A conjecture in lattice theory
Victor Porton
http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
Maybe you can solve?
William Elliot
| I will call center Z(\mathfrak{A}) of a bounded lattice
| mathfrak{A} the sublattice of all complemented elements of a
| mathfrak{A} .
The center of a bounded lattice A,
Z(A) = Z = { x | some y with x + y = 1, xy = 0 }.
. . where 1 = top and 0 = bottom.
Is Z a sublattice?
If a,b in Z, then some x,y with a + x = 1 = b + y, ax = 0 = by.
a + b + x + y = 1 + 1 = 1; ab(x + y) = a0 + b0 = 0 + 0 = 0.
Ok, Z is closed under sup and inf.
| I will call a bounded lattice \mathfrak{A} a lattice with
| seaparable center when
| forall x,y\in\mathfrak{A}: (x\cap y=0\Rightarrow \exists X\in
| Z(\mathfrak{A}):(x\subseteq X\wedge X\cap y = 0)) .
Yicks, TeX is such a mess.
for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0).
Is that correct translation?
| Equivalently a bounded lattice with separable center is such a bounded
| lattice mathfrak{A} that
| forall x,y\in\mathfrak{A}:(x\cap y=0\Rightarrow\exists X,Y\in
| Z(\mathfrak{A}):(x\subseteq X\wedge y\subseteq Y\wedge X\cap Y = 0))
for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0)
Is that correct translation?
.
| Conjecture There exist bounded lattices which are not with separable
| center.
1
|
a
/ \
x y
\ /
0
Z = {0,1}, xy = 0
Victor Porton
> On Fri, 12 Sep 2008, Victor Porton wrote:
> > I don't know how to solve this problem in lattice theory:
> >http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
> Yicks, TeX is such a mess.
>
> for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0).
>
> Is that correct translation?
Yes.
> for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0)
>
> Is that correct translation?
Yes.
> | Conjecture There exist bounded lattices which are not with separable
> | center.
>
> 1
> |
> a
> / \
> x y
> \ /
> 0
>
> Z = {0,1}, xy = 0
Yes, you have solved it. Thanks.
Certainly, I would to think about finite lattices myself before
sending this conjecture to others. I used to use to think only about
infinities, sorry.
The problem was solved too easily. Should we keep it in Open Problem
Garden or I should delete it? See
http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
victor_me...@yahoo.co.uk
> Yes, you have solved it. Thanks.
>
> Certainly, I would to think about finite lattices myself before
> sending this conjecture to others. I used to use to think only about
> infinities, sorry.
>
> The problem was solved too easily.
Do you still want your Abel prize?
Victor Meldrew
"I don't believe it!"
William Elliot
> On Fri, 12 Sep 2008, Victor Porton wrote:
>
> > I don't know how to solve this problem in lattice theory:
> > http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
>
> | mathfrak{A} the sublattice of all complemented elements of a
> | mathfrak{A} .
>
> The center of a bounded lattice A,
> Z(A) = Z = { x | some y with x + y = 1, xy = 0 }.
> . . where 1 = top and 0 = bottom.
>
> Is Z a sublattice?
> If a,b in Z, then some x,y with a + x = 1 = b + y, ax = 0 = by.
> a + b + x + y = 1 + 1 = 1; ab(x + y) = a0 + b0 = 0 + 0 = 0.
Whoops, I've assume A is distributive.
Worse than that, it's nonthink.
Assuming A is distributive
a + b + xy = (a + b + x)(a + b + y) = 1
(a + b)xy = axy + bxy = 0; a + b in Z
ab + x + y = (a + x + y)(b + x + y) = 1
ab(x + y) = abx + aby = 0; ab in Z
> Ok, Z is closed under sup and inf.
Provided A is distributive. Hm, stronger should be possible.
William Elliot
> On Sep 12, 12:59 pm, William Elliot <ma...@hevanet.remove.com> wrote:
> > On Fri, 12 Sep 2008, Victor Porton wrote:
> The problem was solved too easily. Should we keep it in Open Problem
> Garden or I should delete it?
Here's the problem that interests me.
Let A be a bounded lattice.
Let Z = { x | some y with x + y = 1, xy = 1 }
. . = the set of all complemented elements of A.
When A is distributive, then Z is a sublattice as I showed
in another post. Is Z a sublattice when A isn't distributive?
