Monoidal uniqueness theorems for stable homotopy theory

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, \W-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. The equivalences of modules, algebras and commutative algebras in these categories are also considered.