A Banach algebra \(\mathcal{A}\) is called amenable if for every Banach space \(\mathcal{X}\) which is an \(\mathcal{A}\)-module with bounded module actions, all bounded derivations from \(\mathcal{A}\) into the dual module \(\mathcal{X}^*\) are inner [B. E. Johnson, Cohomology in Banach algebras. Providence, RI: American Mathematical Society (AMS) (1972; Zbl 0256.18014)]. Several variants of this concept have appeared in the literature after Johnson’s fundamental work. By replacing the concept of inner derivation with approximately inner derivation, approximate versions of various forms of amenability were introduced and investigated in [F. Ghahramani and R. J. Loy, J. Funct. Anal. 208, No. 1, 229–260 (2004; Zbl 1045.46029); F. Ghahramani et al., J. Funct. Anal. 254, No. 7, 1776–1810 (2008; Zbl 1146.46023)].
Since then, intermediate concepts of amenability were constructed from the combination of former ones. The paper under review is one of this type. Let \(S\) be an inverse semigroup, \(E\) be the semigroup of idempotents of \(S\) and \(\omega\) be a weight on \(S\). Let \(\mathcal{X}\) be an \(\ell^1(S,\omega)\) Banach bimodule which is a commutative \(\ell^1(E,\omega)\) bimodule as well and whose module actions are compatible. Then \(\mathcal{X}\) an \(\ell^1(S,\omega)\)-\(\ell^1(E,\omega)\) module. If for every such module every derivation \(D:\mathcal{A}\longrightarrow \mathcal{X}^*\) which keeps module actions is inner [respectively, approximately inner], then \(\ell^1(S,\omega)\) is called module amenable [repectively, approximately module amenable]. The authors show that if \(\omega\) is bounded away from zero, then module approximate amenability of \(\ell^1(S,\omega)\) implies that \(S\) is amenable. The converse statement holds under an extra assumption. The paper is concluded with several implications among some notions of amenability for \(S\) and some Banach algebras associated with it.