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From: kimle@willow.canberra.edu.au (Kim Le)
Subject: Re: Transitivity for similarity
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Date: 25 Nov 94 04:23:58 GMT
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In <3alern$1v9@bambi.zdv.Uni-Mainz.DE> beckmann@Informatik.Mathematik.Uni-Mainz.DE (Markus Beckmann) writes:


>Can anybody tell me, WHY transitivity is demanded for
>similarity relations? I.e. if I have a domain X and
>a relation R 

>	R(x,z) >= R(x,y) AND R(y,z)       [*]

>should hold for all x,y,z in X.

>If I have tomato, a red sports car and a skateboard I 
>might get:

>	R(tomato,car) > 0           (for the color)
>	R(car,skateboard) > 0       (for the wheels)

>but

>	R(tomato.skateboard) = 0

>so [*] does not hold!!!

>Can you help me?

>Thanks in advance,
>Markus A. Beckmann

>+=========================================================+
>: Markus A. Beckmann               !                      :
>: Johannes Gutenberg-Universitaet  ! Tel.: 06131/39-4358  :
>: Institut fuer Informatik         ! Fax : 06131/39-3534  :
>: Postfach 39 80                   ! i.-nat.: ++49 6131/  :
>: D-55099 Mainz                    !                      :
>+---------------------------------------------------------+
>:   e-mail: beckmann@informatik.mathematik.uni-mainz.de   :
>+=========================================================+
Refer to Bezdek, J.C. "Fuzzy PArtititions and Relations; an Axiomatic
Basis for Clustering" in "Fuzzy Models for Pattern Recognition", IEEE
Press, 1992.
There are three conditions for transitivity: Minimum, product, and
Delta. Minimum and product transitivity are too dense for similarity
relation. Delta transitivity is more appropriate to similarity relation
in reality.
Kim Le, University of Canberra.

