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A semigroup S is called (left, right) absolutely flat if all of its (left, right) S-sets are flat. Let S = ∪{Sβ : β∈Γ} be the least semillatice decomposition of a band S. It is known that if S is left absolutely flat then S is right regular (that is, each Sβ is a right zero). In this paper it is shown that, in addition, whenever α, β∈Γ, α < β, and F is a finite subset of Sβ x Sβ, there exists wSα such that (wu, wv) ∈θR(F) for all (u, v)∈FR(F) denotes the smallest right congruence on S containing F). This condition in fact afford a characterization of left absolute flatness in certain classes of right regular bands (e.g. if Γ is a chain, if all chains contained in Γ have at most two elements, or if S is right normal.