Fair amenability for semigroups

A new flavour of amenability for discrete semigroups is proposed that generalises group amenability and follows from a Følner-type condition. Some examples are explored, to argue that this new notion better captures some essential ideas of amenability. A semigroup S is left fairly amenable if, and only if, it supports a mean m ∈ 𝓁^∞(S)* satisfying m(𝑓) = m(s∗𝑓) whenever s∗𝑓 ∈ 𝓁^∞(S), thus justifying the nomenclature “fairly amenable”.