Let
S
S
be a connected algebraic monoid with group of units
G
G
and lattice of regular
J
\mathcal {J}
-classes
U
(
S
)
\mathcal {U}(S)
. The connection between the solvability of
G
G
and the semilattice decomposition of
S
S
into archimedean semigroups is further elaborated. If
S
S
has a zero and if
U
(
S
)
≤
7
\mathcal {U}(S)\le 7
, then it is shown that
G
G
is solvable if and only if
U
(
S
)
\mathcal {U}(S)
is relatively complemented. If
J
∈
U
(
S
)
J\in \mathcal {U}(S)
, then we introduce two basic numbers
θ
(
J
)
\theta (J)
and
δ
(
J
)
\delta (J)
and study their properties. Crucial to this process is the theorem that for any indempotent
e
e
of
S
S
, the centralizer of
e
e
in
G
G
is connected. Connected monoids with central idempotents are also studied. A conjecture about their structure is forwarded. It is pointed out that the maximal connected submonoids of
S
S
with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If
S
S
is regular, then
S
S
is a Clifford semigroup if and only if for all
f
∈
E
(
S
)
f\in E(S)
, the set
{
e
|
e
∈
E
(
S
)
,
e
≥
f
}
\{ e|e \in E(S),\,e \geq f\}
is finite. Finally the maximal semilattice image of any connected monoid is determined.