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Dual ﬁlters and ideals

by Victor Porton

Email: porton@narod.ru

Web: http://www.mathematics21.org

WARNING: There are errors in this draft. See instead my book.

[TODO: Deﬁne commutatitve diagrams.]

For a lattice Z I denote meets and joins correspondingly as u and t.

In my earlier work I denoted ﬁlters on a poset Z as F( Z) (or just F) and corresponding principal

ﬁlters as P(Z) (or just P).

I will denote X = A n X for a set X ⊆ A.

Filters and ideals are well known concepts:

Filters F are subsets F of A such that:

1. F does not contain the least element of A (if it exists).

2. A u B 2 F , A 2 F ^ B 2 F (for every A; B 2 Z).

Ideals I are subsets F of A such that:

1. F does not contain the greatest element of A (if it exists).

2. A t B 2 F , A 2 F ^ B 2 F (for every A; B 2 Z).

Free stars S are subsets F of A such that:

1. F does not contain the least element of A (if it exists).

2. A t B 2 F , A 2 F _ B 2 F (for every A; B 2 Z).

Mixers M are subsets F of A such that:

1. F does not contain the greatest element of A (if it exists).

2. A u B 2 F , A 2 F _ B 2 F (for every A; B 2 Z).

Proposition 1. A set F is a lower set iﬀ F is an upper set.

Proof. X 2 F ^ Z w X ) Z 2 F is equivalent to Z 2 F ) X 2 F _ Z w X is equivalent

Z 2 F ) (Z w X ) X 2 F ) is equivalent Z 2 F ^ X v Z ) X 2 F .

I will denote dual A where A 2 Z the corresponding element of the dual poset Z

∗

. Also I denote

hdualiX =

def

fdual x j x 2 X g:

Then we have the following bijections between above described four sets:

A t B 2 F , A 2 F _ B 2 F is equivalent to :(A t B 2 F ) , :(A 2 F ^ B 2 F ) is equivalent to

A t B 2 F , A 2 F ^ B 2 F ;

1

A u B 2 F , A 2 F _ B 2 F is equivalent to :(A u B 2 F ) , :(A 2 F ^ B 2 F ) is equivalent to

A u B 2 F , A 2 F ^ B 2 F .

We have the following commutative diagram in category Set, every arrow of this diagram is an

isomorphism, every cycle in this diagram is an identity:

hduali

ideals

:

hduali

free stars

mixers

:

ﬁlters

Figure 1. Diagram Υ

(where : denotes set-theoretic complement).

These isomorphisms are also order isomorphisms if we deﬁne order in the right way.

The above it is deﬁned for lattices only. Generalizing this for arbitrary posets is straigthforward:

Deﬁnition 2. Let A be a poset.

• Filters are sets F without the greatest element of A with A; B 2F ,9Z 2 F :(Z v A ^ Z v B)

(for every A; B 2 Z).

• Ideals are sets F without the least element of A with A; B 2 F ,9Z 2 F : (Z w A ^ Z w B)

(for every A; B 2 Z).

• Free stars are sets F without the greatest element of A with

A; B 2 F , 9Z 2 F : (Z w A ^ Z w B)

A 2/ F ^ B 2/ F , 9Z 2 F : (Z w A ^ Z w B)

A 2 F _ B 2 F , :9Z 2 F : (Z w A ^ Z w B)

• Mixers are lower sets F without the least element of A with :9Z 2 F : (Z v A ^ Z v B) ,

A 2 F _ B 2 F or equivalently 9Z 2 F : (Z v A ^ Z v B) , A 2/ F ^ B 2/ F (for every A; B 2 Z).

Proposition 3. The following are equivelent: [TODO: With one side implications and requirement

to be upper/lower set.]

1. F is a free star.

2. 8Z 2 A: (Z w A ^ Z w B ) Z 2 F ) , A 2 F _ B 2 F for every A; B 2 A and F =/ P A.

Proof. The following is a chain of equivalencies:

9Z 2 F : (Z w A ^ Z w B) , A 2/ F ^ B 2/ F ;

8Z 2 F : :(Z w A ^ Z w B) , A 2 F _ B 2 F ;

8Z 2 A: (Z 2/ F ) :(Z w A ^ Z w B)) , A 2 F _ B 2 F ;

8Z 2 A: (Z w A ^ Z w B ) Z 2 F ) , A 2 F _ B 2 F :

2

Corollary 4. The following are equivelent: [TODO: With one side implications and requirement

to be upper/lower set.]

1. F is a mixer.

2. 8Z 2 A: (Z v A ^ Z v B ) Z 2 F ) , A 2 F _ B 2 F for every A; B 2 A and F does not contain

the least element of A.

1 General isomorphisms

Let θ be an self-inverse order reversing isomorphism of some set P of posets.

We have the following commutative diagram in category Set, every arrow of this diagram is an

isomorphism, every cycle in this diagram is an identity:

hduali

ideals

:

hduali

free stars

mixers

:

F(P )

Figure 2. Diagram Υ

2 Boolean lattices

In the case if Z is a boolean lattice, there is also an alternative commutative diagram in category

Set, every arrow of this diagram is an isomorphism, every loop in this diagram is an identity:

h:i

ideals

:

h:i

free stars

mixers

:

ﬁlters

Figure 3. Diagram Ψ

(here h:iX =

def

fx¯ j x 2 X g).

[TODO: Other deﬁnition where upper/lower set is explicitly said.]

[TODO: Combine two rectangular diagrams into one “cubic”.]

Above we have two diagrams D with ??(every isomorphism). I will denote A!

D

B the unique

bijection from A to B which is a composition of arrows of this diagram.

[TODO: Deﬁne isomorphism of ﬁltrators.]

Deﬁne principal ideals, free stars, mixers as objects isomorphic to principal ﬁlters. [FIXME: There

are two diagrams which provide diﬀerent isomorphisms!] These isomorphisms f have in common

that fa v fb,a wb. We can deﬁne the isomorphism hduali

∗

as a function such that fa v fb,a wb.

It is called order reversing isomorphism. So ﬁrst consider the general case of subsets of ﬁlters,

ideals, ... with an arbitrary antitone isomorphism. Need it to be an involution?

3