On hypersemigroups

Two well known results on semigroups, one of them clue to Calais, the other to Lajos, are generalized in case of hypersemigroups in the way indicated in the present paper. 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 hyper semigroup 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.