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)⁎m∈ℓ∞(S)⁎ satisfying m(f)=m(s⁎f)m(f)=m(s⁎f) whenever s⁎f∈ℓ∞(S)s⁎f∈ℓ∞(S), thus justifying the nomenclature “fairly amenable”.