Re: Generalized Equivalence Relations?

In article
<32158410.1116006215100.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
John Rickert <JPRickert@xxxxxxxxx> wrote:

> ....
> I have a question about what might be called ³generalized equivalence
> relations.² The most primitive question is simply that, whereas an
> equivalence relation consists of a set of ordered pairs (a, b), is there
> a good extension of the key ideas to general n-tuples (a, b, c, ..., k) ?
> ....



Here's a genuine mathematical example of a ternary equivalence
relation in the theory of plane Bachmann groups, a very elegant approach
to geometry (which may or may not be Euclidean).

Each lower-case letter a, b, c, .... represents reflection in a
line, but such reflections are called simply "lines" for short. A
product such as abc represents the composite of a reflection in a,
then in b, then in c. Sometimes such a product may itself be a line
(reflection). This happens if lines a, b, c have a common point or a
common perpendicular, and also if they are suitable hyperbolic parallels.

Anyway, we say (a, b, c) is in a certain ternary relation iff abc
is a line.

This relation is reflexive, in the sense that if any two of a, b,
c are equal then abc is a line.

It is symmetric, in the sense that if abc is a line then so are
acb, bac, bca, cab, cba (but these lines may be different from abc).

It is transitive, in the sense that if a != b, and both abc and
abd are lines, then acd is a line.

The proofs of the first two properties are fairly simple, but the
transitivity is much harder and uses deeper geometrical ideas.

Now, for any two distinct lines a, b, we have an equivalence class
{x : abx is a line}. Such a class is called a pencil. Hence if abc is
a line then we say that a, b, c lie in a pencil. (These are the words
used by Bachmann and his school to describe the ternary equivalence
relation.)

Under a binary equivalence relation of the familiar sort, each
element lies in a unique equivalence class. Under this ternary
relation, each pair of distinct lines lies in a unique pencil.

If you'd like to read more about this beautiful theory, the
standard reference is probably still Friedrich Bachmann, "Aufbau der
Geometrie aus dem Spiegelungsbegriff." AFAIK the best introduction in
English is H. Behnke et al. (editors), "Fundamentals of Mathematics,"
Vol. II "Geometry," Chapter 5. There are shorter treatments, not
getting as far as transitivity, in Günter Ewald, "Geometry: an
Introduction," Chapter 1, and in Michael Henle, "Modern Geometries,"
Chapter 25. Some of those books are out of print.

Ken Pledger.
.

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