6 research outputs found
Representing quantum structures as near semirings
In this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Ćukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a Ćukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of Ćukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Ćukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras
Stone-type representations and dualities for varieties of bisemilattices
In this article we will focus our attention on the variety of distributive
bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and
involutive bisemilattices. After extending Balbes' representation theorem to
bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn
duality and introduce the categories of 2spaces and 2spaces. The
categories of 2spaces and 2spaces will play with respect to the
categories of distributive bisemilattices and De Morgan bisemilattices,
respectively, a role analogous to the category of Stone spaces with respect to
the category of Boolean algebras. Actually, the aim of this work is to show
that these categories are, in fact, dually equivalent
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 eâ”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 pseudoeâ”ect algebras, i.e. non-commutative gener- alizations of lattice eâ”ect algebras can be represented as near semirings. One one side, this result allows the arithmetical treatment of pseudoeâ”ect algebras as total structures; on the other, it shows that near semirings framework can be really seen as the common âgroundâ 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 eâ”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
Algebraic structures from quantum and fuzzy logics
This thesis concerns the wide research area of logic. In particular, the first
part is devoted to analyze different kinds of relational systems (orthogonal
and residuated), by investigating the properties of the algebras associated
to them. The second part is focused on algebras of logic, in particular, the
relationship between prominent quantum and fuzzy structures with certain
semirings is proved. The last chapter concerns an application of group theory
to some well known mathematical puzzles