Towards understanding the Pierce-Birkhoff conjecture via MV-algebras

TL;DR

This paper introduces fMV-algebras, a new algebraic structure linking Łukasiewicz logic with product to the Pierce-Birkhoff conjecture, and shows how their normal form theorem relates to a local version of the conjecture.

Contribution

It defines fMV-algebras and demonstrates their connection to the Pierce-Birkhoff conjecture through a normal form theorem.

Findings

fMV-algebras unify MV-algebras with product and scalar multiplication.

The normal form theorem offers a local perspective on the Pierce-Birkhoff conjecture.

Provides a new algebraic framework for understanding the conjecture.

Abstract

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 \textit{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.