Why?
--
A has separable center when
> > for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0).
> > for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0)
>
Victor Porton
> On Fri, 12 Sep 2008, William Elliot wrote:
> > On Fri, 12 Sep 2008, Victor Porton wrote:
>
> > > I don't know how to solve this problem in lattice theory:
> > >http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
>
> > | I will call center Z(\mathfrak{A}) of a bounded lattice
> > | mathfrak{A} the sublattice of all complemented elements of a
> > | mathfrak{A} .
My mistake also. I should say "bounded distributive lattice" instead
of "bounded lattice".
> > Is Z a sublattice?
> > If a,b in Z, then some x,y with a + x = 1 = b + y, ax = 0 = by.
> > a + b + x + y = 1 + 1 = 1; ab(x + y) = a0 + b0 = 0 + 0 = 0.
>
> Whoops, I've assume A is distributive.
> Worse than that, it's nonthink.
>
> Assuming A is distributive
> a + b + xy = (a + b + x)(a + b + y) = 1
> (a + b)xy = axy + bxy = 0; a + b in Z
>
> ab + x + y = (a + x + y)(b + x + y) = 1
> ab(x + y) = abx + aby = 0; ab in Z
>
> > Ok, Z is closed under sup and inf.
>
> Provided A is distributive. Hm, stronger should be possible.
You can assume that it is distributive.
> > | Conjecture There exist bounded lattices which are not with separable
> > | center.
>
> > 1
> > |
> > a
> > / \
> > x y
> > \ /
> > 0
>
> > Z = {0,1}, xy = 0
Should say "There exist bounded distributive lattices which are not
with separable center."
Oops, is the William Elliot's lattice distributive?
Victor Porton
> On Sep 12, 1:51 pm, William Elliot <ma...@hevanet.remove.com> wrote:
> > On Fri, 12 Sep 2008, William Elliot wrote:
> > > On Fri, 12 Sep 2008, Victor Porton wrote:
> > > > I don't know how to solve this problem in lattice theory:
> > > >http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
I have corrected the problem statement adding the word "distributive"
where necessary. See
http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
The problem is open again.
William Elliot
>
> I have corrected the problem statement adding the word "distributive"
> where necessary. See
> http://garden.irmacs.sfu.ca/?q=op/non_separable_center_of_a_lattice
>
> The problem is open again.
>
Some of your research work interests me.
As it's in pdf formate, I can't read it.
Do you have (yuck) TeX version, or (yah) ASCII version?
Let A be a bounded lattice.
The center of A, Z = Z(A) = { x | some y with x + y = 1, xy = 0 }.
When A is distributive, then Z(A) is a sublattice.
If a,b in Z, then some x,y (also in Z) with
. . a + x = 1 = b + y, ax = 0 = by
Thus a + b, ab in Z, for
. . a + b + xy = (a + b + x)(a + b + y) = 1
. . (a + b)xy = axy + bxy = 0; a + b in Z
. . ab + x + y = (a + x + y)(b + x + y) = 1
. . ab(x + y) = abx + aby = 0; ab in Z
Here's an example of a bounded lattice with a center that's not a
sublattice.
1
/ | \
r a s
| / \ |
x y
\ /
\ /
0
Z = { 0, x,y, r,s, 1 }; x + y = a not in Z.
Open problem. If A is a bounded lattice,
is Z(A) with the inherited order a lattice?
When it is, is it distributive?
No, Z above isn't distributive. Within Z
. . r(x + y) = r1 = r
. . rx + ry = x + 0 = x
-- Speculation
Let L be a lattice and Z(a,b) = Z({ x | a <= x <= b }).
Can L be described by the centers? What significance
does a center hold? What significance does a maximal
center have?
Let L = R. Then Z(a,b) = { a,b } or nulset and every
two element center is maximal.
--
Exercise. A complemented lattice is distributive
iff complements are unique.
--
The center Z is separable when
. . for all x,y in A, (xy = 0 ==> some a in Z with x <= a, ay = 0).
equivalently
. . for all x,y in A, (xy = 0 ==> some a,b in Z with x <= a, y <= b, ab = 0)
Conjecture There exist bounded distributive
lattices which are not with separable center.
1
|
a
/ \
x y
\ /
0
Z = {0,1}, xy = 0
This has two linear sublattices. Thus to check for distributivity,
only expressions with both x and y need to be checked. That is.
