Product MV-algebra Research Articles

Logical Entropy of Partitions in Hyperproduct MV-algebras

In this paper, we introduce the algebraic structure hyperproduct MV-algebras which is a generalization of product MV-algebras. In addition, we study the logical entropy and the logical conditional entropy of partitions in a hyperproduct MV-algebra and provide their properties.

New Kind of MV-Modules

In this paper, by considering the notion of MV-modules, which is the structure that naturally correspond to lu-modules over lu-rings, we investigate some properties of a new kind of MV-modules, that we introduced in Borzooei and Saidi Goraghani, Free MV-modules, J. Intell. Fuzzy Syst. 31 (2016), 151–161 as Ak-modules. With the current situation, it was not easy for us to work on some concepts such as free MV-modules and Noetherian MV-modules. So we limited our scope of work by introducing a new kind of MV-modules. We define and study the notions of free Ak-modules, radical of Ak-modules and Noetherian Ak-modules, where A is a product MV-algebra and k ∈ ℕ. For example, we state a general representation for a free Ak-module, and we obtain conditions in which an Ak-module can be Noetherian.

Logical entropy of dynamical systems in product MV-algebras and general scheme

The present paper is aimed at studying the entropy of dynamical systems in product MV-algebras. First, by using the concept of logical entropy of a partition in a product MV-algebra introduced and studied by Markechová et al. (Entropy 20:129, 2018), we define the logical entropy of a dynamical system in the studied algebraic structure. In addition, we introduce a general type of entropy of a product MV-algebra dynamical system that includes the logical entropy and the Kolmogorov–Sinai entropy as special cases. It is proved that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.

On Injective MV-Modules

In this paper, by considering the notion of MV-module, which is the structure that naturally correspond to lu-modules over lu-rings, we study injective MV-modules and we investigate some conditions for constructing injective MV-modules. Then we define the notions of essential A-homomorphisms and essential extension of A-homomorphisms, where A is a product MV-algebra, and we get some of there properties. Finally, we prove that a maximal essential extension of any A-ideal of an injective MV-module is an injective A-module, too.

R-norm entropy and R-norm divergence in product MV-algebras

The aim of the paper is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in product MV-algebras to the case of R-norm entropy and R-norm divergence. We define the R-norm entropy of finite partitions in product MV-algebras and its conditional version and derive the basic properties of these entropy measures. In addition, we introduce the concept of R-norm divergence in product MV-algebras and we prove basic properties of this quantity. In particular, it is proven that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given product MV-algebra can be obtained as the limits of their R-norm divergence and R-norm entropy, respectively.

MVW-rigs and product MV-algebras

ABSTRACTWe introduce the variety of Many-Valued-Weak rigs (MVW-rigs). We provide an axiomatisation and establish, in this context, basic properties about ideals, homomorphisms, quotients and radicals. This new class contains the class of product MV-algebras presented by Di Nola and Dvurečenskij in 2001 and by Montagna in 2005. The main result is the compactness of the prime spectrum of this new class, endowed with the co-Zariski topology as defined by Dubuc and Poveda in 2010.

Tsallis Entropy of Product MV-Algebra Dynamical Systems.

This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order where of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order where has the property of sub-additivity. This property allows us to define, for the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems.

Rényi Entropy and Rényi Divergence in Product MV-Algebras.

This article deals with new concepts in a product MV-algebra, namely, with the concepts of Rényi entropy and Rényi divergence. We define the Rényi entropy of order q of a partition in a product MV-algebra and its conditional version and we study their properties. It is shown that the proposed concepts are consistent, in the case of the limit of q going to 1, with the Shannon entropy of partitions in a product MV-algebra defined and studied by Petrovičová (Soft Comput. 2000, 4, 41–44). Moreover, we introduce and study the notion of Rényi divergence in a product MV-algebra. It is proven that the Kullback–Leibler divergence of states on a given product MV-algebra introduced by Markechová and Riečan in (Entropy 2017, 19, 267) can be obtained as the limit of their Rényi divergence. In addition, the relationship between the Rényi entropy and the Rényi divergence as well as the relationship between the Rényi divergence and Kullback–Leibler divergence in a product MV-algebra are examined.

