On hypersemigroups

We prove that a nonempty subset $B$ of a regular hypersemigroup $H$ is a bi-ideal of $H$ if and only if it is represented in the form $B=A*C$ where $A$ is a right ideal and $C$ a left ideal of $H$. We also show that an hypersemigroup $H$ is regular if and only if the right and the left ideals of $H$ are idempotent, and for every right ideal $A$ and every left ideal $B$ of $H$, the product $A*B$ is a quasi-ideal of $H$. Our aim is not just to add a publication on hypersemigroups but, mainly, to publish a paper which serves as an example to show what an hypersemigroup is and give the right information concerning this structure. We never work directly on an hypersemigroup. If we want to get a result on an hypersemigroup, then we have to prove it first for a semigroup and transfer its proof to hypersemigroup. But there is further interesting information concerning this structure as well, we will deal with at another time.