We continue from [11] the study of linear algebraic semigroups. Let S be a connected algebraic semigroup defined over an algebraically closed field K. Let
U
(
S
)
\mathcal {U}(S)
be the partially ordered set of regular
J
\mathcal {J}
-classes of S and let
E
(
S
)
E(S)
be the set of idempotents of S. The following theorems (among others) are proved. (1)
U
(
S
)
\mathcal {U}(S)
is a finite lattice. (2) If S is regular and the kernel of S is a group, then the maximal semilattice image of S is isomorphic to the center of
E
(
S
)
E(S)
. (3) If S is a Clifford semigroup and
f
∈
E
(
S
)
f\, \in \,E(S)
, then the set
{
e
|
e
∈
E
(
S
)
,
e
⩾
f
}
\{ \,e\,|\,e\, \in \,E(S),\,e\, \geqslant \,f\}
is finite. (4) If S is a Clifford semigroup, then there is a commutative connected closed Clifford subsemigroup T of S with zero such that T intersects each
J
\mathcal {J}
-class of S. (5) If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in
(
K
n
,
⋅
)
({K^n},\, \cdot )
for some
n
∈
Z
+
n\, \in \,{\textbf {Z}^ + }
. (6) If
ch
⋅
K
=
0
{\text {ch}}\, \cdot \,K\, = \,0
and S is a commutative Clifford semigroup, then S is isomorphic to a direct product of an abelian connected unipotent group and a closed connected subsemigroup of
(
K
n
,
⋅
)
({K^n},\, \cdot )
for some
n
∈
Z
+
n\, \in \,{\textbf {Z}^ + }
. (7) If S is a regular semigroup and
dim
⋅
S
⩽
2
{\text {dim}}\, \cdot \,S\, \leqslant \,2
, then
|
U
(
S
)
|
⩽
4
\left | {\mathcal {U}(S)} \right |\, \leqslant \,4
. (8) If S is a Clifford semigroup with zero and
dim
⋅
S
=
3
{\text {dim}}\, \cdot \,S\, = \,3
, then
|
E
(
S
)
|
=
|
U
(
S
)
|
\left | {E(S)} \right |\, = \,\left | {\mathcal {U}(S)} \right |
can be any even number
⩾
8
\geqslant \,8
. (9) If S is a Clifford semigroup then
U
(
S
)
\mathcal {U}(S)
is a relatively complemented lattice and all maximal chains in
U
(
S
)
\mathcal {U}(S)
have the same number of elements.