I ntern. J. Math. & Math Sci.
Vol. 4 No. 4 (1981) 667-690
667
ON LINEAR ALGEBRAIC SEMIGROUPS III
MOHAN S. PUTCHA
School of Physical and Mathematical
Sciences, Department of Mathematlcs,
North Carolina State University,
Raleigh, North Carolina 27650
(Received July 18,1980)
ABSTRACT.
Using some results on linear algebraic groups, we show that every con-
nected linear algebraic semlgroup S contains a closed, connected dlagonallzable
subsemlgroup T with zero such that E(T) intersects each regular J-class of S.
is also shown that the lattice
rational polytope in some
n
(E(T),<) is isomorphic
It
to the lattice of faces of a
Using these results, it is shown that if S is any
connected semlgroup with lattice of regular J-classes U($), then all maximal chains
in U(S) have the same length.
KEY WORDS AND PHRASES.
Linear algebraic semigroup, idempotent, polytope.
1980 MATHEMATICS SUBJECT CLASSIFICATION CbES.
O.
Primary 20M10, Secondary 20G99
INTRODUCTION.
Z, Q,
Throughout this paper,
+, Z+, Q+ will denote the
sets of reals,
integers, rationals, positive reals, positive integers and positive rationals re-
spectively.
If X is a set then
IXl
denotes the cardlnallty of X.
If X is a
subset of a semlgroup, then <X> denotes the subsemlgroup generated by X.
is a partially ordered set and
{el <e2
<
we define the length of the chain to be n
n
closed field, K
K
<en }
i.
is a finite chain in
(P,<_)
P, then
K will denote a fixed algebraically
K the afflne n-space.
all nxn matrices and GL(n,K) the group of units
If
of
Mn (K)
Mn(K).
note the free commutative semlgroup in the variables X
will denote the set of
F(X l,...,Xn)
l,...,xn
and K[X
will de-
I,...,xn]
H.S. PUTCHA
658
XI,... ,Xn.
the free commutative algebra over K in the variables
[6,(]
notation of
x/g
-1
g E G, then the maps x/xg, x/gx,
If
xg are all automorphisms of the variety S.
group.
a,b a,bES,
If we let
automorphism.
The last one is also a semigroup
1}, then
ab
becomes an algebraic
[5],
Actually, with more general notions of varieties
viewed as an algebraic group
closed submonoid of some
If S
Let S be an algebraic monoid with
for algebraic semigroups.
identity element i and group of units G.
n
By [6, Theorem 1.1], we can assume that S is a
(K)
GL(n,K)
Then clearly G
G, then H is the group of units of
a closed subgroup of
S and dim S
then clearly so is G, G
Lemma
1.9],
If H is
1.
If S is connected,
H.
1
of S and S
1
[7;
is a closed
If S is connected, then
S, it follows from the Lie-Kolchin Theorem [5; Theorem 17.6] that S is
since G
trigonalizable if and only if G is solvable.
*-isomorphic to a closed subsemigroup of
G
connected, then since
a torus
By
[7;
(KP, -)
S, we see that S
3.15],
Corollary
S is a d-semigroup if S is
conjugate to Y if
g-lxg
+.
Z
for some p
If S is
is a d-semigroup if and only if G is
a connected d-semigroup with zero can be
characterized as a connected Clifford semigroup with zero.
Y for some g
X,YCS, then X
If
is
G.
CONNECTED SEMIGROUPS
LEMMA 1.1.
Let S be a connected monoid,
eE(S),
closed connected submonoid S’ of S such that l, e
PROOF.
UV
V
1U
r
where
dim eS
i
e
#
1.
V1,...,Vr
are closed and irreducible.
Then m. < n-l, where n
I
IV.
i
1
is dominant.
of eS such that 0
i
Wi
dim S. Let
and so that dim
-S
/
S\ G.
Let mo
1
Then V
dim V
eS be given by
Let i E {1 ...,r)
(Vi)C eS
So by [5; Theorem .3]
C_ (V
i)I
Then there exists a
S’ and e is the zero of S’.
Let G denote the group of units of S and set V
1,...,r.