. . u(x + y), u = 0, x,y, a, 1
. . x(u + y)
. . y(u + x)
Which is two cases, taking symmetry into account.
Subcases are u = x; u = a,1; u = 0.
Easier is to notice that it doesn't have either of
the forbidden sublattices.
----
Victor Porton
> On Fri, 12 Sep 2008, Victor Porton wrote:
>
> Some of your research work interests me.
> As it's in pdf formate, I can't read it.
> Do you have (yuck) TeX version, or (yah) ASCII version?
William Elliot, I will send you email with my research in the format
"HTML with images (for formulas)".
Does anybody also have problem with reading PDF? If yes, I may
consider publishing my work in HTML format. (To do this is desirable
to write a multiformat-conversion script in Schema (the scripting
language of TeXmacs, http://www.texmacs.org), a programming language I
don't know.)
You can however read PDF with the following free program, please
download it:
http://www.adobe.com/go/gntray_dl_get_reader
William Elliot
> On Sep 13, 10:53 am, William Elliot <ma...@hevanet.remove.com> wrote:
> > On Fri, 12 Sep 2008, Victor Porton wrote:
> >
> > Some of your research work interests me.
> > As it's in pdf formate, I can't read it.
> > Do you have (yuck) TeX version, or (yah) ASCII version?
>
> William Elliot, I will send you email with my research in the format
> "HTML with images (for formulas)".
That will not suffice as I do not have graphic capacity.
> Does anybody also have problem with reading PDF? If yes, I may consider
> publishing my work in HTML format. (To do this is desirable to write a
> multiformat-conversion script in Schema (the scripting language of
> TeXmacs, http://www.texmacs.org), a programming language I don't know.)
>
The prefered math writing software is TeX.
> You can however read PDF with the following free program, please
> download it:
> http://www.adobe.com/go/gntray_dl_get_reader
It's do large to download. pdf...@adobe.com will convert
pdf files to txt files. It works well for simple written
material but not for math formulas, diagrams, etc.
The problem of showing that a complement lattice with unique
complements is distributive, is harder than I expected. Have
you any suggestions?
Victor Porton
> On Sat, 13 Sep 2008, Victor Porton wrote:
> > On Sep 13, 10:53 am, William Elliot <ma...@hevanet.remove.com> wrote:
> > > On Fri, 12 Sep 2008, Victor Porton wrote:
>
> > > Some of your research work interests me.
> > > As it's in pdf formate, I can't read it.
> > > Do you have (yuck) TeX version, or (yah) ASCII version?
OK, William Elliot, I will send you my articles converted to LaTeX.
However I doubt whether a man is able to read these messy formulas in
LaTeX format without special software converting LaTeX to graphical
presentation.
> The prefered math writing software is TeX.
It _was_ TeX in the past. Now the preferred math writing software is
TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org
TeXmacs has the ability to export to LaTeX (and several other
formats).
> The problem of showing that a complement lattice with unique
> complements is distributive, is harder than I expected. Have
> you any suggestions?
No. I'm not interested in this problem. I'm interested only about
lattices which are already known to be distributive.
William Elliot
> On Sep 13, 2:22 pm, William Elliot <ma...@hevanet.remove.com> wrote:
> >
> > > > Some of your research work interests me.
> > > > As it's in pdf formate, I can't read it.
> > > > Do you have (yuck) TeX version, or (yah) ASCII version?
>
> OK, William Elliot, I will send you my articles converted to LaTeX.
> However I doubt whether a man is able to read these messy formulas in
> LaTeX format without special software converting LaTeX to graphical
> presentation.
>
> > The prefered math writing software is TeX.
>
> It _was_ TeX in the past. Now the preferred math writing software is
> TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org
>
> TeXmacs has the ability to export to LaTeX (and several other
> formats).
>
Is TeXmacs like that?
> > The problem of showing that a complement lattice with unique
> > complements is distributive, is harder than I expected. Have
> > you any suggestions?
>
> No. I'm not interested in this problem. I'm interested only about
> lattices which are already known to be distributive.
It's useful to show when a lattice isn't distributive.
----
Victor Porton
> On Sat, 13 Sep 2008, Victor Porton wrote:
> > On Sep 13, 2:22 pm, William Elliot <ma...@hevanet.remove.com> wrote:
>
> > > The prefered math writing software is TeX.
