Connecting the algebras of ？ukasiewicz logic with product: an application of the MV-algebraic tensor product

Using the semisimple tensor product of MV-algebras, we define the tensor PMV-algebra of an MV-algebra and we establish functorial adjunctions between the subcategory of semisimple MV-algebras and the subcategories of structures obtained by adding product operations (Riesz MV-algebras, PMV-algebras, \textit{f}MV-algebras). As consequence we prove the amalgamation property for unital and semisimple PMV-algebras, semisimple Riesz MV-algebras, unital and semisimple \textit{f}MV-algebras. Moreover, we characterize the free PMV-algebra and the free \textit{f}MV-algebra using the tensor product. Finally, we transfer all the results to lattice-ordered structures via categorical equivalence.