Let q
Then
G N S
We say that S is t.rigonalizable if S is *-isomorphic
to a closed semigroup of lower triangular matrices.
i
S\G is closed
N S and
If S is not connected, then
dim G.
1 lies in a unique irreducible component S
connected submonoid of S.
1.
G itself can be
is closed submonoid of S with group of units G
G
l, then 1
1
We use the
Suppose
i’
(x)
W
i
ex.
eS
there exists a non-empty open set 0
_l(x)-
m_-a-<n-q
for all
x0..
i
So
LINEAR ALGEBRAIC SEMIGROUPS
dim(V i
Next suppose W
Let 0
< n-q for all x 60.
1
Then set
-l(x)
@-l(x).
[5;
Theorem
c_ V.m
4.1]
dim D >_n-q.
Since x
y6y.
S
{alaeS,
the
e 0..
1
e
We claim that D
-l(x)
D, I
e}.
ea
gg
-I
e.
N V
i
(2)
# @.
Let x 60.
Then
@-l(x)
such that x
D.
I
Dg
-I
xg
-i
Since x6D, e
Let g6DOG.
Hence ey
e.
xg
-i 6
Y.
C_
Then Y
S
3
e for all a 6 S
and ae
FACT 1.2.
that each A 6
Let
A
A c Mn (K)
3.
So ae
e for all a6 S
ea
such that AB
BA for all A, B
PROOF.
is
(n-l)
We prove by induction on n.
(n-l),
aa6K.
Clearly
minimum polynomial of C
minimum polynomial of C
Let
CC 8 C8C
p-lAp
A
{A
of S
3
MI
such that
MTIcMI
2
such that
3.
A.
Suppose also
is diagonal.
Is
E
}, Aa
for all a,8.
C
C
Since
minimum polynomial of A
has no multiple roots.
So each C
is diagonal for all a.
be
By the dual of
is diagonalizable.
So there exists, by induction, an invertible, lower triangular (n-l)
matrix
2
Then there exists a
is lower triangular and diagonalizable.
lower triangular, invertible matrix P such that
6
Let S
is a closed
2
2.
the above argument, there exists a closed connected submonoid S
e 6 S
e for all
YC_S I.
and S
2
e for a 6 S
and ea
2
So
Let
is a closed submonoid of S and
Thus i, e 6 S
Then
Then Y is closed and irreducible
(unique) irreducible component of I in S
I.
connected submonoid of S.
# @.
Then
t ()
So
is ruled out.
So D N G
V.
Let Y
For suppose D C V.
V.
(2)
i,
x and ey
-i 6y.
Then S
0
Hence D
So eyg
Then yg 6 D.
Since g 6
I
x, xg
Thus eg
Let y 6 y.
Then
Let D be an irreducible component of
for some i.
x.
_
()
Since eS is connected, 0
dim D < n-q, a contradiction.
(g)
eS\W..
0.
for all x
0102... Or.
x
D
@-l(x))
# eS.
i
N
V.1
N
669
Let M
MOIl"
(n-l)
Then
a
670
M.S. PUTCHA
M-IAmM
D
Let
Ea De
a 6
aal,
(all a, 8).
G
.
Then each
Also, since E
c
Let
6
(3),
u_
I
().
0 implies b
is
diagonalizable and
EmE8 EsEm
0
n
2
i
(a).
n,all c
If there exists
(3)
component and 0 elsewhere.
dependent set of eigenvectors of
E
LEMMA 1.3.
.
Let P
#
1
0
let u.=
b
i
1
)/c
1
If there is no such m, let u
0 for all
.
Then u,
for all a
R is lower triangular and invertible.
is diagonal for all
.
By (4), E( u
i and set u
()
such that c
Uo is independent of the choice of
i
th
B
O, all i,e
i
{2,...,n}.
with 1 in i
R-1D R
Ea
is diagonalizable,
i
By
and diagonal
EaE B EsEe,
c.C.b.B.(
b.e.c.8.C
i
i
i
i
Let i
(n-l)
Moreover,
b(C)
n
Since
(n-l)
is
G
Clearly
.
Let e. be the column
1
e2,...,e n
Let R
R-IE R
0
i
is a linearly in-
[u,
e 2,...,e
n].
is diagonal for all
Then
.
MR.
