Hyperbolicity of semigroup algebras

Let A be a finite-dimensional Q-algebra and Γ⊂A a Z-order. We classify those A with the property that and refer to this as the hyperbolic property. We apply this in case A=KS is a semigroup algebra, with K=Q or . A complete classification is given when KS is semi-simple and also when S is a non-semi-simple semigroup.