Journal of Hyperstructures
Volume:6 Issue: 2, Summer and Autumn 2017

After a brief history of Hypergroups, since the beginning around the 40s till today, one gives an excursus of the most recent applications of this topic to Fuzzy Sets and Chinese groups as HX-hypergroups.

In this work, we investigate fuzzy weakly prime Γ-ideal, fuzzy partial weakly prime Γ-ideal and fuzzy semiprime Γ- ideal of a commutative Γ-ring with nonzero identity. We obtained some characterizations of fuzzy weakly prime Γ-ideal, partial weakly prime Γ-ideal and semiprime Γ-ideal of a Γ-ring. Also some properties of these concepts have been studied.

An important problem in the theory of ordered hypersemigroups is to describe the ordered hypersemigroups which enter their properties into their σ-classes. In this respect, we prove the following: If H is a regular, left (right) regular, completely regular, intra-regular, left (right) quasi-regular, semisimple, k-regular, archimedean, weakly commutative, left (right) simple, simple, left (right) strongly simple ordered semigroup and σ a complete semilattice congruence on H then, for each a ∈ H, the σ-class (a)σ of H is, respectively, so.

The Theory of Representations of Hyperstructures was started in mid 80’s but that time there was not any general definition of hyperfield. The Hv-structures, were introduced in 4th AHA Congress 1990, and at the same time, the general definition of the hyperfield, was given. Since then the Theory of Representations is refereed mainly on Hv-groups by Hv-matrices. In Hv-structures the weak axioms replace the ”equality” by the ”non empty intersection”. The characteristic property of Hv-structures, is that a partial order on Hv-structures on the same underline set, is defined. The weak properties increase extremely the number of hyperstructures defined in the same set. In representation theory the researchers have to treat well almost all the classical algebraic structures from semigroups to Lie-algebras. We present the problems, some new results and we give to researchers open problems in mathematics from hyperstructures.

The aim of this paper is to give a definition of an intuitionistic fuzzy 2- absorbing ideal and an intuitionistic fuzzy weakly completely 2- absorbing ideal of commutative rings and to give their properties. Moreover, we give a diagram of transition between definitions of intuitionistic fuzzy 2- absorbing ideals of commutative rings.

Intuitionistic Logic was introduced by L. E. J. Brouwer in[1] and Heyting algebra was defined by A. Heyting to formalize the Brouwer’s intuitionistic logic[4]. The concept of Heyting algebra has been accepted as the basis for intuitionistic propositional logic. Heyting algebras have had applications in different areas. The coHeyting algebra is the same lattice with dual operation of Heyting algebra[5]. Also, co-Heyting algebras have several applications in different areas.
In this paper, we introduced the new concept HH∗− Intuitionistic Heyting Valued Ω-Algebra. The purpose of introducing this new concept is to expand the field of researchers’ area using both membership degree and non-membership degree. This allows us to get more sensitive results.The HH∗− Intuitionistic Heyting valued set, HH∗−Intuitionistic Heyting valued relation, HH∗−Intuitionistic Heyting valued Ω-algebra and the homomorphism over HH∗− Intuitionistic Heyting valued Ω-algebra were defined.

In this study, the fundamental definitions and theorems regarding intuitionistic fuzzy sets and intuitionistic fuzzy ideals of commutative ring R with identity have been given as preliminaries. After the preliminaries, we introduce the notions of intuitionistic fuzzy weakly prime ideals, intuitionistic fuzzy partial weakly prime ideals, intuitionistic fuzzy weakly semiprime ideals of R. Also, we give some relations between intuitionistic fuzzy weakly prime ideals and weakly prime ideals of R.