable.
2. ∀a, b ∈ A : (⋆a ⊆ ⋆b ⇒ a ⊆ b).
3. ∀a, b ∈ A : (a ⊂ b ⇒ ⋆a ⊂ ⋆b).
4. ∀a, b ∈ A : (a ⊂ b ⇒ ⋆a 6= ⋆b).
16
5. ∀a, b ∈ A : (a ⊂ b ⇒ ⋆a + ⋆b).
6. ∀a, b ∈ A : (⋆a ⊆ ⋆b ⇒ a 6 ⊃b).
7. A conforms to Wallman’s disjunction property.
8. ∀a, b ∈ A : (a ⊂ b ⇒ ∃c ∈ A \ {0} : (c ≍ a ∧ c ⊆ b)).
Proof
(1)⇔(2)⇔(3)⇔(4)⇔(5)⇔(6) By the above theorem.
(8)⇒(4) Let the property (8) holds. Let a ⊂ b. Then it exists ele ment c ⊆ b
such that c 6= 0 and c ∩ a = 0. But c ∩ b 6= 0. So ⋆a 6= ⋆b.
(2)⇒(7) Let the property (2) holds. Let a * b. Then ⋆a * ⋆b that is exists
c ∈ ⋆a such that c /∈ ⋆b, in other words c ∩ a 6= 0 and c ∩ b = 0. Let
d = c ∩ a. Then d ⊆ a and d 6= 0 and d ∩ b = 0. So disjunction property
of Wallman holds.
(7)⇒(8) Obvious.
(8)⇒(7) Let b * a. Then a∩b ⊂ b that is a
′
⊂ b where a
′
= a∩b. Consequently
∃c ∈ A \ {0} : (c ≍ a
′
∧ c ⊆ b). We have c ∩ a = c ∩ b ∩ a = c ∩ a
′
. So c ⊆ b
and c ∩ a = 0. Thus Wallman’s disjunction property holds.
3.3. Atomically separable lattices
Proposition 12 “atoms” is a straight monotone map (for any meet-semilattice).
Proof Monotonicity is obvious. The rest follows from the formula
atoms(a ∩ b) = atoms a ∩ atoms b
(the corolla ry 1).
Definition 30 I will call atomically separable such a poset that “atoms” is
an injection.
Proposition 13 ∀a, b ∈ A : (a ⊂ b ⇒ atoms a ⊂ atoms b) iff A is atomically
separable for a poset A.
Proof
⇐ Obvious.
17
⇒ Let a 6= b for example a * b. Then a ∩ b ⊂ a; atoms a ⊃ atoms(a ∩ b) =
atoms a ∩ atoms b and thus atoms a 6= atoms b .
Let atoms a 6= atoms b for example atoms a * atoms b . Then atoms(a ∩
b) = atoms a ∩ atoms b ⊂ atoms a and thus a ∩ b ⊂ a a nd so a * b
consequently a 6= b.
Proposition 14 Any atomistic poset is atomically separable.
Proof We need to prove that atoms a = atoms b ⇒ a = b. But it is obvious
because
a =
[
atoms a and b =
[
atoms b.
Theorem 20 If a lattice with least element is atomic and separable then it is
atomistic.
Proof Suppose the contrary that is a ⊃
S
atoms a. Then, because our lattice
is separable, exists c ∈ A such that c ∩ a 6= 0 and c ∩
S
atoms a = 0. There
exist atom d ⊆ c such that d ⊆ c ∩ a. d ∩
S
atoms a ⊆ c ∩
S
atoms a = 0. But
d ∈ atoms a. Contradiction.
Theorem 21 Any atomistic lattice is atomically separable.
Proof L et A be an atomistic lattice. Let a, b ∈ A, a ⊂ b. Then
S
atoms a ⊂
S
atoms b and consequently atoms a ⊂ atoms b.
Theorem 22 Let A be an atomic meet-semilattice with least element. Then the
following statements are equivalent:
1. A is separable.
2. A is atomically separable.
3. A conforms to Wallman’s disjunction property.
4. ∀a, b ∈ A : (a ⊂ b ⇒ ∃c ∈ A \ {0} : (c ≍ a ∧ c ⊆ b)).