>
> > It _was_ TeX in the past. Now the preferred math writing software is
> > TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org
>
> > TeXmacs has the ability to export to LaTeX (and several other
> > formats).
>
> The advantage of TeX is that it's files are simple ascii files.
> Is TeXmacs like that?
Yes, both TeXmacs texts and TeXmacs styles are plain ASCII.
These are readable and probably more readable that TeX.
William Elliot
> On Sep 13, 2:57 pm, William Elliot <ma...@hevanet.remove.com> wrote:
> > On Sat, 13 Sep 2008, Victor Porton wrote:
> > > On Sep 13, 2:22 pm, William Elliot <ma...@hevanet.remove.com> wrote:
> >
> > > > The prefered math writing software is TeX.
> >
> > > It _was_ TeX in the past. Now the preferred math writing software is
> > > TeXmacs, a WYSIWYG math texts editor, see http://www.texmacs.org
> >
> > > TeXmacs has the ability to export to LaTeX (and several other
> > > formats).
> >
> > The advantage of TeX is that it's files are simple ascii files.
> > Is TeXmacs like that?
>
> Yes, both TeXmacs texts and TeXmacs styles are plain ASCII.
> These are readable and probably more readable that TeX.
>
The web site for them isn't in this thread.
----
Victor Porton
> Hum. What happened to the list of your papers?
> The web site for them isn't in this thread.
http://www.mathematics21.org/algebraic-general-topology.html
William Elliot
Here's a list of your works along with some of your hand waving
bravado. It seems that your research is yet incomplete, thus
immature for climaing a Nobel prize. You are too impatient.
The list is presented here in order of readablity and interest.
I will review the first two first becauase they are in readable
ascii for the most part but in need of editing for typos. As
notions vary, it'd be helpful to include some introductory material
such as are you considering binary relaltions on a single set S,
that is a subset of S^2, or of binary relation between two sets,
X,Y. ie a subset of XxY? The other point to clarify is what
is the domain of a relation and what is a restriction of a relation?
Knowing that, I will review the first and as time allows the second paper.
Have I questions, clarifications, corrections or comments to make, I will
place them in a new thread "Ordering Binary Relations".
[10]Vertical order of binary relations
Defined a partial order relation between binary relations
f and g by the formula g = f|[dom g].
[11]Theorem about binary relation limited to a set
Theorem (with proof), expanding and generalizing the statement:
For two binary relations f and g the formula g = f|[dom
g]is equivalent to conjunction of g being a subset of f
and the formula
g g-1 = f g-1 (or equivalently g g-1 = g f^-1).
--
Set Theoretic Filters (PDF, very preliminary draft)
Considers the lattice of set theoretic filters.
Filters on Posets (PDF, very preliminary draft)
The generalization of the previous article. Currently contains
only a part of the materials of the previous article.
Partially Ordered Categories with Inverses (PDF)
Defined partially ordered category with inverses of morphisms.
For such categories defined monovalued morphisms and entirely
defined morphisms.
Funcoids and Reloids (PDF, draft)
Consider generalizations of proximity spaces and uniform spaces.
Generalized Continuousness (PDF, draft)
Defines continuousness algebraically hiding old epsilon-delta
notion under a smart algebra. Generalizes continuousness,
uniform continuousness, and proximity-continuousness in one
formula.
Connectedness of funcoids and reloids (PDF, draft)
Defined the notion of connectedness for funcoids and reloids.
Shown how connectedness of funcoids is related with
connectedness of reloids.
Convergence of funcoids (PDF, draft)
Defined the notion of convergence and limit for funcoids.
Open Problems in AGT (PDF)
List unsolved problems and conjectures in the field of AGT.
-- Achievements and advantages of AGT
* general topology expressed in simple algebraic operations
* simplicity of operating with infinities, as infinities now can be
comprehended as something "whole", not a mess of parts
* two-three line proofs of some old pages length analysis theorems
* multivalued functions are now so simple to study as single valued
* frees analysis from its messy epsilon-delta notation
* analysis of non-continuous functions
* partially formally unifies math analysis and discrete mathematics
* not limited in any way to metrizable spaces and countable sets
--
AGT isn't a continuation of former functional analysis research,
it is a new beginning almost from scratch requiring little
beyond first level courses to understand it.
