2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140755We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $ϕ$ of $S$ onto a group $G$, we show, under an assumption on $\ker(ϕ)$, that $S$ has weak containment if and only if $G$ is amenable and $\ker(ϕ)$ has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show that all graph inverse semigroups have weak containment and that Nica's inverse semigroup $\mcT_{G,P}$ of a quasi-lattice ordered group $(G,P)$ has weak containment if and only if $(G,P)$ is amenable.14 pages. Corrected an error in Proposition 3.1 (thanks to the referee for finding this). This has led to a slight restructuring of the paper and an extra hypothesis in the main theoremOperator Algebras46L05; 20M18text