Varieties of residuated lattices with an MV-retract and an investigation into state theory

In the shade of the algebraic study of substructural logics, we deal with the variety of residuated lattices. In particular, in the first part of this thesis we will introduce new ways of constructing bounded (commutative, integral) residuated lattices, generalizing in this context, among other things, the notion of disconnected and connected rotation. The class of structures that can be described by this approach is rather large. Some of the generated varieties are well-known algebraic semantics of substructural logics, such as product algebras, Godel algebras, the variety generated by perfect MV-algebras, NM-algebras, n-potent BL-algebras, Stonean residuated lattices, pseudocomplemented MTL-algebras. Furthermore, starting from any variety of (commutative, integral) residuated lattices, via our construction we generate and characterize new varieties of bounded (commutative, integral) residuated lattices, that will hence correspond to new substructural logics. Such varieties will result in having a retraction testified by a term into an MV-algebra, or, as a special case and starting point of our investigation, a Boolean algebra. Moreover, we will give a categorical representation of such generated varieties by means of categories whose objects are triples made of an MV- algebra (or, as a special case, a Boolean algebra), a residuated lattice and an operator intuitively representing the algebraic join of MV-elements (or Boolean elements) and the elements of the residuated lattice. As a corollary, we will 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. Subsequently, we focus on prime filters for the algebras with a Boolean retract. First, we directly exhibit a (weak) Boolean product representation for the algebras our varieties, and then we study the posets of prime lattice filters. For the latter, given any algebra in our varieties, we show an order isomorphism between its poset of prime lattice filters and a structure constructed from the ultrafilters of the Boolean skeleton and the prime lattice filters of the radical of the algebra that aims at dualizing our triple construction. The second part of the thesis is concerned with an investigation into the theory of states for some of the structures studied in the first part. In particular, our first main contribution with respect to state theory consists in introducing and studying states for product logic. We axiomatize a notion of state that results in characterizing Lebesgue integrals of truth-functions of product logic formulas with respect to regular Borel probability measures. We prove that the relation between our states and regular Borel probability measures is one-one. Moreover, and interestingly, we prove that every state belongs to the convex closure of product logic valuations. Then, we will use the algebraic decomposition theorems proved in the first part of the thesis to define states on some interesting subvarieties of MTL-algebras. In particular, we will define a notion of hyperreal-valued state (or hyperstate) of perfect MV-algebras, for which Mundici’s notion trivializes to only one possible state taking just Boolean values, 0 and 1. Such a notion will be generalized to the variety generated by involutive perfect MTL-algebras, which is the variety of involutive MTL-algebras satisfying Di Nola and Lettieri equation 2(x^2) = (2x)^2. In order to do so, we will also define a notion of state of GMTL-algebras (unbounded MTL-algebras). For doing so, we will go through the intuition that any lattice-ordered monoid has an homomorphism into an l-group, using Grothendiek well-known construction