This new research field both generalizes former analysis and gives
new theorems/concepts not having analogs in old theories. Several
different theorems of analysis often collapse into one AGT equation
of which they are obvious consequences.
AGT is very abstract, indeed even the current level of AGT knowledge
often allowed me to find simple solutions of practical tasks (such as
calculations of infinite sums). I have not yet reached the level of
integrals in the synthesis research.
AGT is a kinda thinking with equations. No real numbers analysis
expressiveness with visual images preserved. That is not needed anyway
as the equations of AGT are even more clear than graphics of old
analysis. AGT is simple, natural, and beautiful.
Note that Algebraic General Topology being a generalization of General
Topology has nothing in common (except of the name) with Algebraic
Topology. Math synthesis is a generalization of functional analysis.
--
[23]Algebraic General Topology at WikInfo.
[25]My homepage [26]My math page
10. http://www.mathematics21.org/misc/vertical-order.html
11. http://www.mathematics21.org/misc/limiting-binary-relations-theorem.html
23. http://www.wikinfo.org/index.php/Algebraic_general_topology
25. http://portonvictor.org/
26. http://www.mathematics21.org/index.html
----
Victor Porton
> On Sat, 13 Sep 2008, Victor Porton wrote:
> > On Sep 13, 3:45 pm, William Elliot <ma...@hevanet.remove.com> wrote:
> > > Hum. What happened to the list of your papers?
> > > The web site for them isn't in this thread.
>
> >http://www.mathematics21.org/algebraic-general-topology.html
>
> Here's a list of your works along with some of your hand waving
> bravado. It seems that your research is yet incomplete, thus
> immature for climaing a Nobel prize. You are too impatient.
Not Nobel Prize (there are no Nobel Prize for math works) but Abel
Prize.
In the prize rules there are no requirement that the research should
be complete. They say that Abel Prize is for (among other) those who
are opened new major research areas.
My Algebraic General Topology is opening up a new major research area.
I hope this may be enough to receive Abel Prize.
And philosophically: Can a research be ever completed? Isn't it
infinite.
The biggest gap in my research is yet missing research of objects
similar to compact spaces. Is missing it enough to prevent me from
Abel Prize? ;-)
BTW, I yet plea to nominate me for Abel Prize:
http://www.mathematics21.org/abel-prize.html
You say that I'm to impatient. My impatience is caused mainly by the
desire to receive money to be able to fire from the daily job and
dedicate myself things I deem more important such as math research. So
it is somehow hard for me to to the research before receiving Abel
Prize. This makes the things reverse: first prize and then
research :-)
> ascii for the most part but in need of editing for typos. As
> notions vary, it'd be helpful to include some introductory material
> such as are you considering binary relaltions on a single set S,
> that is a subset of S^2, or of binary relation between two sets,
> X,Y. ie a subset of XxY? The other point to clarify is what
> is the domain of a relation and what is a restriction of a relation?
I'm going to write a book "Filters on Posets" (which will replace the
mentioned article "Set Theoretic Filtes"). There introductory
materials probably will be put to that my book to be able to extend
its size to more that about 100 pages, as publishing companies
require. Not in this version of article however.
> Knowing that, I will review the first and as time allows the second paper.
> Have I questions, clarifications, corrections or comments to make, I will
> place them in a new thread "Ordering Binary Relations".
Better you'd also email me (por...@narod.ru) as I'm not sure that I
will read the thread "Ordering Binary Relations".
Why you call the thread "Ordering Binary Relations"? I don't
understand how this is related with ordering binary relations.
First and second papers are "Set Theoretic Filters" and "Filters on
Posets"? The second is a newer but less complete version of the first.
Think about "Filters on Posets" as a development version of more
stable "Set Theoretic Filters".
I have already said that I am going to write a book "Filters on
Posets" which will supersede both articles.
Denis Feldmann
> On Sep 14, 11:30 am, William Elliot <ma...@hevanet.remove.com> wrote:
>> On Sat, 13 Sep 2008, Victor Porton wrote:
>>> On Sep 13, 3:45 pm, William Elliot <ma...@hevanet.remove.com> wrote:
>>>> Hum. What happened to the list of your papers?
>>>> The web site for them isn't in this thread.
