From ∨e-Semigroups to Hypersemigroups

Abstract A poe-semigroup is a semigroup S at the same time an ordered set having a greatest element “e” in which the multiplication is compatible with the ordering. A ∨e-semigroup is a semigroup S at the same time an upper semilattice with a greatest element “e” such that a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc for every a, b, c ∈ S. If S is not only an upper semi-lattice but a lattice, then it is called le-semigroup. From many results on le-semigroups, ∨e-semigroups or poe-semigroups, corresponding results on ordered semigroups (without greatest element) can be obtained. Related results on hypersemigroups or ordered hypersemigroups follow as application. An example is presented in the present note; the same can be said for every result on these structures. So order-lattices play an essential role in studying the hypersemigroups and the ordered hypersemigroups.