Let S be a connected monoid with identity element l, zero e.
Let G denote the group of units of S.
Suppose G is solvable.
Then for any
maximal torus T of G, e 6 T.
PROOF.
We can assume that S is a closed submonoid of
Kolchin theorem
[5;
lower triangular.
Theorem
Since
17.6]
S,
there exists P
p-Isp
GL(n,K)
is lower triangular.
Mn (K). By
such that
the Lie-
p-1Gp
is
So we can assume
So
LINEAR ALGEBRAIC SEMIGROUPS
that S is lower triangular.
R-lxR
is diagonal.
Clearly
matrix, with the diagonal being that of a.
Mn (K)
$(G)=$(T)
T. So$(G)CS.
{ala
W
submonoid of S.
torus.
a)
S, $(a)
So T
$(G)
and
So there exists a lower triangular
R-1SR
Then
is a torus.
S, $(S) C S.
Since
is closed.
THEOREM 1.h.
T.
$(a) be the nun
$(X)
$(S)
Since
By
W, W
$(e)
Clearly e
E W
Clearly $ is a
[5;
Corollary
$(S).
U
Then S
E1.3C]
Then
is a closed connected
G and H is a
Then T C H
T.
Let S be a connected monoid with group of units G.
Borel subgroup of G.
diagonal
X.
Let W
Let H denote the group of units of W.
H and W
remains lower triangular.
If a E S, then let
So we can assume that X is diagonal.
*-homomorphism of S into
T U (e}.
Let T be a maximal torus of G and set X
Then X satisfies the hypothesis of Fact 1.E.
RE GL(n,K) such that
671
Let B be a
xB--x -1.
xG
PROOF.
{(a,a-l) la E
G
I
then define
Let B
G.
G}
Then G
I
(a,b)(c,d)
is a closed subset of WW.
(ac,db).
{(a,a-1)la
1
B}.
Then B
G1/B1
is a projective variety.
$(a)
aB
Let V
and hence a normal variety.
[1; p. 77].
morphism.
of B
g
1.
Clearly each fibre of
G,g-lag ).
is closed in Y.
some a
So gB
G.
Thus h
contradiction.
(X)
So
Now
Let
$:GI/G1/B1
(a,b)
(X).
y
g-lag E
(X)
denote the projection of Y onto W.
Now [5; Theorem
be the natural projection
By [1; Theorem
6.8],
G1/B1
W G1/B
1
So Y
(a,$(b)).
is an open map.
gb for some b
(X)
GI,
is smooth
is normal
is a surJective
Then
is irreducible and has dimension equal to that
18.],
for some xX,
S, g, h
hB.
1.
Then X is closed in V.
Clearly
(x)= (y)
is a Borel subgroup of G
The same is true for W.
Let :V/Y be given by
So [i; Proposition
1
WG1/B1.
WGl, Y
(a,b),(c,d)
If
Then G is an algebraic group *-isomorphic to
1
El.B],
1.
Mn (K). Let
We can assume that S is a closed submonoid of W
.
So
(X)
So x
X.
Since
is open in
(X),
Suppose
(a,g,g-l),
(x)
{(a,g,g -1 )la
Let X
(y),
m E
y
Y.
Hence (X)
(X).
Then
(a,h,h -I)
(g,g-1)
S,
for
(h,h-1).
,
h-lah b-l(g-lag)bb-lb
a
and (X) is closed. Let e:Y
W G1/B1+ W
Then since G1/B is projective, e is a closed
1
B.
So
672
M.S. PUTCHA
5
map
6.2 ].
Theorem
c_ S.
u gBg
[5;
By
e ( (x))
Hence
is closed in W.
22.2], O C_ e(#(x)).
Theorem
e ( (x))
Clearly
s.
s,e((x))
Since G
This
proves the theorem.
COROLLARY 1.5.
Let S be a connected monoid with zero e and T a maximal torus
in the group of units G of S.
PROOF.
Now T
for some x 6 G.
C__ B
el,...,ek
6
E(S)
x
-I ex E
Hence e is the zero of
such that e
_
k
> e
I
>
2
> e
So assume k > 1.
