Summary of the PhD Thesis “MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Lukasiewicz logic”

The framework. In 1956, G. Birkhoff G. and R.S. Pierce [1] conjectured the following: do the algebra of piecewise polynomial functions with real coefficients and the algebra of functions that can be written as Inf and Sup of finitely many polynomials coincide? The conjecture – that can be equivalently stated asking for the free n-generated lattice-ordered algebra to be isomorphic with the algebra of piecewise polynomial functions in n-variables – is still open. Many authors have approached the problem and the only answer so far is provided by L. Mahé in [11], who proved the conjecture for functions with at most two variables. The aim of the thesis was to consider the problem from a fresh perspective. A seminal link with logic was provided by F. Montagna and G. Panti, that in [14] remarked that the study of free objects in varieties of MV-algebras with product is related to Pierce-Birkhoff conjecture. To give a more precise idea of how the framework of MV-algebras can be an appropriate choice for trying to solve the conjecture, some background is needed. MV-algebras are the algebraic counterpart of Lukasiewicz logic and they are structures (A,⊕,∗ , 0) of type (2, 1, 0) satisfying some appropriate axioms. The variety of MV-algebras is generated by ([0, 1],⊕,∗ , 0) where x ⊕ y = min(1, x + y) and x∗ = 1− x for any x, y ∈ [0, 1]. One of the main achievement in the theory of MV-algebras is the categorical equivalence with Abelian lattice-ordered groups with strong unit established by D. Mundici. A strong unit is a positive element u such that for any x in the group there exists an n ∈ N such that x ≤ nu. As the unit interval [0, 1] is closed under the product of real numbers, a fruitful research direction proved to be the idea of endowing an MV-algebra with products, either internal or scalar. If the real product is interpreted as a binary operation on [0, 1], the corresponding structures are MV-algebras A endowed with an operation · : A×A→ A. These algebras were introduced and studied by A. Di Nola and A. Dvurečenskij under the name of PMV-algebras in [3] and they are categorically equivalent to a class of lattice-ordered rings with strong unit. Particular important subclasses were further investigated by F. Montagna [13, 15], with the aim of studying the quasi-variety generated by [0, 1]PMV = ([0, 1],⊕, ·,∗ , 0). Another approach is to consider the real product on [0, 1] as a multiplication with scalars in [0, 1]. The standard model in this case is [0, 1]RMV = ([0, 1],⊕,∗ , {α |