>>> http://www.mathematics21.org/algebraic-general-topology.html
>> Here's a list of your works along with some of your hand waving
>> bravado. It seems that your research is yet incomplete, thus
>> immature for climaing a Nobel prize. You are too impatient.
>
> Not Nobel Prize (there are no Nobel Prize for math works) but Abel
> Prize.
>
> In the prize rules there are no requirement that the research should
> be complete. They say that Abel Prize is for (among other) those who
> are opened new major research areas.
If your english and your reading abilities are par to your math
abilities... Abel prize is for those who *have* opened etc., i.e for
life works, time-recognized ourstanding achievements, etc. (see what ,
for instance, Serre did to get the first prize in 2003, at
http://en.wikipedia.org/wiki/Abel_Prize ; the achievements of the later
lureates are quite impressive too) So, even if you really were opening
new domains, a try for a Fields medal would be a better guess. As for
we nominating you, why dont you ask noomination for US presidency (or,
better, for world presidency)? Your chances would actually be better, as
some of us are votiers(or at least future voters)...
Victor Porton
wrote:
> Victor Porton a écrit :
>
> > On Sep 14, 11:30 am, William Elliot <ma...@hevanet.remove.com> wrote:
> > In the prize rules there are no requirement that the research should
> > be complete. They say that Abel Prize is for (among other) those who
> > are opened new major research areas.
>
> If your english and your reading abilities are par to your math
> abilities... Abel prize is for those who *have* opened etc., i.e for
It was a misspelling. I know English grammar rules about constructs
like "have opened". Sorry for a misspelling.
Denis Feldmann
> On Sep 14, 1:32 pm, Denis Feldmann <denis.feldmann.sanss...@neuf.fr>
> wrote:
>> Victor Porton a écrit :
>>
>>> On Sep 14, 11:30 am, William Elliot <ma...@hevanet.remove.com> wrote:
>>> In the prize rules there are no requirement that the research should
>>> be complete. They say that Abel Prize is for (among other) those who
>>> are opened new major research areas.
>> If your english and your reading abilities are par to your math
>> abilities... Abel prize is for those who *have* opened etc., i.e for
>
> It was a misspelling. I know English grammar rules about constructs
> like "have opened". Sorry for a misspelling.
Good. Did you mispelled "Abel prise" for "Fields medal" too? And why not
postullate for world president?
William Elliot
>
> > ascii for the most part but in need of editing for typos. As
> > notions vary, it'd be helpful to include some introductory material
> > such as are you considering binary relaltions on a single set S,
> > that is a subset of S^2, or of binary relation between two sets,
> > X,Y. ie a subset of XxY? The other point to clarify is what
> > is the domain of a relation and what is a restriction of a relation?
>
> I'm going to write a book "Filters on Posets" (which will replace the
> mentioned article "Set Theoretic Filtes"). There introductory
> materials probably will be put to that my book to be able to extend
> its size to more that about 100 pages, as publishing companies
> require. Not in this version of article however.
>
> > Knowing that, I will review the first and as time allows the second paper.
> > Have I questions, clarifications, corrections or comments to make, I will
> > place them in a new thread "Ordering Binary Relations".
>
> Better you'd also email me (por...@narod.ru) as I'm not sure that I
> will read the thread "Ordering Binary Relations".
>
thread. Will you answer my questions? Are binary relations a subset of
S^2 or of XxY ? What's an exact definition of domain and restriction?
Wikipedia definitions aren't as exact as I'd like.
> Why you call the thread "Ordering Binary Relations"? I don't
> understand how this is related with ordering binary relations.
>
I'm starting with your two papers on binary relations as mentioned
in my previous post and also mention in your web site in the misc. section
of your works.
After that, I will get to the next papers on filters which protent to be
interesting, if presented in the propriate formate. I'll skip the paper
on catagory theory as catagory theory doesn't interest me.
> First and second papers are "Set Theoretic Filters" and "Filters on
> Posets"? The second is a newer but less complete version of the first.
> Think about "Filters on Posets" as a development version of more
> stable "Set Theoretic Filters".
>
> I have already said that I am going to write a book "Filters on
> Posets" which will supersede both articles.
>
Well, until then, I've your papers to read.
Victor Porton
> On Sun, 14 Sep 2008, Victor Porton wrote:
>
> > Better you'd also email me (por...@narod.ru) as I'm not sure that I
> > will read the thread "Ordering Binary Relations".
