**Algebraic General Topology. Vol 1**:
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**Axiomatic Theory of Formulas**:
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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 </