Journal of the
Korean Mathematical Society JKMS

A useful method of proving the finite decidability of an equationally
definable class \ensuremath{\mathcal V}\ of algebras (i.e., \emph{variety}) %{V}c
is to prove the decidability of the class of finite directly indecomposable
members of \cV. The validity of this method relies on the well-known
result of Feferman-Vaught: \emph{if a class \cK\ of first-order structures
is decidable, then so is the class}
$\{ \prod_{i In this paper we show that the converse of this does not necessarily hold.

Keywords: decidability, direct products, Feferman-Vaught theorem

MSC numbers: 03B25, 08A60