On probabilistic n-ary hypergroups

The T-fuzzy n-ary subhypergroups of an n-ary hypergroup are defined by using triangular norms and some related properties are hence obtained. In particular, we consider the probabilistic version of n-ary hypergroups by using random sets and show that the fuzzy n-ary hypergroups defined by triangular norms are consequences of some probabilistic n-ary hypergroups under certain conditions. Some results on n-ary hypergroups recently given by Davvaz and Corsini are extended.     

Interval-valued fuzzy n-ary subhypergroups of n-ary hypergroups

This paper provides a continuation of ideas presented by Davvaz and Corsini (J Intell Fuzzy Syst 18(4):377–382, 2007). Our aim in this paper is to introduce the concept of quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set. This concept is a generalized concept of quasicoincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup of a n-ary hypergroup. This newly defined interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup is a generalization of the usual fuzzy n-ary subhypergroup. Finally, we consider the concept of implication-based interval-valued fuzzy n-ary subhypergroup in an n-ary hypergroup; in particular, the implication operators in ￡ukasiewicz system of continuous-valued logic are discussed.     

Roughness in n-ary hypergroups

This paper is aimed at studying the rough sets within the context of the commutative n-ary hypergroups. For this approach, the presentation of a detailed study of the various types of n-ary subhypergroups is of the greatest importance. This study was initiated in [V. Leoreanu-Fotea, Several types of n-ary subhypergroups, Italian Journal of Pure and Applied Mathematics, in press]. Subsequently, we shall use the findings in this study to define the upper and lower approximation of a subset with respect to an invertible n-ary subhypergroup of a commutative n-ary hypergroup. In the final part, the concept of n-ary rough subhypergroup is introduced.     

Approximations in <Emphasis Type="Italic">n</Emphasis>-ary algebraic systems

Algebraic systems have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The theory of rough sets has emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy, or incomplete information. This paper is devoted to the discussion of the relationship between algebraic systems, rough sets and fuzzy rough set models. We shall restrict ourselves to algebraic systems with one n-ary operation and we investigate some properties of approximations of n-ary semigroups. We introduce the notion of rough system in an n-ary semigroup. Fuzzy sets, a generalization of classical sets, are considered as mathematical tools to model the vagueness present in rough systems.     

Strongly Hamiltonian laceability of the even k-ary n-cube

The interconnection network considered in this paper is the k-ary n-cube that is an attractive variance of the well-known hypercube. Many interconnection networks can be viewed as the subclasses of the k-ary n-cubes include the cycle, the torus and the hypercube. A bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every two vertices which are in distinct partite sets. A bipartite graph G is strongly Hamiltonian laceable if it is Hamiltonian laceable and there exists a path of length N − 2 joining each pair of vertices in the same partite set, where N = |V(G)|. We prove that the k-ary n-cube is strongly Hamiltonian laceable for k is even and n  2.     

Generalizations of -fuzzy subalgebras in BCK/BCI-algebras

More general form of the notion of quasi-coincidence of a fuzzy point with a fuzzy set is considered, and generalizations of results in the papers [Y. B. Jun, On (α,β)-fuzzy subalgebras of BCK/BCI-algebras, Bull. Korean Math. Soc. 42 (4) (2005) 703–711; Y. B. Jun, Fuzzy subalgebras of type (α,β) in BCK/BCI-algebras, Kyungpook Math. J. 47 (2007) 403–410] are discussed. The notions of -fuzzy subalgebras and -fuzzy subalgebras in a BCK/BCI-algebra X are introduced, and several properties are investigated. Characterizations of -fuzzy subalgebra in a BCK/BCI-algebra X are discussed.     

Fuzzy <Emphasis Type="Italic">R</Emphasis>-subgroups with thresholds of near-rings and implication operators

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (α, β)-fuzzy R-subgroup of a near-ring where α , β are any two of {∈, q, ∈∨q , ∈∧q} with α ≠ ∈∧q is introduced and related properties are investigated. We also introduce the notion of a fuzzy R-subgroup with thresholds which is a generalization of an ordinary fuzzy R-subgroup and an (∈, ∈∨q)-fuzzy R-subgroup. Finally, we give the definition of an implication-based fuzzy R-subgroup.     

Intuitionistic (<Emphasis Type="Italic">S</Emphasis>, <Emphasis Type="Italic">T</Emphasis>)-fuzzy hyperquasigroups

The concept of intuitionistic fuzzy subhyperquasigroups in a hyperquasigroup with respect to an s-norm and a t-norm on intuitionistic fuzzy sets is introduced and their properties of such hyperquasigroups are studied. Intuitionistic (S, T)-fuzzy relations on a hyperquasigroup G are discussed. In particular, we investigate connections hyperquasigroups with binary quasigroups.     

Fuzzy n-ary hypergroups related to fuzzy points

In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called $(\in,\in \vee q)In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup. By using this new idea, we consider the ( ? , ? úq)(\in,\in\vee q)-fuzzy n-ary sub-hypergroup of a n-ary hypergroup. This newly defined ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup is a generalization of the usual fuzzy n-ary sub-hypergroup. Finally, we consider the concept of implication-based fuzzy n-ary sub-hypergroup in an n-ary hypergroup and discuss the relations between them, in particular, the implication operators in £\poundsukasiewicz system of continuous-valued logic are discussed.     

Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice

This paper considers the relationships among L-fuzzy sets, rough sets and n-ary hypergroup theory. Based on a complete residuated lattice, the concept of (invertible) L-fuzzy n-ary subhypergroups of a commutative n-ary hypergroup is introduced and some related properties are presented. The notions of lower and upper L-fuzzy rough approximation operators with respect to an L-fuzzy n-ary subhypergroup are introduced and studied. Then, a new algebraic structure called (invertible) L-fuzzy rough n-ary subhypergroups is defined, and the (strong) homomorphism of lower and upper L-fuzzy rough approximation operators is studied.     

