Towards understanding the Pierce-Birkhoff conjecture via MV-algebras

Our main issue was to understand the connection between Łukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of fMV-algebras, which are MV-algebras endowed with an internal binary product and with 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.