If S is a left amenable semigroup, let
dim
⟨
M
l
(
S
)
⟩
\dim \langle Ml(S)\rangle
denote the dimension of the set of left invariant means on S. Theorem. If S is left amenable, then
dim
⟨
M
l
(
S
)
⟩
=
n
>
∞
\dim \langle Ml(S)\rangle = n > \infty
if and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer for S countable or left cancellative. Moreover, when S is infinite, left amenable, and either left or right cancellative, we show that
dim
⟨
M
l
(
S
)
⟩
\dim \langle Ml(S)\rangle
is at least the cardinality of S. An application of these results shows that the radical of the second conjugate algebra of
l
1
(
S
)
{l_1}(S)
is infinite dimensional when S is a left amenable semigroup which does not contain a finite ideal.