Atanassov’s intuitionistic (S, T)-fuzzy n-ary sub-hypergroups and their properties

In this paper, we define a new kind of intuitionistic fuzzy n-ary sub-hypergroups of an n-ary hypergroup. This definition, which is based on Atanassov’s intuitionistic fuzzy sets, t-norms and t-conorms, includes earlier definitions of (n-ary) sub-hypergroups, (intuitionistic) fuzzy (n-ary) sub-hypergroups. Then some related properties are investigated. Also, intuitionistic fuzzy relations with respect to t-norms and t-conorms on n-ary hypergroups are discussed.     

Product structure of the fuzzy n-ary factor polygroup

Fuzzy n-ary hypergroups were introduced as suitable generalizations of fuzzy polygroups and a special case of fuzzy n-ary hyper groups. The aim of this paper is to introduce the notion of fuzzy n-ary factor polygroups of a polygroup. Based on the fuzzy n-ary factor group, we also study the product structures of the generating fuzzy factor n-ary groups. At the end of the paper, we prove the fundamental theorem of isomorphism of fuzzy n-ary groups.     

Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Also, we obtain some results of T-product (S-product) of the interval-valued fuzzy subsets for infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨-distributive t-norms T (∧-distributive t-conorms S).     

Generalized fuzzy bi-ideals of semigroups

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of (∈, ∈ ∨q)-fuzzy subgroups introduced by Bhakat is one among them. In this paper, using the relations between fuzzy points and fuzzy sets, the concept of a fuzzy bi-ideal with thresholds is introduced and some interesting properties are investigated. The acceptable nontrivial concepts obtained in this manner are the (∈, ∈ ∨q)-fuzzy bi-ideals and -fuzzy bi-ideals, which are extension of the concept of a fuzzy bi-ideal in semigroup.     

(∈, ∈ ∨ q)-fuzzy subnear-rings and ideals

Our aim in this paper is to introduce and study the new sort of fuzzy subnear-ring (ideal and prime ideal) of a near-ring called (∈, ∈ ∨q)-fuzzy subnear-ring (ideal and prime ideal). These fuzzy subnear-rings (ideals) are characterized by their level ideals. Finally, we give a generalization of (∈, ∈ ∨q)-fuzzy subnear-rings (ideals).     

Characterizations of regular ordered semigroups in terms of (α, β)-fuzzy generalized bi-ideals

In this paper, the concept of an (α, β)-fuzzy generalized bi-ideal in an ordered semigroup is introduced, which is a generalization of the concept of a fuzzy generalized bi-ideal in an ordered semigroup. Using this concept, some characterization theorems are provided. The upper/lower parts of an (∈, ∈ ∨ q)-fuzzy generalized bi-ideal are introduced and some characterizations of regular ordered semigroups are given. Also, we consider the concept of implication-based fuzzy generalized bi-ideals in an ordered semigroup. In particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Very and more or less in non-commutative fuzzy logic

In this paper we consider fuzzy subsets of a universe as L-fuzzy subsets instead of [ 0, 1 ]-valued, where L is a complete lattice. We enrich the lattice L by adding some suitable operations to make it into a pseudo-BL algebra. Since BL algebras are main frameworks of fuzzy logic, we propose to consider the non-commutative BL-algebras which are more natural for modeling the fuzzy notions. Based on reasoning with in non-commutative fuzzy logic we model the linguistic modifiers such as very and more or less and give an appropriate membership function for each one by taking into account the context of the given fuzzy notion by means of resemblance L-fuzzy relations.     

A Performance Model of Adaptive Wormhole Routing in k-Ary n-Cubes in the Presence of Digit-Reversal Traffic

Several analytical models of fully adaptive routing in wormhole-routed k-ary n-cubes under the uniform traffic pattern have recently been proposed in the literature. Although the uniform reference model has been widely used by researchers, it is not always true in practice as there are many applications that exhibit traffic nonuniformity. There has hardly been any study that describes an analytical model of fully adaptive routing under nonuniform traffic conditions. This paper describes the first analytical model of fully adaptive routing in k-ary n-cubes in the presence of nonuniform traffic generated by the digit-reversal permutation, which is an important communication operation found in many matrix computation problems. Results obtained through simulation experiments confirm that the model predicts message latency with a good degree of accuracy under different working conditions.     

(?, T)-fuzzy rough approximation operators and the TL-fuzzy rough ideals on a ring

In this paper, we consider a ring as a universal set and study (?, T)-fuzzy rough approximation operators with respect to a TL-fuzzy ideal of a ring. First, some new properties of generalized (?, T)-fuzzy rough approximation operators are obtained. Then, a new fuzzy algebraic structure - TL-fuzzy rough ideal is defined and its properties investigated. And finally, the homomorphism of (?, T)-fuzzy rough approximation operators is studied.     

Parallel Lagrange interpolation on <Emphasis Type="Italic">k</Emphasis>-ary <Emphasis Type="Italic">n</Emphasis>-cubes with maximum channel utilization

This paper proposes an efficient parallel algorithm for computing Lagrange interpolation on k-ary n-cube networks. This is done using the fact that a k-ary n-cube can be decomposed into n link-disjoint Hamiltonian cycles. Using these n link-disjoint cycles, we interpolate Lagrange polynomial using full bandwidth of the employed network. Communication in the main phase of the algorithm is based on an all-to-all broadcast algorithm on the n link-disjoint Hamiltonian cycles exploiting all network channels, and thus, resulting in high-efficiency in using network resources. A performance evaluation of the proposed algorithm reveals an optimum speedup for a typical range of system parameters used in current state-of-the-art implementations.