>
> The advantage of posting here is that others may participate in the
> thread. Will you answer my questions? Are binary relations a subset of
> S^2 or of XxY ? What's an exact definition of domain and restriction?
> Wikipedia definitions aren't as exact as I'd like.
One variant of understanding binary relations, is to consider it as
sets of pairs of elements of a certain (fixed) set (which may be
called "the universal set"). This definition of binary relation well
suits for my "Theorem about limiting binary relations".
The domain of a binary relation f is dom f={a | (a;b)\in f}.
The restriction of a binary relation f to a set A is
f|_{A}={(a;b)\in f | a\in A}.
> > Why you call the thread "Ordering Binary Relations"? I don't
> > understand how this is related with ordering binary relations.
>
> I'm starting with your two papers on binary relations as mentioned
> in my previous post and also mention in your web site in the misc. section
> of your works.
Now I understood that you a telling about "Misc" section of my site,
not the "General Topology" session.
Misc session does not deserve Abel Prize.
My serious works are in "General Topology" section:
http://www.mathematics21.org/algebraic-general-topology.html
However recently I have invented how my "Theorem about binary relation
limited to a set" probably may be used in Algebraic General Topology.
(Not now a time to speak about this however.)
> After that, I will get to the next papers on filters which protent to be
> interesting, if presented in the propriate formate. I'll skip the paper
> on catagory theory as catagory theory doesn't interest me.
My research is not category theory based like some other
generalizations of point-set topology.
It however significantly use category theory to define the notion of
continuousness:
http://www.mathematics21.org/binaries/continuousness.pdf
If you want to deal with anything related to continuousness in my
theory you need basic category theory. (I do not use advanced category
theory indeed, only the basic.)
William Elliot
> The domain of a binary relation f is dom f={a | (a;b)\in f}.
>
Don't you mean f = { a | some b with (a,b) in f } ?
When f subset XxY, dom f = p1(f), where
p1 is the first or X projection of XxY.
> The restriction of a binary relation f to a set A is
> f|_{A}={(a;b)\in f | a\in A}.
f|A = f /\ AxY, /\ is intersection.
> Now I understood that you a telling about "Misc" section of my site,
> not the "General Topology" session.
First things first. Besides it was readable while your others are not.
Well mostly readable. It is missing many important math symbols. Most
of which I've been able to desern. Here's a section I cannot understand
Will you fill in the missing symbols?
Proposition. f g <=> f* g* <=> f* g*.
Theorem. f g iff x:( f*(x) = g*(x) f*(x) = ).
Theorem. Vertical order is a meet-semilattice (that is has infimum of
any two binary relations).
Proof. This follows from f g <=> f* g* and that any two elements of
the set of functions, whose images are sets of sets, have the infimum
(namely set theoretic intersection). The operation * is a partial
order isomorphism and so maps a semilattice to semilattice. Vertical
infimum of two binary relations f and g can be so restored by the
formula (f g)* = f* g*. End of proof..
Except for this snafu, I expect to quickly finish with your two misc.
papers. The results I will post within a day or two after your reply. It
will be in the thread "Ordering Binary Relations". Look for it. You'll
see major revisions and proofs in much greater detail and in a simpler
fashion because of the use of algebraic set theory.
----
Victor Porton
> On Sun, 14 Sep 2008, Victor Porton wrote:
> > The domain of a binary relation f is dom f={a | (a;b)\in f}.
>
> Don't you mean f = { a | some b with (a,b) in f } ?
Yes, I mean this.
> When f subset XxY, dom f = p1(f), where
> p1 is the first or X projection of XxY.
>
> > The restriction of a binary relation f to a set A is
> > f|_{A}={(a;b)\in f | a\in A}.
>
> f|A = f /\ AxY, /\ is intersection.
Yes.
> First things first. Besides it was readable while your others are not.
> Well mostly readable. It is missing many important math symbols. Most
> of which I've been able to desern. Here's a section I cannot understand
> Will you fill in the missing symbols?
Oh, now I noticed that my XML processor was wrongly configured.
So there were some missing symbols in
http://www.mathematics21.org/misc/vertical-order.html
and
http://www.mathematics21.org/misc/limiting-binary-relations-theorem.html
I have corrected these, now the Unicode symbols are in their places.