Corollary 1.5.
monoids
SI,...,Sk
all a
S i, i
,k.
submonoid of S and
S1,
S1,...,Sk
the zero of S.
If k
Let G
1
C V.
E S, ae
k
eka
Then ae
ek).
denote the group of units of W.
1
and G
1
T
LEMMA 1.7.
there exists
e
for
k
Then V is a closed
So e
k
is
By our induction hypothesis,
such that e 1,...,ek_
1
6
T
Then
1.
By Corollary
el,...,ek
6
T.
Jl’’’’’Jk e (S), Jl >J2 > >Jk" Then
that
e.l J’’l i l,...,k and el>e2> ...>e
Let S be a semigroup,
el,...,ek
THEOREM 1.8.
Let
Let W be the (unique) irreducible component of
Let T C T where T is a maximal torus of G.
1.
1
By [7; Lemma 1.3] we have,
6
eka
k
W and W is a closed connected submonoid of S.
there exists a maximal torus T
1.5, e k
By Lemma 1.3, e 6 T.
l, we are done by Lemma 1.1 and
i
{ala
Let V
k
Exx-I
By Lemma 1.1, there exist closed connected subi
S
e
Then there exists a maximal torus
k.
of S such that e. is the zero of S..
1
Then
1.4,
T
6
We prove by induction on k.
PROOF.
.
By heorem
Let S be a connected monoid with group of units G.
T of G such that e 1,...,e
1 in V.
.
for some Borel subgroup B of G.
So e
COROLLARY 1.6.
Then e 6 T.
6
E(S)
such
Let S be a connected monoid with group of units G.
Then
(1)
All maximal closed connected d-submonoids of S are conjugate.
(2)
All maximal closed connected d-submonoids with zeroes, of S, are con-
Jugate.
(3)
Let Y be a maximal closed connected d-submonoid with zero, of S.
U
geG
gE(Y)g-1
E(S).
Then
In particular E(Y) N J # @ for all J 6U(S). More-
673
LINEAR ALGEBRAIC SEMIGROUPS
PROOF.
Since the group of units of a maximal closed connected d-submonoid
of S is a maximal torus in G, (1) follows from
(2)
S.
Let
i
l, 2.
e.l
f
H.I
Then
HiCTi,_
T
a maximal torus of G, i
i
(i)
SiCUi,
i, 2.
=U
i
BM
submonoid S
I
Then e. > f
I
> e
2
> e
>
el
> e
submonoid
Let W
i"
2
BM
2
the
2
{ala6Vi,
Since VI, V are
2
Since
SiCW.l’
i
are conjugate.
1
i=l
i
maximalitM of S i
By Corollary 1.6, e e S for some closed connected d1
2
of S
Clearly
>
> e
k
2
I
S.
there exists a closed connected d-
such that e 6 S
E(S).
For if
By Corollary 1.6,
of
3.16],
# @.
By (2)
2.
Next let
ei>f>ei+I,
el,... ,eke
By [7; Theorem 3.16],
xS2x-I C__
Jl > J2
e e E(Ji)
i
By Lemma 1.7, there exist
k.
is a maximal chain in
contradiction.
U
1
f.1 EU i’ i
Hence E(Y) N J e
U(S).
maximal chain in
e
Hence U
By [7; Theorem
of S.
So xex -I 6 Y.
G.
i--
=T
i
2
submonoid with zero, S
x
2
Lemma i.i,
Let e e E(S).
(3)
W
so are W
1
i=l
i’
Let V
l, 2.
the (unique) irreducible component of i in W..
}, U
conjugate by
Si
21.3A]
Corollary
Let S l, S be two maximal closed connected d-submonoids with zeroes of
2
be the group of units of Si,
be the zero of S i, i
l, 2. Let
Let fi be the minimum idempotent of V i"
af.
[5;
f
E(S),
y for some
>
>
Jk
be a
such that
then J
e
>Jf>J el+1
a
i
for some closed connected d-
el,...,ek
6
M
2
for some closed
M for some maximal
1. So el,...,ek
3
connected d-submonoid with zero, M of S. Since (5) is maximal in E(S), it is
3
dim Y.
k-1. By (2) dim M
By [7; Theorem 3.17], dim M
maximal in
3
3
connected d-submonoid with zero,
of M
M
2
E(M3).
