Thomas Vougiouklis

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.

Commutative Rings Obtained from Hyperrings (H v -rings) with α∗-Relations

The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hyperring was introduced by Vougiouklis at the fourth AHA congress. The fundamental relation on a hyperring (H v -ring) is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring. Note that the commutativity with respect to both sum and product in the (fundamental) ring are not assumed. Now, in this article we would like the (fundamental) ring to be commutative with respect to both sum and product, that is, the fundamental ring should be an ordinary commutative ring. Therefore we introduce a new strongly regular equivalence relation on hyperrings (H v -rings). If we consider this relation on a hperring (H v -ring), then the set of quotients is a commutative ring. Some properties of such rings are investigated.

Groups in Hypergroups

We study the fundamental relation introduced by Koskas as the transitive closure of a basic relation in a given hypergroup. We use the quotient set, which is a group, in order to define a semi-direct hyperproduct of two hypergroups. We obtain an extension of hypergroups by hypergroups. Moreover we reveal some hypergroups in a given hypergroup.

H v -groups defined on the same set

If a hyperoperation is weak associative then every greater hyperoperation, defined on the same set, is also weak associative. Using this property, the set of all Hv-groups with a scalar unit, defined on a set with three elements is determined.

Convolutions on WASS hyperstructures

Abstract In a hypergroup, especially in the generalization called H v -group, several convolutions can be defined. A convolution is given and a hypergroup algebra is obtained which is a H v -ring. Examples and applications on known classes of hyperstructures are also investigated.

Representations of hypergroups by generalized permutations

Special classes of generalized permutations are introduced and studied. An interpretation of them is given and they are used to represent finite hypergroups.

On Hv-rings and Hv-representations

Abstract The problem of the H v -representations of H v -groups by H v -matrices, is presented. Classes of H v -rings useful in the theory of representations are introduced. The advantages of some classes of the H v -matrices in the representation theory is pointed out.

Generalization ofP-hypergroups

Si definiscono e si studiano larghe classi di ipergruppi derivanti da comuni semi-gruppi.

Multi-valued hypermathematics for characterization of matter and antimatter systems

The hyperstructures used in Hadronic Mechanics, more precisely on the characterization of matter-antimatter systems are presented. Some topics of hyperstructures related on the topic as the H_v-structures, e-hypernumbers, e-hypervector spaces, P-hyperoperations are extensively presented. The theory of Santillis isomathematics is also presented. A new realization is used to represent a part of the Santillis theory on matter and anti-matter.

Hypermatrix representations of finite H v -groups

We present some results on the representation theory on the large class of hyperstructures called H v -structures. Some related classes of hyperstructures, especially in the finite case, are presented which are involved in this theory.