Unitariness in Ordered Semigroups

We introduce the concepts of unitary, almost unitary and strongly almost unitary subset of an ordered semigroup. For the notions of almost unitary and strongly almost unitary subset of an ordered semigroup, we use the notion of translational hull of an ordered semigroup. If (S,⋅,≤) is an ordered semigroup having an element e such that e ≤ e² and U is a nonempty subset of S such that u ≤ eu, u ≤ ue for all u ∈ U, we show that U is almost unitary in S if and only if U is unitary in . Also if (S,⋅,≤) is an ordered semigroup, e ∉ S, U is a nonempty subset of S, S^e:= S ∪ {e} and U^e:= U ∪ {e}, we give conditions that an (“extension” of S) ordered semigroup and the subset U^e of S^e must satisfy in order for U to be almost unitary or strongly almost unitary in S (in case U is strongly almost unitary in S, then the given conditions are equivalent).