Thursday, 4 August, 11:00-12:00, Room 4102


Zlil Sela (Hebrew University)


The First Order Theory of Free Products of Groups


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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