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

Categories of Residuated Lattices

We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on extended Priestley duality in which the ternary relation dualizing a residuated multiplication may be viewed as the graph of a partial function. We also present a new Esakia-like duality for Sugihara monoids in the spirit of Dunn\u27s binary Kripke-style semantics for the relevance logic R-mingle

Residuated structures and orthomodular lattices

The variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., ℓ-groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated ℓ-groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated ℓ-groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated ℓ-groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices

Deciding Equations in the Time Warp Algebra

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover

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

Embedding theorems and finiteness properties for residuated structures and substructural logics

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.Paper 1. This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. Paper 2. It is proved that the variety of relevant disjunction lattices has the finite embeddability property (FEP). It follows that Avron’s relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron’s result that RMImin is decidable. Paper 3. An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity t is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form a variety SCIL, which is not locally finite, but it is proved that SCIL has the FEP. More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains. Paper 4. Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x (x t) x. Since the semiconic algebras in IP are locally finite, it follows that when an axiomatic extension of RMO has ((p t) p) p among its theorems, then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized