- [1)] Closure:
*A*and*B*in the group implies*A*+*B*in the group. - [2)] Associativity:
*A*,*B*,*C*in the group implies that*A*+ (*B*+*C*) = (*A*+*B*) +*C*. - [3)] Identity: There exists an element
*E*in the group such that*A*+*E*=*E*+*A*=*A*for all*A*in the group. - [4)] Inverse: For every element
*A*in the group there exists another element in the group such that .

- [5)] Commutativity: For all
*A*and*B*in the group*A*+*B*=*B*+*A*.

For our spring example the elements in the group are springs parameterized
by their compliance 1/*k*.
The binary operator ``+'' represents connecting two springs in series
and replacing the result with an equivalent single spring.
Note that in terms of compliance (the ``spring-group'' representation)
``+'' behaves exactly like standard addition.
Given this definition,
closure, associativity, and commutativity are obvious enough.
The identity element is just the infinitely stiff spring with 1/*k* = 0.
In order to have inverses as required we must allow 1/*k* < 0;
to get the inverse of a given spring just change the sign
on the spring compliance 1/*k*.

It appears that if we allow springs with negative *k* into our set
the ``spring group'' is indeed a formal Abelian group.
Is there any practical reason to do so?
The group notation does not seem particularly enlightening for the
trivial case of springs in series,
but the group concept of inverse elements is quite a useful one
for more complex equivalent-medium systems such as the Schoenberg-Muir
calculus.
Inverse elements form the theoretical basis for layer stripping and
more generally allow us to decompose a bulk equivalent
(what we can actually observe) back into candidate heterogeneous models
that may represent the real earth.

11/17/1997