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##### Algebraic Analysis of Demodalised Analytic Implication with Francesco Paoli and Michele Pra Baldi Journal of Philosophical Logic 48 (6): 957-979. 2019.
The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn as a variation on a time-honoured logical system by C.I. Lewis’ student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is “analytically contained” in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several differen…Read more
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##### Stone-Type Representations and Dualities for Varieties of Bisemilattices Studia Logica 106 (2): 417-448. 2018.
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 bi…Read more
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##### Expanding Quasi-MV Algebras by a Quantum Operator with Roberto Giuntini and Francesco Paoli Studia Logica 87 (1): 99-128. 2007.
We investigate an expansion of quasi-MV algebras ([10]) by a genuine quantum unary operator. The variety of such quasi-MV algebras has a subquasivariety whose members—called cartesian—can be obtained in an appropriate way out of MV algebras. After showing that cartesian . quasi-MV algebras generate ,we prove a standard completeness theorem for w.r.t. an algebra over the complex numbers.
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##### Completion and amalgamation of bounded distributive quasi lattices with Majid Alizadeh and Hector Freytes Logic Journal of the IGPL 19 (1): 110-120. 2011.
In this note we present a completion for the variety of bounded distributive quasi lattices, and, inspired by a well-known idea of L.L. Maksimova [14], we apply this result in proving the amalgamation property for such a class of algebras
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##### MV-Algebras and Quantum Computation with Martinvaldo Konig, Francesco Paoli, and Roberto Giuntini Studia Logica 82 (2): 245-270. 2006.
We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.
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##### The Toffoli-Hadamard Gate System: an Algebraic Approach with Maria Luisa Dalla Chiara, Antonio Ledda, Giuseppe Sergioli, and Roberto Giuntini Journal of Philosophical Logic 42 (3): 467-481. 2013.
Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. The basic algebraic properties of this system have been studied in Dalla Chiara et al. (Foundations of Physics 39(6):559–572, 2009), where we have introduced the notion of Shi-Aharonov quantum computational structure. In this paper we propose an algebraic abstraction from the Hilbert-space quantum computational structures, by introducing the notion o…Read more
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##### On Certain Quasivarieties of Quasi-MV Algebras with T. Kowalski and F. Paoli Studia Logica 98 (1-2): 149-174. 2011.
Quasi-MV algebras are generalisations of MV algebras arising in quantum computational logic. Although a reasonably complete description of the lattice of subvarieties of quasi-MV algebras has already been provided, the problem of extending this description to the setting of quasivarieties has so far remained open. Given its apparent logical repercussions, we tackle the issue in the present paper. We especially focus on quasivarieties whose generators either are subalgebras of the standard square…Read more
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##### The Lattice of Subvarieties of $${\sqrt{\prime}}$$ quasi-MV Algebras with T. Kowalski, F. Paoli, and R. Giuntini Studia Logica 95 (1-2): 37-61. 2010.
In the present paper we continue the investigation of the lattice of subvarieties of the variety of ${\sqrt{\prime}}$ quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D r is not finitely based, and we provide an infinite equational basis for the same variety
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##### Entanglement as a semantic resource with Maria Luisa Dalla Chiara, Roberto Giuntini, Antonio Ledda, Roberto Leporini, and Giuseppe Sergioli Foundations of Physics 40 (9-10): 1494-1518. 2010.
The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregi…Read more
• ##### On some properties of quasi-MV algebras and $\sqrt{^{\prime }}$ quasi-MV algebras with Francesco Paoli, Roberto Giuntini, and Hector Freytes Reports on Mathematical Logic 31-63. 2009.
We investigate some properties of two varieties of algebras arising from quantum computation - quasi-MV algebras and $\sqrt{^{\prime }}$ quasi-MV algebras - first introduced in \cite{Ledda et al. 2006}, \cite{Giuntini et al. 200+} and tightly connected with fuzzy logic. We establish the finite model property and the congruence extension property for both varieties; we characterize the quasi-MV reducts and subreducts of $\sqrt{^{\prime }}$ quasi-MV algebras; we give a representation of semisimple…Read more
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##### A New View of Effects in a Hilbert Space with Roberto Giuntini and Francesco Paoli Studia Logica 104 (6): 1145-1177. 2016.
We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-…Read more
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##### The Lattice of Subvarieties of √′ quasi-MV Algebras with T. Kowalski, F. Paoli, and R. Giuntini Studia Logica 95 (1-2). 2010.
In the present paper we continue the investigation of the lattice of subvarieties of the variety of √′ P quasi-MV algebras, already started in [6]. Beside some general results on the structure of such a lattice, the main contribution of this work is the solution of a long-standing open problem concerning these algebras: namely, we show that the variety generated by the standard disk algebra D r is not finitely based, and we provide an infinite equational basis for the same variety
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