Scalar extensions for algebraic structures of Lukasiewicz logic

In this paper we study the tensor product for MV-algebras, the algebraic structures of \L ukasiewicz $\mathrm{\infty }$-valued logic. Our main results are: the proof that the tensor product is preserved by the categorical equivalence between the MV-algebras and abelian lattice-order groups with strong unit and the proof of the scalar extension property for semisimple MV-algebras. We explore consequences of this results for various classes of MV-algebras and lattice-ordered groups enriched with a product operation.