“ Varieties of residuated lattices with an MV-retract and an investigation into state theory”
Advisor: Andrea Sorbi
Co-Advisor: Tommaso Flaminio
My PhD thesis has two main topics:
- In the shade of the algebraic study of substructural logics, we introduce new ways of constructing bounded (commutative, integral) residuated lattices, generalizing the notion of disconnected and connected rotation. The varieties generated have a retraction testified by a term into an MV-algebra, or, as a special case a Boolean algebra. The class characterized is quite large, some of the examples are given by: Gödel algebras, product algebras, Stonean residuated lattices, perfect MV-algebras, NM-algebras, n-contractive BL-algebras. We give a categorical representation by means of categories whose objects are triples made of an MV-algebra (or a Boolean algebra), a residuated lattice and an operator representing the algebraic join of MV-elements (or Boolean elements) and the elements of the residuated lattice. As a corollary, we obtain categorical equivalences between the classes of involutive algebras generated by our construction from one side, and the ones generated by liftings of residuated lattices from the other side. Moreover, we directly exhibit a (weak) Boolean product representation for the algebras of our varieties, and then we study the posets of prime lattice filters for which we show an order isomorphism with a structure constructed from the ultrafilters of the Boolean skeleton and the prime lattice filters of the radical that aims at dualizing our triple construction.
- The second part of the thesis is concerned with an investigation into the theory of states, namely a generalization of probability theory to the many-valued context, for some of the structures studied in the first part. In particular, we axiomatize a notion of state for the Lindenbaum algebra of product logic, that results in characterizing Lebesgue integrals of truth-functions of product logic formulas with respect to regular Borel probability measures. The relation between our states and regular Borel probability measures is one-one. Moreover every state belongs to the convex closure of product logic valuations.
Then, we define a notion of hyperreal-valued state of perfect MV-algebras. Such a notion is generalized to the variety generated by involutive perfect MTL-algebras. In order to do so, we also define a notion of state of prelinear semihoops (unbounded MTL-algebras), going through the intuition that any lattice-ordered monoid has an homomorphism into an l-group, using Grothendiek well-known construction.
“A categorical equivalence for product algebras”
Advisor: Franco Montagna
We obtain a categorical equivalence between the algebraic category of product algebras and a category whose objects are triplets made of a Boolean algebra, a cancellative hoop (i.e. a negative cone of a lattice ordered abelian group) and an operator that intuitively represents the join between boolean and cancellative elements.
“Qual è la logica dei matematici?” (Which is the logic of mathematicians?)
Advisor: Franco Montagna
We develop a logic whose purpose is to represent mathematical reasoning, combining classical and intuitionistic logic. In order to elaborate it, we use modal logic S4, with Kripke models for the semantics.