Let S be a connected semigroup.
THEOREM i. 9.
(S)
Then all maximal chains in
have the same length.
PROOF. U(S) has mximum element
G(eSe)
{JNeSe
by Theorem
J 6 (S)}
_U(S)
J0"
Let e 6
E(J0).
BY [7;
Now eSe is a connected monoid.
Lemma 1.3, 1.7].
We are done
1.8 (3).
THEOREM i.i0.
Let S be a connected monoid such that for all a,b E S,
alb
674
M.S. PUTCHA
implies a
2
b
i
for some i E Z
J e U(S).
are,
Let Y be a maximal closed connected d-submonoid
The hypothesis implies by
Let J e U(S).
Then J N y
E(Y).
bHf in Y for some e, f
Since J is completely simple, e
[8]
# @
by Theorem
So e, f
J.
1.8.
Let a, b e J N Y. Then
Since e, f
e y, ef
fe.
So aHb in Y and J N y is a subgroup of Y.
f.
have
Let S be a connected semigroup such that for all a, b 6 S,
COROLLARY 1.11.
a21b i
In particular
that J is a subsemigroup of S for all
Now applying the proof of Theorem 1.9 we
implies
J 6U(S).
Then JOy is a subgroup of Y for all
with zero, of S.
PROOF.
+
for some i
e Z+
Then
((S)
<)
(E(Y)
<) for some
connected
d-monoid with zero, Y.
Let S be a connected monoid such that the group of units G of
THEOREM 1.12.
S is nilpotent.
PROOF.
By
Then
[5;
E(S)
is finite.
Proposition
19.2], G
has a unique maximal torus T.
the unique maximal closed connected d-submonoid of S.
So T is
By Theorem 1.8, E(S) C T.
So E(S) is finite.
EXAMPLE 1.13.
S
is an example of a connected
a
0
monoid satisfying the hypothesis of Theorem 1.12.
CONJECTURE iolh.
Then the group of units of S is solvable.
finite.
EXAMPLE 1.15.
with zero and
2.
Let S be a connected monoid with zero such that E(S) is
Let S
E(S)
2.
a, b 6 S
Then S is a connected monoid
The group of units of S is solvable but not nilpotent.
CONNECTED d-SEMIGROUPS WITH ZEROS
Let S be a connected d-semigroup with zero, dim S > 0.
[7;
Theorem
X(1)
l,
that
By a character of S, we mean a *-homomorphism x:S/K such
2.7].
X(0)
Then S is a monoid
0.
We let (S) denote the commutative semigroup of all characters
of S with pointwise multiplication.
It is clear that if S l, S 2 are connected d-
LINEAR ALGEBRAIC SEMIGROUPS
dim S > 0
semigroups with zeros
then S
675
to S implies (S
2
I *-isomorphic
I)
(S 2)
A commutative semigroup W is said to be totally cancellative if W is
cancellative and for all a, b E W,
[3;
the following result of Grillet
THEOREM A [Grillet].
n E Z
+
a
Theorem
n
b
n
implies a
We will need
b.
2.2].
Let W be a finitely generated commutative semigroup.
Then W can be embedded in a free commutative semigroup if and only if W is totally
cancellative and idempotent-free.
LEMMA 2.1.
dim S > 0.
Let S be a connected d-semigroup with zero 0 and identity l,
Then
()
(s) #
(2)
If e E
>_
g
.
E(S),
# 0,
e
x(g)
e implies
XE(S)
then there exists
g_e
l,
implies
x(g)
such that for all
gEE(S),
0.
(3)
(S)
is idempotent- free and totally cancellative.
(4)
(S)
is linearly independent in the vector space of all functions from
S into K.
PROOF.
Let G denote the group of units of S.
Mn (K)
closed submonoid of
E
(S),
a(f)
0 for
for some n E Z
fEE(S),
#
f
#
Let X
x’S
Define
e.
E
@(S)
m E Z
So a E
S
X1
m
X2
on S.
for some
and
S
t
t(S)
E Y.
for some
Xl(a)
#
So
@(S)
{I
Let Y
S= {ala E S, Xl(a)
connected, S
?(2
X3
X2
m
such that
E Y, let
Xl
S,
Since
+
So #(S) is idempotent-free.
