A semigroup
S
S
is called (left, right) absolutely flat if all of its (left, right)
S
S
-sets are flat. Let
S
=
∪
{
S
γ
:
γ
∈
Γ
}
S = \cup \{ {S_\gamma }:\gamma \in \Gamma \}
be the least semilattice decomposition of a band
S
S
. It is known that if
S
S
is left absolutely flat then
S
S
is right regular (that is, each
S
γ
{S_\gamma }
is right zero). In this paper it is shown that, in addition, whenever
α
,
β
∈
Γ
,
α
>
β
\alpha ,\beta \in \Gamma ,\alpha > \beta
, and
F
F
is a finite subset of
S
β
×
S
β
{S_\beta } \times {S_\beta }
, there exists
w
∈
S
α
w \in {S_\alpha }
such that
(
w
u
,
w
v
)
∈
θ
R
(
F
)
(wu,wv) \in {\theta _R}(F)
for all
(
u
,
v
)
∈
F
(
θ
R
(
F
)
(u,v) \in F({\theta _R}(F)
denotes the smallest right congruence on
S
S
containing
F
F
). This condition in fact affords a characterization of left absolute flatness in certain classes of right regular bands (e.g. if
Γ
\Gamma
is a chain, if all chains contained in
Γ
\Gamma
have at most two elements, or if
S
S
is right normal).