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.     

Characterizations of ordered semigroups in terms of (∈, ∈ ∨ q)-fuzzy interior ideals

In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals (resp. (∈, ∈ ∨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨q)-fuzzy ideals and (∈, ∈ ∨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy ideals (resp. (∈, ∈ ∨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

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.     

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.     

(∈, ∈ ∨ 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).     

Generalized fuzzy interior ideals of semigroups

The concept of $(\overline{\in},\overline{\in} \vee \overline{q})The concept of ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals, (∈, ∈ ∨ q)-fuzzy interior ideals and ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups. Finally, we give some characterization of [F] _t by means of (∈, ∈ ∨ q)-fuzzy interior ideals.     

Fuzzy -ary polygroups related to fuzzy points

Recently, fuzzy n-ary sub-polygroups were introduced and studied by Davvaz, Corsini and Leoreanu-Fotea [B. Davvaz, P. Corsini, V. Leoreanu-Fotea, Fuzzy n-ary sub-polygroups, Comput. Math. Appl. 57 (2008) 141–152]. Now, in this paper, the concept of (,q)-fuzzy n-ary sub-polygroups, -fuzzy n-ary sub-polygroups and fuzzy n-ary sub-polygroup with thresholds of an n-ary polygroup are introduced and some characterizations are described. Also, we give the definition of implication-based fuzzy n-ary sub-polygroups in an n-ary polygroup, in particular, the implication operators in Łukasiewicz 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.     

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.     

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).     

A new view of fuzzy hypernear-rings

In this survey article, we first introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. This concept is a generalized concept of quasi-coincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval valued (∈,∈∨q)-fuzzy sub-hypernear-rings (hyperideals) of a hypernear-ring, and hence, a generalization of a fuzzy sub-near-ring (ideal) is given. Some related properties of fuzzy hypernear-rings are described. Finally, we consider the concept of implication-based interval valued fuzzy sub-hypernear-rings (hyperideals) in a hypernear-ring, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

The three-dimensional fuzzy sets and their cut sets

In this paper, a new kind of L-fuzzy set is introduced which is called the three-dimensional fuzzy set. We first put forward four kinds of cut sets on the three-dimensional fuzzy sets which are defined by the 4-valued fuzzy sets. Then, the definitions of 4-valued order nested sets and 4-valued inverse order nested sets are given. Based on them, the decomposition theorems and representation theorems are obtained. Furthermore, the left interval-valued intuitionistic fuzzy sets and the right interval-valued intuitionistic fuzzy sets are introduced. We show that the lattices constructed by these two special L-fuzzy sets are not equivalent to sublattices of lattice constructed by the interval-valued intuitionistic fuzzy sets. Finally, we show that the three-dimensional fuzzy set is equivalent to the left interval-valued intuitionistic fuzzy set or the right interval-valued intuitionistic fuzzy set.     

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.     

Redefined fuzzy implicative filters

Using the belongs to relation (∈) and quasi-coincidence with relation (q) between fuzzy points and fuzzy sets, the concept of (θ, ψ)-fuzzy implicative filters where θ, ψ are any two of {∈, q, ∈ ∨ q, ∈ ∧ q} with θ ≠ ∈ ∧ q is introduced, and related properties are discussed. Relations between (∈ ∨ q, ∈ ∨ q)-fuzzy implicative filters and (∈, ∈ ∨ q)-fuzzy implicative filters are investigated, and conditions for an (∈, ∈ ∨ q)-fuzzy implicative filter to be an (∈, ∈)-fuzzy implicative filter are provided. Characterizations of (∈, ∈ ∨ q)-fuzzy implicative filters are given, and conditions for a fuzzy set to be a (q, ∈ ∨ q)-fuzzy implicative filter are provided.     

Special sequences as subcodes of reed-solomon codes

We consider sequences in which every symbol of an alphabet occurs at most once. We construct families of such sequences as nonlinear subcodes of a q-ary [n, k, n − k + 1] _q Reed-Solomon code of length n ≤ q consisting of words that have no identical symbols. We introduce the notion of a bunch of words of a linear code. For dimensions k ≤ 3 we obtain constructive lower estimates (tight bounds in a number of cases) on the maximum cardinality of a subcode for various n and q, and construct subsets of words meeting these estimates and bounds. We define codes with words that have no identical symbols, observe their relation to permutation codes, and state an optimization problem for them.     

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.     

Special classes of separable Goppa codes with improved parameter estimates

We show that subclasses of q-ary separable Goppa codes Γ(L, G) with L = {α ∈ GF(q ^2ℓ): G(α) ∈ 0} and with special Goppa polynomials G(x) can be represented as a chain of equivalent and embedded codes. For all codes of the chain we obtain an improved bound for the dimension and find an exact value of the minimum distance. A chain of binary codes is considered as a particular case with specific properties.     

A new type of fuzzy n-ary hyperstructures

The aim of this paper is to consider and study a new kind of fuzzy n-ary hyperstructures, such as fuzzy n-ary hypergroups and fuzzy (m,n)-ary hyperrings.     

(?, 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.