## Geometries
with intransitive equivalence relation

## Yuri A. Rylov

*Institute for Problems in Mechanics, Russian
Academy of Sciences*

* 101-1 ,Vernadskii Ave., Moscow, 119526, Russia*

* email: rylov@ipmnet.ru*

Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm

or mirror Web site: http://gasdyn-ipm.ipmnet.ru/~rylov/yrylov.htm

Updated July 13, 2008

#### abstract

One considers
geometry with the intransitive equaivalence relation. Such a geometry is a
physical geometry, i.e. it is described completely by the world function, which
is a half of the squared distance function. The physical geometry cannot be
axiomatized, in general. It is obtained as a result of deformation of the
proper Euclidean geometry. Class of physical geometries is more powerful, than
the class of axiomatized geometries. The physical geometry admits one to
describe such geometric properties as discreteness, granularity and limited
divisibility. These properties are important in application to the space-time.
They admits one to explain the discrimination properties of the space-time,
which generate discrete parameters of elementary particles. Mathematical
formalism of a physical geometry is very simple. The physical geometry is
formulated in geometrical terms (in terms of points and world function) without
a use of means of description (coordinate system, space dimension, manifold,
etc.).

There is
text of the paper in English (pdf, ps) and in Russian (ps, pdf)