History and new possible research directions of hyperstructures

We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research

The Lie-Santilli admissible hyperalgebras of type An

The largest class of hyperstructures is the one which satisfy the weak properties. These are called H v -structures introduced in 1990 and they proved to have a lot of applications on several applied sciences. In this paper we present a construction of the hyperstructures used in the Lie-Santilli admissible theory on square matrices

THE TRANSPOSITION AXIOM IN HYPERCOMPOSITIONAL STRUCTURES

The hypergroup (as defined by F. Marty), being a very general algebraic structure, was subsequently quickly enriched with additional axioms. One of these is the transposition axiom, the utilization of which led to the creation of join spaces (join hypergroups) and of transposition hypergroups. These hypergroups have numerous applications in geometry, formal languages, thetheory of automata and graph theory. This paper deals with transposition hypergroups. It also introduces the transposition axiom to weaker structures, which result from the hypergroup by the removal of certain axioms, thus defining the transposition hypergroupoid, the transposition semi-hypergroup and the transposition quasi-hypergroup. Finally, it presents hypercompositional structures with internal or external compositions and hypercompositions, in which the transposition axiom is valid. Such structures emerged during the study of formal languages and the theory of automata through the use of hypercompositional algebra

On Further Properties of Fully Zero-Simple Semihypergroups

Let $mfF_0$ the class of fully zero-simple semihypergroups. In this paper we study the main properties of residual semihypergroup $(H_+, star)$ of a semihypergroup $(H, circ)$ in $mfF_0$. We prove that the quotient semigroup $H_+/eta^*_{H_+}$ is a completely simple and periodic semigroup. Moreover, we find the necessary and sufficient conditions for $(H_+, star)$ to be a torsion group and, in particular, an Abelian $2$-group

MULTIVALUED FUNCTIONS, FUZZY SUBSETS AND JOIN SPACES

One has considered the Hypergroupoid Η Γ = associated with a multivalued function Γ from H to a set D, defined as follows:∀ x ∈ H, x ο Γ x = ⎨y⏐ Γ(y) ∩ Γ(x) ≠ ∅⎬ ,∀ (y,z) ∈ H 2 , y ο Γ z = y ο Γ y ∪ z ο Γ z ,and one has calculated the fuzzy grade ∂(Η Γ ) for several functions Γ defined on sets H, such that ⎮H⎮ ∈ ⎨3, 4, 5, 6, 8, 9, 16⎬

On hypercyclic fully zero-simple semihypergroups

Let I be the class of fully zero-simple semihypergroups generated by a hyperproduct. In this paper we study some properties of residual semihypergroup (H_+; star) of a semihypergroup (H; \u25e6)in I. Moreover, we find sufficient conditions for (H; \u25e6) and (H_+; star) to be cyclic