Proof
(1)⇔(3)⇔(4) Proved above.
(2)⇒(4) Let our semilattice be atomically separable. Let a ⊂ b. Then atoms a ⊂
atoms b and so exists c ∈ atoms b such that c /∈ atoms a. c 6= 0 and c ⊆ b;
c * a, from which (taking in account that c is an atom) c ⊆ b and c∩a = 0.
So our semilattice conforms to the formula (4).
18
(4)⇒(2) Let formula (4) holds. Then for any elements a ⊂ b exists c 6= 0 such
that c ⊆ b and c∩a = 0. B ecause A is atomic there exists atom d ⊆ c . d ∈
atoms b and d /∈ atoms a. So atoms a 6= atoms b and a toms a ⊆ atoms b.
Consequently atoms a ⊂ atoms b.
4. Filtrators
Definition 31 I will call a filtrator a pair (A; Z) of a poset A and its subset
Z ⊆ A. I call A the base of a filtrator and Z the core of a filtrator.
Definition 32 I will call a lattice filtrator a pair (A; Z) of a lattice A and its
subset Z ⊆ A.
Definition 33 I will call a complete lattice filtrator a pair (A; Z) of a com-
plete lattice A and its subset Z ⊆ A.
Definition 34 I will call a central filtrator a filtrator (A; Z(A)) where Z(A)
is the center of a bounded lattice A.
Remark 8 One use of filtra tors is the theory of filters where the base lattice
(or the lattice o f principal filters) is essentially considered as the core of the
lattice of filters. See below for a more exact formulation. Our primary interest
is the properties o f filters on sets (that is the filtrator of filters on a set), but
instead we will research more general theor y of filtrators.
Remark 9 An other important example of filtrators is filtrator of funcoids
whose base is the set of funcoids [11] and whose core is the set of binary relations
(or discrete funcoids).
Definition 35 I will call el ement of a filtrator an element of its base.
Definition 36 up a = {c ∈ Z | c ⊇ a} where a ∈ A.
Definition 37 down a = {c ∈ Z | c ⊆ a} where a ∈ A.
Obvious 8 “up” and “down” are dual.
The main purpose of this text is knowing properties of the core of a filtrator
to infer pro perties of the base of the filtrator, specifically properties of up a for
every element a.
Definition 38 I call a filtrator with join-closed core such filtrator (A; Z) that
S
Z
S =
S
A
S whenever
S
Z
S exists for S ∈ PZ.
19
Definition 39 I call a filtrator with meet-closed core such filtrator (A; Z) t hat
T
Z
S =
T
A
S whenever
T
Z
S exists for S ∈ PZ.
Definition 40 I call a filtrator with finitely join-closed core such filtrator
(A; Z) that a ∪
Z
b = a ∪
A
b whenever a ∪
Z
b exists for a, b ∈ Z.
Definition 41 I call a filtrator with fini tely meet-closed core such filtrator
(A; Z) that a ∩
Z
b = a ∩
A
b whenever a ∩
Z
b exists for a, b ∈ Z.
Definition 42 Filtered filtrator is a filtrator (A; Z) such that ∀a ∈ A : a =
T
A
up a.
Definition 43 Prefiltered filtrator is a filtrator (A; Z) such that “up” is in-
jective.
Definition 44 Semifiltered filtrator is a filtrator (A; Z) such that
∀a, b ∈ A : (up a ⊇ up b ⇒ a ⊆ b).
Obvious 9 • Every filtered filtrator is semifiltered.
• Every semifiltered filtrator is prefiltered.
Obvious 10 “up” is a straight map from A to the dual of the poset PZ if (A; Z)
is a semifiltered filtrator.
Theorem 23 Each semifiltered filtrator is a filtrator with join-closed core.
Proof Let (A; Z) be a semifiltered filtrator. Let S ∈ PZ and
S
Z
S is defined.
We need to prove
S
A
S =
S
Z
S. That
S
Z
S is an upper bound for S is obvious.
