Fundamental relations in simple and 0-simple semihypergroups of small size

We consider the fundamental relations β and γ in simple and 0-simple semihypergroups, especially in connection with certain minimal cardinality questions. In particular, we enumerate and exhibit all simple and 0-simple semihypergroups having order 3 where β is not transitive, apart of isomorphisms. Moreover, we show that the least order for which there exists a strongly simple semihypergroup where β is not transitive is 4. Finally, we prove that γ is transitive in all simple semihypergroups, and determine necessary and sufficient conditions for a 0-simple semihypergroup to have γ transitive. The latter results obviously hold also for simple and 0-simple semigroups.