A General View on Normal Form Theorems for Łukasiewicz Logic with Product

In this survey paper we explore the connection between the Pierce- Birkhoff conjecture and Lukasiewicz logic with product. Conservative extensions of Lukasiewicz logic can be defined by adding an internal product or a multiplication with scalars from [0; 1]. The corresponding models reflect an algebraic hierarchy of lattice-ordered structures, from groups to algebras. We prove a general version of the normal form theorem and we state a local version of the Pierce-Birkhoff conjecture