Thus G
E Y.
C__
L
{1,0}
$5.
m
is totally cancellative.
X2, X B
E(eS)
(S)
So
Now let
such
X2
If aEG, then
#
xi(a)
S, S
XI(1)
(3).
k
X3
XI,X2 E(S),
Then Y is finite.
1).
This proves
E
x3(a).
x2(a
In particular, i
0.
contradicting the fact
Let X l,
Since G
Then
(2).
is cancellative.
E K,
det ao
0 for all f E
8(f)
0 and so
x2(a)}.
(a)
Let e E E(S), e
This proves
X(S)
Then
X.
If a E G, then
XIX2=XIX3
on G.
2
#
such that
8(ea).
x(a)
K as
such that X
that S is connected.
that
/
.
If a E S, let
So (S)
1.
@(eS)
Then by the above, there exists 8 E
with f
+.
We can assume that S is a
If
0 for i=l, 2.
S.
X2(1)
Since S is
.
So
676
M.S. PIPrCHA
Now let
Let
dependent.
Xm E
I’
@(S)
be distinct characters of S which are linearly
denote the restriction of
@i
linearly dependent homomorphism of G.
#
i
S,
G
Since
j.
(0,...,0)
such that 0
(2)
@(S)
(3)
If a, b
n,
(K
0
Z
that e
(0,
i
,n.
1
+
(1).
of
is the zero of S
.,0).
denote the identity of S.
’an
0, say i
So a. # 0, i
i
closed submonoid of
n,
(K
-).
1.
This proves
n,
E1
’n E
Let
(S)"
denote the i
Let X
@(S).
th
2
C
i
Kn-1
i’
So
contradicting the
l, i
1,...,n and S is a
(1).
-)
(1,...,1),
with identity 1
projection of S into K.
Since
So
Then n > 1 and S C {0}
So m.
i
1,...,n.
Let S be a closed submonoid of (K
0).
1
So without loss of generality we can assume
1.
(l,
Let f
S and set S
Let e denote the zero of
represents a *-isomorphism between S and S l,
Suppose some m.
I
minimality of n.
(0,
+
-) for some n E Z
@(S) such that (a) # (b).
S is *-isomorphic to a closed subsemigroup of (Kn-1
0
n,
(K
Then
We can assume that S is closed subsemigroup of
n minimal.
Then a <--> a-e
(0,... ,0)
for some
S
S, s # b, then there exists
for some n
{a-ela
1,...,m are
is finitely generated.
First we prove
S).
@i’ i
@i @J
This proves the lemma.
S is *-isomorphic to a closed submonoid S
PROOF.
Then
Let S be a connected d-semigroup with zero, dim S > 0.
LEMMA 2.2.
(1)
to G.
[5; Lemma 16.1],
So
contradiction.
Xj,
i
Xi
(0)
0,
So there exist
ml"’’’mt F(XI"’’’Xn) al
t
(a)
[ eii(a) for all a e S.
zero
Then clearly
does not have a constant term.
m
n
K such that
i=l
I ii(Xl,..., Xn).
So X
Lemma 2.1(h), X
B
i=l
(S)
<X1,...,Xn
all X 6
(S).
>.
Then a
This proves
(2).
i(Xl,...,Xn
Let a, b e S such that
(Xl(a),...,Xn(a))
(Xl(b),...,Xn(b))
for some i.
x(a)
b.
x(b)
So
for
This proves
(3).
LEMMA 2.3.
0
(0,...,0).
n,
Let S be a closed connected submonoid of (K
Then there exist
Ul,...,ut, Vl,...,vt6 F
.) with zero
(Xl,...,Xn)
such that for
LINEAR ALGEBRAIC SEMIGROUPS
K,
a
S if and only if
a
PROOF
Let
ui(a)..
K[,...,Xn] f(S)
F(Xl,...,Xn)}. We claim that
(D)
f
.(a,
i
i
l,
’t
denote the i th projection of S into K.
i
{flf
Let I
v
v
677
Since f(0)
0
0}.
I
there exist
{flf
Let D
Then
I, f
(D).