Let a ∈ A be a n upper bound for S. Enough to prove tha t
S
Z
S ⊆ a. Really,
c ∈ up a ⇒ c ⊇ a ⇒ ∀x ∈ S : c ⊇ x ⇒ c ⊇
[
Z
S ⇒ c ∈ up
[
Z
S;
so up a ⊆ up
S
Z
S and thus a ⊇
S
Z
S because it is semifiltered.
4.1. Core part
Definition 45 The core part of an element a ∈ A is Cor a =
T
Z
up a.
Definition 46 The dual core part of an element a ∈ A is Cor
′
a =
S
Z
down a.
Obvious 11 Cor
′
is du al of Cor.
Theorem 24 Cor a ⊆ a whenever Cor a exists for any element a of a filtered
filtrator.
20
Proof Cor a =
T
Z
up a ⊆
T
A
up a = a.
Corollary 6 Cor a ∈ down a whenever Cor a exists for any element a of a
filtered filtrator.
Theorem 25 Cor
′
a ⊆ a whenever Cor
′
a exists for any element a of a filtrator
with join-closed core.
Proof Cor
′
a =
S
Z
down a =
S
A
down a ⊆ a.
Corollary 7 Cor
′
a ∈ down a whenever Cor
′
a exists for any element a of a
filtrator with join-closed core.
Proposition 15 Cor
′
a ⊆ Cor a whenever both Cor a and Cor
′
a exist for any
element a of a filt rator with join-closed core.
Proof Cor a =
T
Z
up a ⊇ Cor
′
a because ∀A ∈ up a : Co r
′
a ⊆ A.
Theorem 26 Cor
′
a = Cor a whenever both Cor a and Cor
′
a exist for any ele-
ment a of a filtered filtrator.
Proof It is with join-closed core because it is semifiltered. So Cor
′
a ⊆ Cor a.
Cor a ∈ down a. So Cor a ⊆
S
Z
down a = Cor
′
a.
Obvious 12 Cor
′
a = max down a for an element a of a filtrator with join-
closed core.
4.2. Filtrators with separable core
Definition 47 Let A be a filtrator. A is a filtrator w ith separable core when
∀x, y ∈ A : (x ≍
A
y ⇒ ∃X ∈ up x : X ≍
A
y).
Proposition 16 Let A be a filtrator. A is a filtrator with separable core iff
∀x, y ∈ A : (x ≍
A
y ⇒ ∃X ∈ up x, Y ∈ up y : X ≍
A
Y ).
Proof
⇒ Apply the definition twice.
⇐ Obvious.
Definition 48 Let A be a filtrator. A is a filtrator with co-separable core
when
∀x, y ∈ A : (x ≡
A
y ⇒ ∃X ∈ down x : X ≡
A
y).
21
Obvious 13 Co-separability is the dual of s eparability.
Proposition 17 Let A be a filtrator. A is a filtrator with co-separable core iff
∀x, y ∈ A : (x ≡
A
y ⇒ ∃X ∈ down x, Y ∈ down y : X ≡
A
Y ).
Proof By duality.
4.3. Intersecting and joining with an element of the core
Definition 49 I call down-aligned filtrator such a filtrator (A; Z) that A and
Z have common least element. (Let’s denote it 0.)
Definition 50 I call up-aligned filtrator such a filt rator (A; Z) that A and Z
have common greatest element. (Let’s denote it 1.)
Theorem 27 For a filtrator (A; Z) where Z is a boolean lattice, for every B ∈ Z,
A ∈ A:
1. B ≍
A
A ⇔
B ⊇ A if it is down-aligned, with finitely meet-closed and sepa-
rable core;
2. B ≡
A
A ⇔
B ⊆ A if it is u p-aligned, with finitely join-closed and co-
separable core.
Proof We will prove only the first as the second is dual.
B ≍
A
A ⇔
∃A ∈ up A : B ≍
A
A ⇔
∃A ∈ up A : B ∩
A
A = 0 ⇔
∃A ∈ up A : B ∩
Z
A = 0 ⇔
∃A ∈ up A :
B ⊇ A ⇔
B ∈ up A ⇔
B ⊇ A.
