Riesz MV-algebras Research Articles

Models, coproducts and exchangeability: Notes on states on Baire functions

Abstract We discuss exchangeability and independence in the setting of σ-complete Riesz MV-algebras. We define and link to each other the notions of exchangeability and distribution law for a sequence of observables (i.e. non classical random variables), as well as the notion of independence for a sequence of algebras. We obtain two categorical dualities for σ-complete Riesz MV-algebras endowed with states and we define a “canonical” state on the coproduct of a sequence of probability Riesz tribes, giving a weak version of de Finetti’s result. Finally, we discuss statistical models.

De Finetti's coherence and exchangeability in infinitary logic

We continue the investigation towards a logic-based approach to statistics within the infinitary conservative extension of Łukasiewicz logic IRL and prove versions of de Finetti's theorems on coherence and exchangeability. In particular we will prove a coherence criterion for a subclass of the variety of σ-complete Riesz MV-algebras in the conditional and unconditional case, and discuss de Finetti's exchangeability in a special case.

Expanding Lattice Ordered Abelian Groups to Riesz Spaces

Abstract First we give a necessary and sufficient condition for an abelian lattice ordered group to admit an expansion to a Riesz space (or vector lattice). Then we construct a totally ordered abelian group with two non-isomorphic Riesz space structures, thus improving a previous paper where the example was a non-totally ordered lattice ordered abelian group. This answers a question raised by Conrad in 1975. We give also a partial solution to another problem considered in the same paper. Finally, we apply our results to MV-algebras and Riesz MV-algebras.

Dualities and algebraic geometry of Baire functions in non-classical logic

AbstractIn this paper we aim at completing the study of $\sigma $-complete Riesz MV-algebras that started in Di Nola et al. (2018, J. Logic Comput., 28, 1275–1292). To do so, we discuss polynomials, algebraic geometry and dualities in the infinitary variety of such algebras. In particular, we characterize the free objects as algebras of Baire-measurable functions and we generalize two dualities, namely the Marra–Spada duality and the Gelfand duality, obtaining a duality with basically disconnected compact Hausdorff spaces and an equivalence with Rickart $C^*$-algebras.

An approach to stochastic processes via non-classical logic

Within the infinitary variety of σ-complete Riesz MV-algebras RMVσ, we introduce the algebraic analogue of a random variable as a homomorphism defined on the free algebra in RMVσ. After a preliminary study of the proposed notion, we use it to define stochastic processes in the framework of non-classical logic (Łukasiewicz logic, more precisely) and we define stochastic independence.

Some remarks on divisible polyhedral MV-algebras

AbstractBuilding on similar notions for MV-algebras, polyhedral DMV-algebras are defined and investigated. For such algebras dualities with suitable categories of polyhedra are established, and the relation with finitely presented Riesz MV-algebras is investigated. Via hull-functors, finite products are interpreted in terms of hom-functors, and categories of polyhedral MV-algebras, polyhedral DMV-algebras and finitely presented Riesz MV-algebras are linked together. Moreover, the amalgamation property is proved for finitely presented DMV-algebras and Riesz MV-algebras, and for polyhedral DMV-algebras.

IoT Devices Signals Processing Based on Shepard Local Approximation Operators Defined in Riesz MV-Algebras

The interconnection network considered in this paper is the complete WK-Recursive network that demonstrates many attractive properties, such as high degree of regularity, symmetry and efficient communication. Chen and Duh have proposed a distributed stack-base broadcasting algorithm for the complete WK-Recursive networks [Networks, 24 (1994) 303-317]. To perform this algorithm, a stack of O(log N) elements, where N is the number of nodes, to keep the labels of links is included in each message. Moreover, as a node receives the message, a series of O(log N) pop and push operations on the stack is required. In this paper, we present a novel broadcasting algorithm for the complete WKRecursive network, which is much simpler and requires only constant data included in each message and constant time to determine the neighbors to forward the message. Povzetek: Opisan je nov algoritem razposiljanja v rekurzivni mreži.

