(2n)-Weak Module Amenability of Triangular Banach Algebras on Semigroup Algebras

‎Let $S$ be a commutative (not necessary unital) inverse semigroup with the set of idempotents $E$‎. Consider semigroup algebras $ell^1(S)$ and  $ell^1(E)$ and triangular Banach algebras ‎$mathcal{T}=begin{bmatrix}ell^1(S) &ell^1(S) /M_0&ell^1(S)end{bmatrix}$ and ‎$mathfrak{T}={begin{bmatrix}alpha &0&alphaend{bmatrix}: alpha in ell^1(E)]}$, where $M_0$ be the closed linear span of ${delta_{es}-delta_s‎: ‎ein E‎, ‎sin S}$. ‎Recently‎, the author of this paper along with Pourabbas shown that for every $nin N$, $(2n+1)$-weak module amenability of ‎$mathcal{T}} (as a ‎$mathfrak{T}$-module) and $(2n+1)$-weak module amenability of ‎$ell^1(S)} (as a ‎$ell^1(E)$-module), are equal. In this paper, we extend this result and prove that the result is also true for the even state (2n)-weak module amenability, in the non-unitary state of these algebras.