Remarks on the order-theoretical and algebraic properties of quantum structures

This thesis is concerned with the analysis of common features and distinguishing traits of algebraic structures arising in the sharp as well as in the unsharp approaches to quan- tum theory both from an order-theoretical and an algebraic perspective. Firstly, we recall basic notions of order theory and universal algebra. Furthermore, we introduce fundamental concepts and facts concerning the algebraic structures we deal with, from orthomodular lattices to e↵ect algebras, MV algebras and their non-commutative gener- alizations. Finally, we present Basic algebras as a general framework in which (lattice) quantum structures can be studied from an universal algebraic perspective. Taking advantage of the categorical (term-)equivalence between Basic algebras and Lukasiewicz near semirings, in Chapter 3 we provide a structure theory for the lat- ter in order to highlight that, if turned into near-semirings, orthomodular lattices, MV algebras and Basic algebras determine ideals amenable of a common simple description. As a consequence, we provide a rather general Cantor-Bernstein Theorem for involutive left-residuable near semirings. In Chapter 4, we show that lattice pseudoe↵ect algebras, i.e. non-commutative gener- alizations of lattice e↵ect algebras can be represented as near semirings. One one side, this result allows the arithmetical treatment of pseudoe↵ect algebras as total structures; on the other, it shows that near semirings framework can be really seen as the common “ground” on which (commutative and non commutative) quantum structures can be studied and compared. In Chapter 5 we show that modular paraorthomodular lattices can be represented as semiring-like structures by first converting them into (left-) residuated structures. To this aim, we show that any modular bonded lattice A with antitone involution satisfying a strengthened form of regularity can be turned into a left-residuated groupoid. This condition turns out to be a sucient and necessary for a Kleene lattice to be equipped with a Boolean-like material implication. Finally, in order to highlight order theoretical peculiarities of orthomodular quantum structures, in Chapter 6 we weaken the notion of orthomodularity for posets by introduc- ing the concept of the generalized orthomodularity property (GO-property) expressed in terms of LU-operators. This seemingly mild generalization of orthomodular posets and its order theoretical analysis yields rather strong applications to e↵ect algebras, and orthomodular structures. Also, for several classes of orthoalgebras, the GO-property yields a completely order-theoretical characterization of the coherence law and, in turn, of proper orthoalgebras.