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Point free funcoids as a generalization of frames

by Victor Porton

Email: porton@narod.ru

Web: http://www.mathematics21.org

August 29, 2013

Abstract

I deﬁne an injection from the set of frames to the set of pointfree endo-funcoids.

1 Preliminaries

This article is a rough partial draft of a future longer writting.

Read my book [2] before this article. The preprint of [2] is not ﬁnal, s o theorem numbers may

change.

2 Confession of a non-professiona l

I am not a professional mathematician. I just recently started my study of pointfree topology by

the book “Stone Spaces”.

I don’t understand how to prove the statement I use below that every frame can be embedded

into a boolean lattice. (I take it below as granted, without under standing the proof.)

Despite of all that, this article presents a new branch of mathematics discovered by me: rela-

tionships between fram e s and locales on one side and pointfree funcoids (p ointfree funcoids are

also my discovery) on an other side. (Frames and locales can be cons idered as a spe cial kind of

pointfree funcoids.) I expect that analyzing pointfree funcoids will turn to be much more easy that

customary ways of doing pointfree topology.

3 Deﬁnitions

3.1 Pointfree funcoid induced by a co-frame

Let L is a co-frame.

We will deﬁne pointfree funcoid ⇑L.

Let B(L) is a boolean lattice whose co-subframe L is. (That this mapping exis ts follows from [1],

page 53.) There may be probably more than one such mapping, but we just choose one B arbitrarily.

Deﬁne cl(A) =

d

{X ∈ L | X ⊒ A}.

Here

d

can be taken on either L or B(L) as they are the same.

Obvious 1. cl ∈ L

B(L)

.

cl(A ⊔ B) =

d

{X ∈ L | X ⊒ A ⊔ B} =

d

{X ∈ L | X ⊒ A, X ⊒ B } =

d

{X

1

⊔ X

2

| X

1

⊒ A,

X

2

⊒ B } =

d

{X

1

| X

1

⊒ A} ⊔

d

{X

2

| X

2

⊒ B} = cl A ⊔ cl B.

cl 0 = 0 is obvious.

Hence we are under conditions of the theo rem 14.26 in my book.

So there exists a unique pointfree endo-funcoid ⇑L ∈ FCD(F(B(L)); F(B(L))) such that

h⇑LiX =

l

F(B(L))

hcliup

(F(B(L));P(B(L)))

X

for every ﬁlter X ∈ F(B(L)).

1

3.2 Co-frame induced by a pointfree funcoid

The co- frame ⇓f for some pointfree endo-funcoids f will be deﬁ ned to be the reverse of ⇑. See

below for exact meaning of being reverse.

Let restore the co-frame L from the pointfree funcoid ⇑L.

Let poset ⇓f for every pointfree funcoid f is deﬁned by the formula:

⇓f = {X ∈ Z(Ob f) | hf iX = X }.

Remark 2. It seems that ⇓ is not a mo novalued function from pfFCD to Ob(Frm).

3.3 Isomorphis m of co-frames through pointfree funcoids

Remark 3. P(B(L)) = Z(F(B(L))) (theo rem 4.137 in [2]).

Theorem 4. L

⇓⇑L (where L ranges all sm all frames) is an order isomorphism.

Proof. Let A

′

∈ ⇓⇑L. The n there exists A ∈ B(L) s uch that A

′

= ↑

B(L)

A.

hf iA

′

= ↑

B(L)

cl A.

hf iA

′

= A

′

that is ↑

B(L)

cl A = A

′

= ↑

B(L)

A. So cl A = A and thus A ∈ L.

Let now A ∈ L. Then take A

′

= ↑

B(L)

A. We have hf iA

′

= cl A = ↑

B(L)

A = A

′

. So A

′

∈ ⇓⇑L.

We have proved that it is a bijection.

Because A and A

′

are related by the equation A

′

= ↑

B(L)

A it is obvious that this is an order

embedding.

4 Postface

Pointfree func oids are a massive generalization of locales and frames: They don’t only require the

lattice of ﬁlters to be boolean but these can be even not lattices of ﬁlters at all but just arbitrary

posets. I think a new era in pointfree topology s tarts.

Much work is yet needed to relate diﬀerent properties o f frames and locales with corresponding

properties of pointfree funcoids.

Bib liography

[1] Peter T. Johnstone. Stone Spaces. Cambridge University Press, 1982.

[2] Victor Porton. Algebraic General Topology. Volume 1. 2013.

2 Section