Search Results (19)

Although algebraic structures are frequently analyzed using unary and binary operations, they can also be effectively defined and unified using ternary operations. In this context, we introduce structures that contain two constants and a ternary operation. We demonstrate that these structures are isomorphic to various significant algebraic systems, including Boolean algebras, de Morgan algebras, MV-algebras, and (near-)rings of characteristic two. Our work highlights the versatility of ternary operations in describing and connecting diverse algebraic structures. Full article

The logic sqŁ^* is closely related to complex fuzzy sets. In this paper, we continue our study on sqŁ^* by establishing a system that includes all formulas with values $\mathbf{0}$ in sqŁ^*. This system has paraconsistent formulas within sqŁ^*. Moreover, we show that this logical system is both sound and weakly complete. Full article

In this paper, $(\odot ,\vee )$-multiderivations on an MV-algebra A are introduced, the relations between $(\odot ,\vee )$-multiderivations and $(\odot ,\vee )$-derivations are discussed. The set $\mathrm{MD}\left(A\right)$ of $(\odot ,\vee )$-multiderivations on A can be equipped with a preorder, and $\left(\mathrm{MD}\right(A)/\sim ,\preccurlyeq )$ can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain $L_n$, $(\mathrm{MD}\left(L_n\right)/\sim ,\preccurlyeq )$ becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of $\mathrm{MD}\left(L_n\right)$. Full article

The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too. Full article

The widespread adoption of massively parallel processors over the past decade has fundamentally transformed the landscape of high-performance computing hardware. This revolution has recently driven the advancement of FPGAs, which are emerging as an attractive alternative to power-hungry many-core devices in a world increasingly concerned with energy consumption. Consequently, numerous recent studies have focused on implementing efficient dense and sparse numerical linear algebra (NLA) kernels on FPGAs. To maximize the efficiency of these kernels, a key aspect is the exploration of analytical tools to comprehend the performance of the developments and guide the optimization process. In this regard, the roofline model (RLM) is a well-known graphical tool that facilitates the analysis of computational performance and identifies the primary bottlenecks of a specific software when executed on a particular hardware platform. Our previous efforts advanced in developing efficient implementations of the sparse matrix–vector multiplication (SpMV) for FPGAs, considering both speed and energy consumption. In this work, we propose an extension of the RLM that enables optimizing runtime and energy consumption for NLA kernels based on sparse blocked storage formats on FPGAs. To test the power of this tool, we use it to extend our previous SpMV kernels by leveraging a block-sparse storage format that enables more efficient data access. Full article

In this paper, W-graph, called the notion of graphs on Wajsberg algebras, is introduced such that the vertices of the graph are the elements of Wajsberg algebra and the edges are the association of two vertices. In addition to this, commutative W-graphs are also symmetric graphs. Moreover, a graph of equivalence classes of Wajsberg algebra is constructed. Meanwhile, new definitions as complement annihilator and ∆-connection operator on Wajsberg algebras are presented. Lemmas and theorems on these notions are proved, and some associated results depending on the graph’s algebraic properties are presented, and supporting examples are given. Furthermore, the algorithms for determining and constructing all these new notions in each section are generated. Full article

BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with $2^r+1$ elements, $r\ge 2,$ and BL-chains with k elements, $k\ge 4.$ In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size ($2\le n\le 5$). Full article

The mean-variance (MV) portfolio optimization targets higher return for investment period despite the unknown stochastic behavior of the future asset returns. That is why a risk is explicitly considering, quantified by algebraic characteristics of volatilities and co-variances. A new probabilistic definition of portfolio risk is the Value at Risk (VaR). The paper makes explicit inclusion and minimization of VaR as a quantitative measure of financial sustainability of a portfolio problem. Thus, the portfolio weights as problem solutions will respect not only the MV requirements for risk and return, but also the additional minimization of risk defined by VaR level. The portfolio problem is defined in a new, bi-level form. The upper level minimizes and evaluates the VaR value. The lower level evaluates the optimal assets weights by minimizing portfolio risk and maximizing the return in MV form. The bi-level model allows to have extended set of portfolio solutions with the portfolio weights and the value of VaR. Graphical interpretation of this bi-level definition of the portfolio problem explains the differences with the MV portfolio definition. Thus, the bi-level portfolio problem evaluates the optimal weights, which makes maximization of portfolio return and minimization of the risk in its algebraic and probabilistic form of definition. Full article

For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state $\mathbf{m}$. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability. Full article

Two categories of lower and upper lattice-valued F-transforms with fuzzy relations as morphisms are introduced, as generalisations of standard categories of F-transforms with maps as morphisms. Although F-transforms are defined using special structures called spaces with fuzzy partitions, it is shown that these categories are identical to the relational variants of the two categories of semimodule homomorphisms where these fuzzy partitions do not occur. This a priori independence of the F-transform on spaces with fuzzy partitions makes it possible, for example, to use a simple matrix calculus to calculate F-transforms, or to determine the image of F-transforms in relational morphisms of the two categories. Full article

Many-valued (MV; the many-valued logics considered by Łukasiewicz)-algebras are algebraic systems that generalize Boolean algebras. The MV-algebraic probability theory involves the notions of the state and observable, which abstract the probability measure and the random variable, both considered in the Kolmogorov probability theory. Within the MV-algebraic probability theory, many important theorems (such as various versions of the central limit theorem or the individual ergodic theorem) have been recently studied and proven. In particular, the counterpart of the Kolmogorov strong law of large numbers (SLLN) for sequences of independent observables has been considered. In this paper, we prove generalized MV-algebraic versions of the SLLN, i.e., counterparts of the Marcinkiewicz–Zygmund and Brunk–Prokhorov SLLN for independent observables, as well as the Korchevsky SLLN, where the independence of observables is not assumed. To this end, we apply the classical probability theory and some measure-theoretic methods. We also analyze examples of applications of the proven theorems. Our results open new directions of development of the MV-algebraic probability theory. They can also be applied to the problem of entropy estimation. Full article

This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order $\alpha ,$ where $\alpha \in (0,1)\cup (1,\infty ),$ 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 $\alpha ,$ where $\alpha >1,$ has the property of sub-additivity. This property allows us to define, for $\alpha >1,$ 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. Full article

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. Full article

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. Full article