2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142458Let $A$ be a finite dimensional $Q-$algebra and $Γsubset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(Γ)$. We call this last property the hyperbolic property. We apply this in the case that $A = KS$ a semigroup algebra with $K = Q$ or $K = Q(\sqrt{-d})$. In particular, when $KS$ is semi-simple and has no nilpotent elements, we prove that $S$ is an inverse semigroup which is the disjoint union of Higman groups and at most one cyclic group $C_n$ with $n \in \{5,8,12\}$.This article corresponds to the second chapter of the third author PhD ThesisRings and AlgebrasGroup Theory16U60, 20M25text