Suppose not.
ml
,m r
l’’’’’n (S)"
u-v for some u,
Then there exists fI,
F(X1
’Xn)
a
r
)
such that f
i=l
f(a)
r
Of all such f choose one with r minimal
aimi
(0
,a r
l,
So
[ aimi (a)
+/-=l
0 for all a
S.
So
r
[
i=l
(x,...,)
21(4),
By Lemma
Assume p
l, q
o
p(Xl,...,Xn) q(XI,...,Xn)
m.+/-(a) m2(a)
So
2.
for some p, q 6
for all a 6 S.
Thus
{l,...,r}, p # q.
_+/--a
6
D.
Now
r
(i + 2)2
By minimality of r,
(D).
I
I
So
[ s’’11
+
=i(I 2
f
i=3
f-al(mI m2)
e (D).
6
e(D),
So f
By the Hilbert Basis Theorem there exist
(fl,...,ft).
(D1)= I. This
(D),
Since I
(Dl)
fl,...,ft
I
a contradiction
fl"’" ’ft 6
I such that
for some finite subset D
0(S I)
/
S
by
2
be a -homomorphism such that
0*(X)
1
of D.
proves the lemma.
Let S l, S be connected d-semigroups with zeros, dim S. > O, i
2
0:S I
Thus
X o 0-
0(1)
"
Next assume that
Then we claim:
for all a 6 S
0(0)
i,
0(S 2)
I, there exists unique b
/
6 S
2
O.
0(S I)
l, 2. Let
Then define
*’($2)/
is a homomorphism.
such that
(6)
()
Assume (6).
for
/
S
for all a
e
Then we can define
xC@Ca))
Next we
(())(a)
(@(x))Ca)
2
(a)
as
X e
SI,
b.
Then
(7)
($2)
claim
is a W-homomorphism,
We now prove
2.2(3).
@:S 1
(6), (8). Note
(i)
I, #(0)
that the uniqueness of b in
By Lemma 2.2(1) we can assume that S
2
(8)
0
(6) follows
from Lemma
is a closed submonoid of
n,
(K
.)
678
M. So PUTCI-IA
(0,...,0).
with zero 0
Lemma 2.2,
X1,...,Xn
Let
($2)
6
Xi
and
denote the
(S 2)
<
.th
projection of S
X1,...,n >.
2
into K.
By Lemma 2.3, there
F(X1,...,Xn) such that for b Kn, b S 2
only if ui(b)
, ...,Xn) vi(x1,...,xn)
vi(b), i 1,...,t. So u.(Xl
Hence ui((Xl),..., (Xn))
vi((X 1),...,(xn)), i 1,...,t. Thus
exist
,vt
v
Ul,...,ut,
I
6
ui(((Xl))(a),...,((Xn))(a))
:Sl+ S 2
(0)
(((X1))(a),...,((Xn))(a)) e S2
(a)
(((X1))(a),...,(((n))(a)).
as
0.
(S 2)
<
vi(((l))(a),...,((Xn))(a))
So
i=l,...,t.
Clearly
Xi((a))
l,...,n >, (7)
(((i))(a)
and hence
(6)
if and
i=l,...,t.
for all a
for all a 6 S
So
Then by
1.
S1,
Define
(1)=l,
is a *-homomorphism,
for all aS, i=l,...,t.
Since
It is clear from (7) that
is true.
(9)
Now let
ae SI,
@’Sl+S 2
by
be a *-homomorphism,
@(1)
@(0)
l,
0.
Then for all X
($2),
(7),
x(( )(a))= ( (x))(a)= x((a))
By Lemma
2.2(3),
(i0)
@
@
THEOREM 2.4.
i
i, 2, 3.
(i)
If
Si > 0,
Then
(S 2)/(S I)
If i’S
/
1
S
@(0)
is a *-homomorphism with
@:SI/S 2
@
(2)
Let S I, S 2, S be connected d-semigroups with zeros, dim
3
1
is a homomorphism and
is the identity map then i
O, @(i)
@
i, then
@.
-(S1)/(S1)
is the identity
map.
(3)
If
@:(S 2) / @(S I)
with
(4)
If
$(0)
0,
is a homomorphism, then
(I)
i’(S1)/(Sl)
i.
