Semihypergroups obtained by merging of 0-semigroups with groups

Abstract

We consider the class of 0-semigroups (H,) that are obtained by adding a zero element to a group (G, •) so that for all x, y ∈ G it holds x y 0 ⇒ x y = xy. These semigroups are called 0-extensions of (G, •). We introduce a merging operation that constructs a 0-semihypergroup from a 0-extension of (G, •) by a suitable superposition of the product tables. We characterize a class of 0-simple semihypergroups that are merging of a 0-extension of an elementary Abelian 2-group. Moreover, we prove that in the finite case all such 0-semihypergroups can be obtained from a special construction where (H,) is nilpotent.

On hypercyclic fully zero-simple semihypergroups

“…In [16] the authors found all simple and zero-simple semihypergroups of size 3 , such that the fundamental relation β is not transitive, apart from isomorphisms. This semihypergroups of size 3 were used in [8][9][10][11][12] to characterize the fully simple semihypergroups and the fully zero-simple semihypergroups having all hyperproducts of size ≤ 2 .…”