I ntern. J. Mh. & Mth. Sci.
(1982) 205-207
Vol. 5 No.
205
CORRIGENDUM
ON LINEAR ALGEBRAIC SEMIGROUPS III
MOHAN $. PUTCHA
School of Physical and Mathematical
Sciences, Department of Mathematics,
North Carolina State University,
Raleigh, North Carolina 27650
There are some errors in the above paper. There is a line missing at the
bottom of page 672. Also, pages 681-683 are organized incorrectly.
These errors are corrected as follows:
Replace the last sentence on page 672 with:
gE(Y)g
gG
-I
E(S).
J #
In particular E(Y)
@
over the length of any maximal chain in U(S) equals dim
Replace page 681 beginning from line 14
(from the
U(S).
for all J
More-
Y..
top),
the entire page
of 682 and the first seven lines (from the top) of page 683 with:
"PROOF.
We can assume that e is the identity element of S (otherwise we work
with eSe).
By Lemma i.i we are reduced to the case when f is the zero of S.
Corollary 1.5, we are reduced to the case when S is also a d-semigroup.
By Lemma
2.2 and Theorem 2.7, we can assume that S is as in Theorem 2.7, with
e-- (i
S
1
@(a)
V
(0
i), f
I.
Then e, f
(ml(a
connected.
0).
SI,
dim S
a),...,mn(a
Let V
1
1
i,
,a)).
This proves the theorem.
{(ml(a
SIC_
S.
a)
Define 8:K
Then @ is a
,a))la
n(a
/
S
1
K},
as
*-homomorphism.
So S
By
1
is
M.S. PUTCHA
206
3.
POLYTOPES
If X C_
n,
n
vex hull of a finite set in
I.
(0, i), a + (i
P, e
Then dimension of P
PI’ P2 have
Two polytopes
X(P 2) (see [4; p. 38]).
Then [4; p. 21],
(length of any maximal chain
By [4; p. 244], every polytope of dimension
If u
(i
an),
X(PI
the same combinatorial type if
(81
v
!
3 has the
However this is not true in
same combinatorial type as some rational polytope.
general [4: p. 94].
X if
)a
Dimension of P is defined to be the dimension of
is a finite lattice.
the afflne hull of P [4; p.3].
X(P))
P, then X is said
Let X(P) denote the set of all faces of P.
X.
and only if a, b
If X
polytope.
[4; p. 25] if for all a, b
to be a face of P
in
ra.tional
The con-
If the vertices of P
[4].
is called a polytope
are rational, then P is said to be a
(X(P),C_)
_
then we let C(X) denote the convex hull of X (see[4]).
,8
n
n)
then let u
v
n
iSi
denote the inner product of u and v.
An ideal I of S is said to be semIprlme if for all
Let S be a semlgroup.
a C S, a
or b
I.
2
I.
I implies a
S, ab
I is prime if for all a, b
Let
I(S)
{All ideals of S}
A(S)
{All principal ideals of S}
F(S)
{All semiprime ideals of S}
A(S)
{All prime ideals of S}
X(S)
{S\III
D(S)
Maximal semilattice image of S.
_ __
’
{}
{@}.
C A(S)}.
It is easy to see that (A(S),
(A((S)),
is a complete lattice.
finitely generated, then (S) is finite and so (A(S),
THEOREM 3.1.
F((S)) as (I)
as 8(W)
I implies a
{ala
bijections and 8
is a finite lattice.
Let S be a connected d-semigroup with zero.
{XIX E
S, x(a)
e
-i
(S), x(a)
I}.
0 for all a
0 for all X
Moreover e(A(S))
W}.
Then
Define :[(S) +
Define 8:F((S)) + (S)
eare
A((S)).
If S is
inclusion reversing
I
_
LINEAR ALGEBRAIC SEMIGROUPS
PROOF.
e
X
B((I)) such that a
(I) and f
B((I)),
Let P C A((S)).
is true for P
A(S), (I) A((S))
i, (e)
and B((I))
O.
So
So
(12)
I
A(S) and (B(P))
# (S).
So assume P
B((I)).
I, f
Then f
So
a contradiction.
O) C S and that (S)
S into K, i
e.l
1
n.
1 if
A,
Xi
and u(e)
O.
# v(e)
Clearly
Then F
P.
By Lemma 2.1, this
(S)\P is a subsemigroup of
ei
2
,X
n > where i is
X1
{XilX iC
0 if
Since u(e)
U(Xl
<
Let A
by Lemma 2.3, there exist u, v
v(e)
E(S).
A((S)).
’Then there exists
(S) such that x(f)
We calim that B(P)
(S).
Suppose not.
eS for
By Lemma 2.2 we can assume that S is a closed submonoid of some (Kn .)
(0
where
Let a Hf, f
Then I
It follows that (I)
B((I)).
By Lemma 2.1 (2), there exists X
f.
(S).
I.
A(S).
Let I
0}.
(S), x(e)
We claim that I
for all I
0
{XIX
So (I)
B((I)).
Clearly I
a
Clearly ,B are inclusion reversing.
E(S).
some e
207
A.
Xi
F(X
1
u(e) and
n)
F}.
v( 1
Then <A>
F.
We claim that e
X
n)
such that u(a)
v(e)2
Xn).
the i
th
projection of
(e I
Let e
S.
e
Suppose not.
v(a) for all a
v(e) we can assume that u(e)
Since u(e)
i, u(X
I
,Xn
n
Then
S
i,