Complex vector lattices via functional completions

We show that the Fremlin tensor product C(X)⊗¯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.