  We show that the Fremlin tensor product $C(X)\bar{\otimes}C(Y)$ is not square
mean complete when X and Y are uncountable metrizable compact spaces. This
motivates the definition of complexification of Archimedean vector lattices,
the Fremlin tensor product of Archimedean complex vector lattices, and a theory
of powers of Archimedean complex vector lattices.
