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Re: equivalence relation (quotient relation)

From: Yuzhong Qu <yzqu@seu.edu.cn>
Date: Fri, 11 Feb 2005 18:24:26 +0800
Message-ID: <011101c51023$de0de700$fd0b77ca@xobjects>
To: "Henry Story" <henry.story@bblfish.net>
Cc: "SWIG" <semantic-web@w3.org>
The definition of the quotientOf(p) [denoted as q] as follows:

x q y  iff there exists z such that x p z and y p z. (suppose p is functional)


You can Google it for a more formal definition.


Yuzhong Qu


----- Original Message ----- 
From: "Henry Story" <henry.story@bblfish.net>
To: "Yuzhong Qu" <yzqu@seu.edu.cn>
Sent: Friday, February 11, 2005 6:05 PM
Subject: Re: equivalence relation (quotient relation)


> Do you have a link to a good definition of the quotient relation you
> are thinking of?
> 
> I was not myself so much interested in the role of the
> 'owner' relation in the graph. I had just added it to make
> it clearer that there was a simplification that had taken place,
> and that relations that had previously applied to _blank, also applied
> to <http://bblfish.net/>. This is clearly true if "relation" is just
> the identity property, since two identical things have
> all the same properties.
> 
> Henry
> 
> 
> On 11 Feb 2005, at 10:28, Yuzhong Qu wrote:
> 
> >  From your Example:
> > ----------------------------------
> >
> >  _blank ---relation---> <http://bblfish.net/>
> >    |------owner-------> "Henry Story"
> >
> >       deduce the graph
> >
> >    <http://bblfish.net> ----owner----> "Henry Story"
> >
> > ------------------------------------
> >
> > I think that a "quotient relation" of a property (e.g. "owner") might 
> > be what you needed.
> >
> > Just use the following triple:
> >
> > :yourRelation rdfs:subPropertyOf  quotientOf(owner)
> >
> >
> > Note that quotientOf is a mathematical construct, and not yet been 
> > supported by OWL DL.
> >
> > BTW, is there any Description Logic supporting "quotient relation" ?
> >
> >
> > Yuzhong Qu
> > Dept.Computer Science & Engineering
> > Southeast University
> > Nanjing, China, 210096
> >
> >
> >
> >
> > ----- Original Message -----
> > From: "Henry Story" <henry.story@bblfish.net>
> > To: <semantic-web@w3.org>
> > Sent: Monday, February 07, 2005 9:57 PM
> > Subject: Re: equivalence relation
> >
> >
> >>
> >> I just realized that not everyone may know (including me btw.)
> >> what I mean by an equivalence relation .
> >>
> >> I think this is the concept I am getting at:
> >>
> >> <http://en.wikipedia.org/wiki/Equivalence_relation>
> >>
> >> Henry Story
> >>
> >>
> >> On 7 Feb 2005, at 09:02, Henry Story wrote:
> >>
> >>>
> >>> I am looking for a way to state that a relation is an equivalence
> >>> relation [1]. I want to know this so that I can starting from a graph
> >>> such as
> >>>
> >>> _blank ---relation---> <http://bblfish.net/>
> >>>   |------owner-------> "Henry Story"
> >>>
> >>> deduce the graph
> >>>
> >>> <http://bblfish.net> ----owner----> "Henry Story"
> >>>
> >>>
> >>> My thought was that a relation that is functional, symmetric and
> >>> transitive
> >>>  is just such a relation. Here is how I come to this conclusion.
> >>>
> >>> 1) Functional and symmetric
> >>>
> >>>  If a relation is functional and symmetric, then it is also
> >>>  inverse functional. It is a 1 to 1 mapping.
> >>>
> >>> 2) If it is functional, inverse functional and symmetric
> >>>
> >>>    then for all aRb we also have bRa
> >>>
> >>>    this still allows a and b to be different
> >>>
> >>> 3) if it is transitive then for any a, b and c, where
> >>>
> >>>    [1] aRb
> >>>    [2] bRc
> >>>
> >>>    then
> >>>
> >>>    [3] aRc
> >>>
> >>>    but since R is symmetric
> >>>
> >>>    from [2] bRc we deduce that
> >>>
> >>>    [4] cRb
> >>>
> >>>     and since R is inverse functional
> >>>
> >>>    from  [1] aRb and [4] cRb we deduce that a==c
> >>>
> >>>    similarly from [3] aRb, [1] aRc and the functional nature of R
> >>>    we deduce that c == b.
> >>>
> >>>    So a == c and c == b and so a == b.
> >>>
> >>> Is this reasoning ok?
> >>> I was hoping it would be, cause then I can just specify in OWL that
> >>> properties
> >>> are functional, symmetric and transitive if I want them to be
> >>> equivalence relations (or is there a shorthand for this)
> >>>
> >>> Henry Story
> >>>
> >>> [1] http://en.wikipedia.org/wiki/Equivalence_relation

> >>>
> >>
> >>
> >>
> 
> 
Received on Friday, 11 February 2005 10:24:17 UTC

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