MV-algebras Research Articles

Some applications of idempotent elements in MV algebras

"In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group $\left( \mathbb{Z}% _{n},+\right) $, where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true."

On Quantum-MV Algebras - Part I: The Orthomodular Algebras

We prove that almost all the properties of quantum-MV algebras are verified by orthomodular algebras, the new algebras introduced in a previous paper. We put a special insight on transitive antisymmetric orthomodular (taOM) algebras, generalizations of MV algebras. We make the connection with IMTL and NM algebras. In memoriam Dragos. Vaida (1933 – 2020)

Some new algebraic structures of soft sets

In this paper, firstly, we introduce the XOR product and XNOR product of soft sets. In practice, these products lead to simplicity and brevity. We discuss new algebraic structures such as group, ring, isomorphic rings, lattice and MV algebra on the soft sets using the emerging soft products and some present soft operations like intersection, union and complement. Moreover, the axioms for uncertainty measurements among the soft sets are proposed, and then, novel categories of the measures of similarity and distance between two soft sets are presented. The comparison and performance analysis of new measures with some of soft set measures in the literature are provided; thus, it is shown that the proposed ones decide correctly. Finally, we deal with the soft cryptosystem to illustrate the potential application of the new structures.

Ideals in residuated lattices

"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

Quasi-MV algebras for complex fuzzy logic

<abstract><p>Complex fuzzy logic (CFL) is an emerging topic of research in fuzzy logic. Due to the circle structured codomain of the complex membership function, algebraic structure (including MV-algebra) in traditional fuzzy logic cannot be easily transplanted to CFL. Quasi-MV algebras are almost identical to MV algebras, except $ \alpha\oplus\textbf{0} = \alpha $ does not always hold. In this paper, our goal is to derive some algebraic structures for CFL. We first construct a quasi-MV algebra in complex fuzzy logic by introducing negation and truncated sum in the unit disc of the complex plane $ \textbf{S} $. Next we construct a $ \sqrt{\neg} $ quasi-MV algebra over $ \textbf{S} $ by adding an operation of square root of the negation. Moreover, implication connective and some derived connectives on $ \textbf{S} $ are introduced. Furthermore, we construct a quasi-Wajsberg algebra over $ \textbf{S} $ in which implication is a primitive connective. These algebraic structures are suitable for CFL.</p></abstract>

On transitive modal many-valued logics

This paper is focused on the study of modal logics defined from valued Kripke frames, and particularly, on computability and expressivity questions of modal logics of transitive Kripke frames evaluated over certain residuated lattices. It is shown that a large family of those logics -including the ones arising from the standard MV and Product algebras- yields an undecidable consequence relation. Later on, the behavior of transitive modal Łukasiewicz logic is compared with that of its non-transitive counterpart, exhibiting some particulars concerning computability and equivalence with other logics. We conclude the article by showing the undecidability of the validity and the local SAT questions over transitive models when the Δ operation is added to the logic.

MV-algebras with pseudo MV-valuations

MV-algebras with pseudo MV-valuations

Algebraic and Shannon entropies of commutative hypergroups and their connection with information and permutation entropies and with calculation of entropy for chemical algebras

Studying the evolution of a system and dealing with its complexity are key issues in analyzing and predicting its future behavior. In this respect, the uncertainty problem for a wide variety of mathematical structures such as hyper MV–algebras and stochastic processes (information sources) that provide models for varied systems has been studied. This paper presents the algebraic and Shannon entropies for hypergroupoids and commutative hypergroups, respectively, and studies their fundamental properties. These notions are established in a way that they are technically feasible to be adapted to other algebraic hyperstructures. Moreover, it is investigated that how these two types of entropies are connected for the case of commutative hypergroups. In addition, conditions under which the algebraic entropy is connected with the information and permutation entropies are detected. In the end, the algebraic entropy is calculated for some hypergroupoids and chemical algebras.

Coordinatizing some concrete MV algebras and a decomposition theorem

MV algebras are the Lindenbaum–Tarski algebras of Łukasiewicz many-valued logics. From work of D. Mundici, countable such algebras correspond to certain AF C*-algebras. M. Lawson and P. Scott gave a coordinatization theorem for them, representing any countable MV-algebra as the lattice of principal ideals of an AF Boolean inverse monoid. In this note, we give two concrete examples of such a coordinatization, one for $${\mathbb {Q}}\cap [0,1]$$ and another for the so-called Chang algebra. We also discuss Bratteli diagram techniques to further the coordinatization program.

Corrigendum to “Semiring and semimodule issues in MV-algebras, Communications in Algebra 41:3 (2013), 1017–1048”

ABSTRACTWith the present note the authors intend to correct a mistake in the paper Semiring and semimodule issues in MV-algebras, Communications in Algebra 41:3 (2013), 1017–1048, by the same authors.

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.

