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