  This paper aims at connecting the various classes that provide an algebraic
semantics for three different conservative expansions of Lukasiewicz logic,
using algebraic and category-theoretical techniques. We connect such classes of
algebras by adjunctions, using the tensor product of MV-algebras and defining
the tensor PMV-algebra of a semisimple MV-algebra, inspired by the construction
of the tensor algebra of a vector space. We further apply the main results to
prove amalgamation properties and, via categorical equivalence, we transfer all
results to the framework of lattice- ordered groups.
