pseudo-amenability and biflatness of weighted algebras and their second dual on inverse semigroups

This paper proves that when s is an inverse semigroup, pseudo-amenability, amenability and approximately amenability of ℓ1(s;w) are equivalent, e(s) is finite and every maximal subgroup of s is amenable. in addition, for a discrete inverse semigroup of s, we prove when (e(s);_) is uniformly locally finite, then ℓ1(s;w) __ is pseudo-amenable if and only if ℓ1(s), it is pseudo-amenable and s is finite. also, we prove that for discrete inverse semigroup s, if ℓ1(s; w) has a bounded approximate identity, then ℓ1(s;w) __ is amenable if and only if ℓ1(s) is biprojective and s is finite. moreover, we show that for an archimedean semigroup s, if ω be bounded on every maximal subgroup g of s, pseudo-amenability, amenability and approximate amenability of ℓ1(s; w) are equivalent. finally, we investigate the amenability and pseudo-amenability of ℓ1(s; w), where s is a left (right) zero semigroup or it is a rectangular band semigroup.