Wieslaw A. Dudek

Fuzzy h-ideals of hemirings

A characterization of an h-hemiregular hemiring in terms of a fuzzy h-ideal is provided. Some properties of prime fuzzy h-ideals of h-hemiregular hemirings are investigated. It is proved that a fuzzy subset @z of a hemiring S is a prime fuzzy left (right) h-ideal of S if and only if @z is two-valued, @z(0)=1, and the set of all x in S such that @z(x)=1 is a prime (left) right h-ideal of S. Finally, the similar properties for maximal fuzzy left (right) h-ideals of hemirings are considered.

Interval-valued fuzzy graphs

We define the Cartesian product, composition, union and join on interval-valued fuzzy graphs and investigate some of their properties. We also introduce the notion of interval-valued fuzzy complete graphs and present some properties of self-complementary and self-weak complementary interval-valued fuzzy complete graphs.

Intuitionistic fuzzy hypergraphs with applications

Hypergraphs are considered a useful tool for modeling system architectures and data structures and to represent a partition, covering and clustering in the area of circuit design. In this paper, we apply the concept of intuitionistic fuzzy set theory to generalize results concerning hypergraphs. For each intuitionistic fuzzy structure defined, we use cut-level sets to define an associated sequence of crisp structures. We determine what properties of the sequence of crisp structures characterize a given property of the intuitionistic fuzzy structure. We also present applications of intuitionistic fuzzy hypergraphs.

Regular bipolar fuzzy graphs

We introduce the concepts of regular and totally regular bipolar fuzzy graphs. We prove necessary and sufficient condition under which regular bipolar fuzzy graph and totally bipolar fuzzy graph are equivalent. We introduce the notion of bipolar fuzzy line graphs and present some of their properties. We state a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We examine when an isomorphism between two bipolar fuzzy graphs follows from an isomorphism of their corresponding bipolar fuzzy line graphs.

On intuitionistic fuzzy sub-hyperquasigroups of hyperquasigroups

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. In this paper, we consider the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigate some properties of such sub-hyperquasigroups. In particular, we investigate some natural equivalence relations on the set of all intuitionistic fuzzy sub-hyperquasigroups of a hyperquasigroup.

(α,β)-fuzzy ideals of hemirings

We give a characterization of different types of (@a,@b)-fuzzy ideals of hemirings, where @a,@b@?{@?,q,@?@?q,@?@?q} and @a @?@?q. Special attention is paid to (@?,@?@?q)-fuzzy prime and semiprime ideals.

A Generalization of n-Ary Algebraic Systems

One of the aims of this article is to extract, whenever possible, the common elements of several seemingly different types of algebraic hypersystems. In achieving this, one discovers general concepts, constructions, and results which not only generalize and unify the known special situations, thus leading to an economy of presentation, but, being at a higher level of abstraction, can also be applied to entirely new situations, yielding significant information and giving rise to new directions. We shall consider a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups, and n-ary semigroups. This new class of hypersystems is called n-ary hypersemigroups and properties of such hypersemigroups are investigated. On an n-ary hypersemigroup, we describe the smallest equivalence relation β* whose quotient is an ordinary n-ary semigroup.

NEUTRAL ELEMENTS, FUNDAMENTAL RELATIONS AND n-ARY HYPERSEMIGROUPS

The main tools in the theory of n-ary hyperstructures are the fundamental relations. The fundamental relation on an n-ary hypersemigroup is defined as the smallest equivalence relation so that the quotient would be the n-ary semigroup. In this paper we study neutral elements in n-ary hypersemigroups and introduce a new strongly compatible equivalence relation on an n-ary hypersemigroup so that the quotient is a commutative n-ary semigroup. Also we determine some necessary and sufficient conditions so that this relation is transitive. Finally, we prove that this relation is transitive on an n-ary hypergroup with neutral (identity) element.

Intuitionistic fuzzy left k -ideals of semirings

We introduce the notion of intuitionistic fuzzy left k-ideals of semirings and investigate their properties and connections with left k-ideals of the corresponding semirings. Next we give some important characterizations of intuitionistic fuzzy left k-ideals of different type and describe various methods of constructions of such intuitionistic fuzzy sets. Finally, we propose some natural classification of intuitionistic fuzzy left k-ideals.

Around the Hosszú-Gluskin theorem for n-ary groups

We survey the results related to the important Hosszu-Gluskin Theorem on n-ary groups adding several new results and comments. The aim of this paper is to write all such results in uniform and compressive forms. Therefore some proofs of new results are only sketched or omitted if their completing seems to be not too difficult for readers. In particular, we show as the Hosszu-Gluskin Theorem can be used for evaluation how many different n-ary groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the mentioned theorem can also be used for investigation of Q-independent subsets of semiabelian n-ary groups for some special families Q of mappings.