Logical Divergence, Logical Entropy, and Logical Mutual Information in Product MV-Algebras.

In the paper we propose, using the logical entropy function, a new kind of entropy in product MV-algebras, namely the logical entropy and its conditional version. Fundamental characteristics of these quantities have been shown and subsequently, the results regarding the logical entropy have been used to define the logical mutual information of experiments in the studied case. In addition, we define the logical cross entropy and logical divergence for the examined situation and prove basic properties of the suggested quantities. To illustrate the results, we provide several numerical examples.

Kullback–Leibler Divergence and Mutual Information of Partitions in Product MV Algebras

The purpose of the paper is to introduce, using the known results concerning the entropy in product MV algebras, the concepts of mutual information and Kullback–Leibler divergence for the case of product MV algebras and examine algebraic properties of the proposed measures. In particular, a convexity of Kullback–Leibler divergence with respect to states in product MV algebras is proved, and chain rules for mutual information and Kullback–Leibler divergence are established. In addition, the data processing inequality for conditionally independent partitions in product MV algebras is proved.

Equational type characterization for σ-complete MV-algebras

In the framework of algebras with infinitary operations, an equational base for the category of σ-complete MV-algebras is given. In this way, we study some particular objects as simple algebras, directly irreducible algebras, injectives, etc. A completeness theorem with respect to the standard MV-algebra, considered as σ-complete MV-algebra, is obtained. Finally, we apply this result to the study of σ-complete Boolean algebras and σ-complete product MV-algebras.

MV-algebras with operators (the commutative and the non-commutative case)

In the present paper we define the (pseudo) MV-algebras with n-ary operators, generalizing MV-modules and product MV-algebras. Our main results assert that there are bijective correspondences between the operators defined on a pseudo MV-algebra and the operators defined on the corresponding ℓ-group. We also provide a categorical framework and we prove the analogue of Mundici's categorical equivalence between MV-algebras and abelian ℓ-groups with strong unit. Thus, the category of pseudo MV-algebras with operators is equivalent to some category of ℓ-groups with operators.

On Product MV-Algebras

In this paper we apply the notion of the product MV-algebra in accordance with the definition given by B. Riečan. We investigate the convex embeddability of an MV-algebra into a product MV-algebra. We found sufficient conditions under which any two direct product decompositions of a product MV-algebra have isomorphic refinements.

On the entropy of dynamical systems in product MV algebras

Recently, the so-called product MV algebras have been introduced and the entropy of partitions in these algebras have been studied. In this paper the previous results are applied to dynamical systems.

A note on the conditional probability on product MV algebras

A conditional probability (state) of an element from a product MV algebra with respect to a product MV valued observable is formulated and a version of the martingale convergence theorem is proved.

On the entropy of partitions in product MV algebras

Generalizing the entropy of fuzzy partitions, the entropy of a partition in recently introduced product MV algebra has been studied. The least common refinement of two partitions is defined and the algebraic properties of the entropies and conditional entropies are examined.

A Note on the Individual Ergodic Theorem on Product MV Algebras

Classical ergodic theory was built on σ-algebras. The aim of this paper is toprove the individual ergodic theorem on more general structures—mon productMV algebras.

On the L p Space of Observables on Product MV Algebras

A weakly σ-distributive product MV algebra M is considered as a base of aquantum structure model. A state is a morphism from M to the unit interval, andan observable is a morphism from the system of all Borel sets to M. It is provedthat the subspace Lp of the space of observables is a complete pseudometricspace. This result generalizes the previous result; the proof is new.

ON THE SUM OF OBSERVABLES IN PRODUCT MV ALGEBRAS

As an alternative to quantum logics MV algebras have been suggested2,20. In a quite general class of MV algebras there have been constructed sum of any two observables10,14,20. Recently the notion of product MV algebra has been introduced4,5,12,17. In any weakly σ-distributive product MV algebra the joint observable of any observables can be constructed and on the base also the sum of any observables. In this paper we show that the two concepts of constructing of the sum are equivalent.