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Around 1956 R. Vaught asked the following
natural question. Let A,B,C,D
be arbitrary groups.
Suppose that A and B have the same first order
theory (such groups
are called elementarily equivalent), and so do C
and D. Do A*C and
B*D have the same first order theory? (i.e., is
elementary equivalence
preserved under free products of groups?)
A similar question
for (generalized) direct products (of general
structures) was
answered affirmatively by Mostowski in 1952,
and later generalized by Feferman
and Vaught in 1959. On the
other hand Olin proved in 1974
that the answer to Vaught's
question is negative
if we replace groups by semigroups.
We develop a
geometric structure theory that is based on the tools that
were developed to
solve Tarski's problem on the first order theory of a
free group to answer
Vaught's problem affirmatively. This structure
theory suggests a
generalization of Tarski's problem to free products of
arbitrary groups, as
well as other (somehwat surprising) results in model
theory over groups. It suggests
open questions, and will probably have
Generalizations in quite a few
directions.
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