Towards understanding the Pierce-Birkhoff conjecture via MV-algebras

Our main issue was to understand the connection between \L ukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of $\mathit{\text{f}}$MV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from $[0,1]$. The proper quasi-variety generated by $[0,1]$, with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture.