1 Preliminaries 1 . 1 An overview of MV-algebras with product and Lukasiewicz logic

This paper aims at connecting the various classes that provide an algebraic semantics for three different conservative expansions of Lukasiewicz logic, using algebraic and categorical theoretic techniques. We connect such classes of algebras by adjunctions, using the tensor product of MValgebras 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 latticeordered groups.