A Novel Clustering Algorithm in a Neutrosophic Recommender System for Medical Diagnosis

Decision-making processes have been extensively used in artificial intelligence and cognitive sciences to explain and improve individual and social perception. As one of the most typical decision-making problems, medical diagnosis is used to analyze the relationship between symptoms and diseases according to uncertain and inconsistent information. It is essential to investigate the structure of a set of records on different levels such that similar patients can be treated concurrently within a group. In this paper, we propose a novel clustering algorithm in a neutrosophic recommender system for medical diagnosis. First, we define new algebraic structures for the system such as lattices, De Morgan algebra, Kleen algebra, MV algebra, BCK algebra, Stone algebra, and Brouwerian algebra. Based on these algebraic structures, we construct a neutrosophic recommender similarity matrix and a neutrosophic recommender equivalence matrix. A consecutive series of compositions between the neutrosophic recommender similarity matrices is performed to obtain the neutrosophic recommender equivalence matrix. From this matrix, a λ-cutting matrix is defined to conduct clustering among the neutrosophic recommender systems. Regarding the values of clustering validity indices, the Davies-Bouldin (DB) of the proposed method is approximately 20% better than those of the methods of Sahin (Neutrosophic Sets and Systems. 2014;2:18–24), Ye (J Intell Syst. 2014;23(4):379–89), and Ye (Soft Computing 2016;1–7). Analogously, the IFV and simplified silhouette with criterion (SSWC) of the proposal are better than those of the relevant methods with the improvement percentages being 30 and 70%, respectively. The results show that the proposed method is better than the related algorithms in terms of clustering quality whilst its computational time is slightly slower. The contributions of this research is significant in both algorithmic aspects of computational intelligence and and practical applications.

Weak QMV algebras and some ring-like structures

In this work, we propose a new quantum structure--weak quantum MV algebras (wQMV algebras)--and define coupled bimonoids and strong coupled bimonoids. We find that the coupled bimonoids and strong coupled bimonoids are ring-like structures corresponding to lattice-ordered wQMV algebras and lattice-ordered QMV algebras, respectively. Using an automated reasoning tool, we give the smallest 4-element wQMV algebra but not a QMV algebra. We also show that lattice-ordered wQMV algebras are the real nondistributive generalization of MV algebras. Certainly, most important properties of quantum MV algebras (QMV algebras) are preserved by wQMV algebras. Furthermore, we can conclude that lattice-ordered wQMV algebras are the simplest unsharp quantum logical structures by far, based on which computation theory could be set up.

RS-BL-algebras are MV-algebras

RS-BL-algebras are MV-algebras

Projectable ℓ-groups and algebras of logic: Categorical and algebraic connections

P.F. Conrad and other authors launched a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex ℓ-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. In this broader perspective, we revisit the Galatos–Tsinakis categorical equivalence between integral generalized MV algebras and negative cones of ℓ-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Upon recalling that projectable integral generalized MV algebras and negative cones of projectable ℓ-groups can be endowed with a positive Gödel implication, and turned into varieties by including this implication in their signature, we prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications.

On free MV algebras and a problem of Tarski

In this paper we prove that the first order theories of the free MV algebras with finitely many generators are different from each other. This answers to the MV algebra analogue of a well‐known (now solved) problem of Tarski for groups.

A Neutrosophic Recommender System for Medical Diagnosis Based on Algebraic Neutrosophic Measures

Neutrosophic set has the ability to handle uncertain, incomplete, inconsistent, indeterminate information in a more accurate way. In this paper, we proposed a neutrosophic recommender system to predict the diseases based on neutrosophic set which includes single-criterion neutrosophic recommender system (SC-NRS) and multi-criterion neutrosophic recommender system (MC-NRS). Further, we investigated some algebraic operations of neutrosophic recommender system such as union, complement, intersection, probabilistic sum, bold sum, bold intersection, bounded difference, symmetric difference, convex linear sum of min and max operators, Cartesian product, associativity, commutativity and distributive. Based on these operations, we studied the algebraic structures such as lattices, Kleen algebra, de Morgan algebra, Brouwerian algebra, BCK algebra, Stone algebra and MV algebra. In addition, we introduced several types of similarity measures based on these algebraic operations and studied some of their theoretic properties. Moreover, we accomplished a prediction formula using the proposed algebraic similarity measure. We also proposed a new algorithm for medical diagnosis based on neutrosophic recommender system. Finally to check the validity of the proposed methodology, we made experiments on the datasets Heart, RHC, Breast cancer, Diabetes and DMD. At the end, we presented the MSE and computational time by comparing the proposed algorithm with the relevant ones such as ICSM, DSM, CARE, CFMD, as well as other variants namely Variant 67, Variant 69, and Varian 71 both in tabular and graphical form to analyze the efficiency and accuracy. Finally we analyzed the strength of all 8 algorithms by ANOVA statistical tool.

ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS

AbstractIn this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

Soft Vectorial MV–Algebras

In this paper the soft vectorial MV-algebra has been studied. In first, we defined the concept of soft vectorial MV-algebra and give some examples. Then we establish some basic properties and investigate the effect of the extended intersection, restricted intersection, ∧-intersection, extended union, restricted union, ∨-union and soft function on them. Keywords: Soft Set, Soft Vectorial MV-Algebra, Vectorial MV-Algebra

Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications

Eighteen open problems posed during FSTA 2014 (Liptovský Ján, Slovakia) are presented. These problems concern fuzzy logics, fuzzy partitions, copulas, triangular norms and related aggregation functions. Some open problems concerning effect and MV algebras are also included.