hyperstructures

An application of hyperstructure theory on social sciences is presented. In social sciences when questionnaires are used, there is a new tool, the bar instead of Likert scale. The bar has been suggested by Vougiouklis & Vougiouklis in 2008, who have proposed the replacement ofLikert scales, usually used in questionnaires, with a bar. This new tool, gives the opportunity to researchers to elaborate the questionnaires in different ways, depending on the filled questionnaires and of course on the problem. We study these filled questionnaires using hyperstructure theory. The hyperstructure theory is being related with questionnaires and we study the obtained hyperstructures which are used as an organized device of the problem and we focus on special problems.

A hyperproduct on non-square ordinary matrices can be defined by using the helix-hyperoperation. Therefore, the helix-hyperoperation (abbreviated hope ) is based on a classical operation and was introduced in order to overcome the non-existing cases. We study the helixhyperstructures on the special type of matrices, the Shelix matrices, used on the small dimension representations. In this paper, we introduce and focus our study on the class of S-helix matrices called k-overlap helix matrices. The reason is that their hyper-vector spaces can represent n-dimensional spaces which have independent both, single valued dimensions and multivalued dimensions.

I am working on algebraic hyperstructures from 1995. During the last twenty years, I together with my students and co-authors studied and developed the theory of algebraic hyperstructures in many directions. In particular, we tried to find real examples of hyperstructures in nature. In this paper we review some parts of these works such as (1) Fundamental relations on hyperstructures; (2) Fuzzy sets and hyperstructures; (3) Rough sets and hyperstructures; (4) Topology and hyperstructures; (5) Number theory and hyperstructures; (6) n-ary hypergroups and there extension to hyperrings and hypermodules; (7) Applications of hyperstructures in biology, physics and chemistry.

The largest class of hyperstructures is the Hv-structures, introduced in 1990, which proved to have a lot of applications in mathematics and several applied sciences, as well. Hyperstructures are used in the Lie-Santilli theory focusing to the hypernumbers, called e-numbers. We present the appropriate e-hyperstuctures which are defined using any map, in the sense the derivative map, called theta-hyperstructures.

In this paper, we attempt to investigate the connection between various types of ideals (for examples $(m, n)$-ideal, bi-ideal, interior ideal, quasi ideal, prime ideal and maximal ideal) of an ordered semigroup $(S,cdot ,leq)$ and the correspond hyperideals of its EL hyperstructure $(S,*)$ (if exists). Moreover, we construct the class of EL-$Gamma$ semihypergroup associated to a partially ordered $Gamma$-semigroup.

We study a new class of Hv -structures called Fundamentally Very Thin. This is an extension of the well known class of the Very Thin hyperstructures. We present applications of these hyperstructures.

Abstract. The largest class of hyperstructures is the one which satisfy the weak properties. These are called Hv-structures introduced in 1990 and they proved to have a lot of applications on several applied sciences. In this paper we present an introduction on the special classes of hyperstructures used inthe Lie-Santilli theory called e-hyperstructures. Several extensions of these hyperstructures are also presented