No abstract

In this paper we determine a family P_\sigma(H) of subsets of a hypergroup H such that the geometric space \ud (H, P_\sigma(H)) is strongly transitive and we use this fact to characterize the hypergroups such that the derived hypergroup D(H) of H coincides with an element of P_\sigma(H). In this case a n-tuple (x_1, x_2,...,x_n)\in H^n exists such that D(H) = B(x_1, x_2,...,x_n) = \ud {x\inH | \exist \sigma \in S_n, x\in x_\sigma(1)...x_sigma(n)}. Moreover, in the last section, we prove that in every semigroup the transitive closure \gamma* of the relation \gamma is the smallest congruence such that G/\gamma* is a commutative semigroup. We determine a necessary and sufficient condition such that the geometric space (G, P_\sigma(G)) of a 0-simple semigroup is strongly transitive. Finally, we prove that if G is a simple semigroup, then the space (G, P_\sigma(G)) is strongly transitive and the relation \gamma of G is transitive

We generalize the classical definition of hypergroups of type U on the right to semihypergroups, and we prove some properties of their subsemihypergroups and subhypergroups. In particular, we obtain that a finite proper semihypergroup of type U on the right can exist only if its order is at least 6. We prove that one such semihypergroup of order 6 actually exists. Moreover, we show that there exists a hypergroup of type U on the right of cardinality 9 containing a proper non-trivial subsemihypergroup. In this way, we solve a problem left open in [D. Freni, Sur les hypergroupes de type U et sous-hypergroupes engendrés par un sous-ensemble, Riv. Mat. Univ. Parma 13 (1987) 29-41]

In this paper we consider the class of semihypergroups H such that all subsemihypergroups K ⊆ H are simple and, when |K| ≥ 3 the fundamental relation β_K is not transitive. For these semihypergroups we prove that hyperproducts of elements in H have size ≤ 2 and the quotient semigroup H/β∗ is trivial. This last result allows us to completely characterize these semihypergroups in terms of a small set of simple semihypergroups of size 3. Finally, we solve a problem on strongly simple semihypergroups introduced in [11]

We study existence and possible uniqueness of special semihypergroups of type U on the right. In particular, we prove that there exists a unique proper semihypergroup of this kind having order 6, apart of isomorphisms; the least order for a hypergroup of type U on the right to have a stable part which is not a subhypergroup is 9; and the minimal cardinality of a proper semihypergroup of that kind whose heart and derived semihypergroup are proper and nontrivial is 12.Contextually, we analyze properties of the kernel of homomorphisms g : H → G, where H is a finite semihypergroup of type U on the right and G is a group. In this way, we obtain results that are immediately applicable both to the heart and to the derived of such semihypergroups.