On the Riesz structures of a lattice ordered abelian group

Abstract A Riesz structure on a lattice ordered abelian group G is a real vector space structure where the product of a positive element of G and a positive real is positive. In this paper we show that for every cardinal k there is a totally ordered abelian group with at least k Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same l-group with strong unit. This gives a solution to a problem posed by Conrad in 1975. Finally we apply the main result to MV-algebras and Riesz MV-algebras.

Risk analysis via Łukasiewicz logic

In this paper, we apply logical methods to risk analysis. We study generalized events, i.e. not yes-no events but continuous ones. We define on this class of events a risk function and a measure over it to analyse risk in this context. We use Riesz MV-algebras as algebraic structures and their associated logic in support of our research, thanks to their relations with other applications. Moreover, we investigate on decidability of consequence problem for our class of risk.

IoT Devices Signals Processing based on Multi-dimensional Shepard Local Approximation Operators in Riesz MV-algebras

In this article we continue the study started in [8] to use Riesz MValgebras for IoT devices signals processing. The Shepard local approximation operators introduced in [8] were defines such that to approximate single variable functions. In real industrial usage, the signals coming from IoT devices may be influenced by mode than a parameter, and thus we introduce generalized Shepard local approximation operators that can approximate multi-dimensional functions and some numerical experiments are considered.

An analysis of the logic of Riesz spaces with strong unit

We study Łukasiewicz logic enriched by a scalar multiplication with scalars in [0,1]. Its algebraic models, called Riesz MV-algebras, are, up to isomorphism, unit intervals of Riesz spaces with strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of DMV-algebras and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective.

Riesz–McNaughton functions and Riesz MV-algebras of nonlinear functions

We focus on Riesz MV-algebras, which are MV-algebras equipped with a multiplication by numbers in the real interval [0,1]. We consider for every integer n the Riesz MV-algebra of all continuous functions from the n-th power of [0,1] to [0,1] and the Riesz MV-subalgebras thereof. In particular we study the Riesz MV-subalgebras isomorphic to free Riesz MV-algebras with finitely many generators, possibly different from the usual linear models given by what we call Riesz–McNaughton functions (and which generalize McNaughton functions used in the case of MV-algebras). In doing this we characterize zerosets of Riesz–McNaughton functions by means of polyhedra, and we extend to Riesz MV-algebras a duality for MV-algebras exposed in a paper by Marra and Spada.

The Riesz Hull of a Semisimple MV-Algebra

Abstract MV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

Some Notes on Elimination Properties for The Theory of Riesz MV-Chains

Abstract Riesz MV-algebras are a variety of algebras strongly connected to Riesz spaces. In this short article we investigate some elimination properties of the first-order theory RMV of linearly ordered Riesz MV-algebras and show that RMV admits elimination of quantifiers and uniform elimination of imaginary elements. In the process, we also prove several other results such as modelcompleteness, o-minimality, definability of Skolem functions, and a version of the Di Nola Representation Theorem for Riesz MV-algebras.

Łukasiewicz logic and Riesz spaces

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from \([0,1]\). Extending Mundici’s equivalence between MV-algebras and \(\ell \)-groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C\(^*\)-algebras. The propositional calculus \({\mathbb R}{\mathcal L}\) that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz \(\infty \)-valued propositional calculus and is complete with respect to evaluations in the standard model \([0,1]\). We prove a normal form theorem for this logic, extending McNaughton theorem for Ł ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in \({\mathbb R}{\mathcal L},\) and relate them with the analogue of de Finetti’s coherence criterion for \({\mathbb R}{\mathcal L}\).

Elementary calculus in Riesz MV-algebras

We introduce and study a topological structure over vectorial and Riesz MV-algebras, similar to metric spaces. Continuous functions with values in these spaces are studied and a fix point theorem is obtained. Moreover we introduce the concept of derivative and integral of MV-valued functions; i.e. functions with values in a Riesz MV-algebra. Finally we present two applications.