5. Filters
5.1. Filters on posets
Let A be a poset (pa rtially orde red set) with the partial order ⊆. I will call
it the base poset.
Definition 51 Filter base is a nonempty subset F of A such that
∀X, Y ∈ F ∃Z ∈ F : (Z ⊆ X ∧ Z ⊆ Y ).
22
Obvious 14 A nonempty chain is a fi lter base.
Definition 52 Upper set is a subset F of A such that
∀X ∈ F, Y ∈ A : (Y ⊇ X ⇒ Y ∈ F ).
Definition 53 Filter is a subset of A which is both filter base and upper set . I
will denote the s et of filt ers f.
Proposition 18 If 1 is the maximal element of A then 1 ∈ F for any filter F .
Proof If 1 6∈ F then ∀K ∈ A : K 6∈ F and so F is empty what is impossible.
Proposition 19 Let S be a filter base. If A
0
, . . . , A
n
∈ S (n ∈ N), then
∃C ∈ S : (C ⊆ A
0
∧ ... ∧ C ⊆ A
n
).
Proof It can be easily proved by induction.
The dual of filters is called ideals. We do not use ideals in this work however.
5.2. Filters on meet-semilattice
Theorem 28 If A is a m eet-semilattice and F is a nonempty subset of A then
the following conditions are equivalent:
1. F is a filter.
2. ∀X , Y ∈ F : X ∩ Y ∈ F and F is an u pper set.
3. ∀X , Y ∈ A : (X, Y ∈ F ⇔ X ∩ Y ∈ F).
Proof
(1)⇒(2) Let F be a filter. Then F is a n upper set. If X , Y ∈ F then Z ⊆
X ∧ Z ⊆ Y for s ome Z ∈ F . Because F is an upper set and Z ⊆ X ∩ Y
then X ∩ Y ∈ F.
(2)⇒(1) Let ∀X, Y ∈ F : X ∩ Y ∈ F and F is an upper set. We need to prove
that F is a filter base. But it is obvious taking Z = X ∩ Y (we have also
taken in account that F 6= ∅).
(2)⇒(3) Let ∀X, Y ∈ F : X ∩ Y ∈ F and F is an upper set. Then
∀X, Y ∈ A : (X, Y ∈ F ⇒ X ∩ Y ∈ F).
Let X ∩ Y ∈ F ; then X, Y ∈ F because F is an upper set.
23
(3)⇒(2) Let
∀X, Y ∈ A : (X, Y ∈ F ⇔ X ∩ Y ∈ F).
Then ∀X, Y ∈ F : X ∩ Y ∈ F . Let X ∈ F a nd X ⊆ Y ∈ A. Then
X ∩ Y = X ∈ F. Consequently X, Y ∈ F. So F is an upper set.
Proposition 20 Let A be a meet-semilattice. Let S be a filter base. If A
0
, . . . , A
n
∈
S (n ∈ N), then
∃C ∈ S : C ⊆ A
0
∩ ... ∩ A
n
.
Proof It can be easily proved by induction.
Proposition 21 If A is a meet-semilattice and S is a filter base, A ∈ A, then
hA∩i S is also a filter base.
Proof hA∩i S 6= ∅ because S 6= ∅.
Let X, Y ∈ hA∩i S. Then X = A ∩ X
′
and Y = A ∩ Y
′
where X
′
, Y
′
∈ S.
Exists Z
′
∈ S such that Z
′
⊆ X
′
∩Y
′
. So X ∩Y = A∩X
′
∩Y
′
⊇ A∩Z
′
∈ hA∩i S.
5.3. Characterization of finitely meet-closed filtrators
Theorem 29 The following are equivalent for a filtrator (A; Z) whose core is a
meet-semilattice such that ∀a ∈ A : up a 6= ∅:
1. The filtrator is finitely meet-closed.
2. up a is a filter on Z for every a ∈ A.
Proof
(1)⇒(2) Let X, Y ∈ up a. Then X ∩
Z
Y = X ∩
A
Y ⊇ a. That up a is an upper
set is </