A family of 0-simple semihypergroups related to sequence A000070

For any integer n 65 2, let R_0(n + 1) be the class of 0-semihypergroups H of size n + 1 such that {y} 86 xy 86 {0, y} for all x, y 08 H - {0}, all subsemihypergroups K 86 H are 0-simple and, when |K| 65 3, the fundamental relation \u3b2_K is not transitive. We determine a transversal of isomorphism classes of semihypergroups in R0(n + 1) and we prove that its cardinality is the (n + 1)-th term of sequence A000070 in [21], namely, 11 _{k=0}^n p(k), where p(k) denotes the number of non-increasing partitions of integer k

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

Semihypergroups obtained by merging of 0-semigroups with groups

We consider the class of 0-semigroups (H;star) that are obtained by adding a zero element to a group (G; cdot) so that for all x,yin G it holds x star y not=0 Rightarrow x star y = xy. These semigroups are called 0-extensions of (G; cdot). We introduce a merging operation that constructs a 0-semihypergroup from a 0-extension of (G; cdot) 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;star) is nilpotent