Moreover
@:SI/S 2
is a *-homomorphism
@
is the identity map, then
i’Sl+S 1
is the identity
map.
(5)
If
i
@l’Sl+S2 @2"S2/S 3
l, 2, then (@2 o
@l
are *-homomorphism with
1 o @2
@i(0)
0,
@i(1)
i,
LINEAR ALGEBRAIC SEMIGROUPS
(6)
If
I’($2)/(S1), 2:($3)/(S 2)
679
are homomorphisms, then
i o 2 2i
(7)
S
1
is *-isomorphic to S
(i), (3) follow from
PROOF.
if and only if
2
the equations
(2), (4), (5), (6).
(7) follows
from
prove (5).
Let X e
(S3).
is isomorphic to
(2), (4)
(6)-(10).
So we need only prove
Then for all
())(a)
((2 o i
(S l)
a
($2).
are trivial.
(5), (6).
First we
S I,
($2(1(a)))
(2(X))(,l(a))
(,l(,2(X)))(a)
((1 o 2)(X))(a)
So
(2
i o $2
i
0
Next we prove (6).
S,
Let a
(S3)-
X
Then by equation
(7),
((1 o 2)(X))(a)
X(l o 2(a))
(l(2(X)))(a)
(2(X)) (el(a))
X(2(l(a)))
X(2 o el(a))
By Lemma
2.2(3),
i o 2 2 o i’
THEOREM 2.5.
CKn
Let e
Set
Moreover
,e
I
(S)
<
Define
el,
i denote
<Xl,...,Xn>
u(b)
1
.,Xm)
Let V
{(el(a)
,e
m}
n (a))laEK
the i
e:(Km, .) (Kn, .)
/
v(b)
th
So S
e(a)
as
for all b
F(Y1,...,Yn).
S.
So
(el(a)""’en(a))"
is connected.
projection of S into K.
Let u, v
(Kn,
).
e >
*-homomorphism with image V.
Then
F(X
n
Then S is a closed connected d-submonoid with zero, of
PROOF.
Let
proving the theorem.
Clearly 1
Then by Lemma 2.2,
Suppose
Then 8 is a
e(1),
0
e(0) S.
(S)
U(Xl,...,Xn) v(Xl,...,Xn).
.
680
S. PUTCHA
m
v(ml(a),...,mn(a)) for all a
U(l(a)’’’’’mn(a))
,Xm).
u(col ’con) v(col," ’con) in F(X1
u(col"" "’con) v(col’" ,con in F(Xl,...,Xm). Then
Since K is infinite,
suppose
u(b)
v(b) for
It follows that (S)
v(x1,...,Xn).
S, u(b)
Since V
all bEV.
Conversely
(ii) is true.
v(b) for all bES.
=<X1,...,n >
(ll)
K
So
u(Xl,...,Xn )=
So
<col,’’’,mn >"
By Theorem A, Lemmas 2.1, 2.2, Theorems 2.4, 2.5, we have.
Let N
THEOREM 2.6.
be the category of connected d-semigroups with zeros of
1
dimension > 0 with morphism being *-homomorphisms
N2 be the
Cwith (0)
0,
(1)
is a contravariant equivalence between
Then
(Kn, -)
Let S be a closed connected submonoid of
+
Then for some mZ
col"’’’n
(Xl’’’’’Xm)’
(0,.. ,0).
S
Xi denote the i
<X1,...,Xn >. By Theorem
V.
1
th
Let
..,Xm)
col’" ’con E (X1
u(c)
v(c)
V(l
be S 1.
Thus
Since K is infinite,
u(c)
v(c)
(0,...,0),
<
col’’’’’con >
Then
all bEV.
S.
By Lemma 2.3,
S I, u(b)
Since V
S
v(b) for all b
u(b)
Then there exist
Conversely, for any
closed connected submonoid of
(Kn, ")
Z
for some m
+
’n (a))laKm) and set
Let u, v(Y1,...,Yn). Suppose
v(x1,...,Xn). So u(col,. ,con)
U(l,...,Xn)
v(b) for
Then by Lemma 2.2,
S
for all a
So
S
for all
e Km
U(Xl,...,n
v(Xl,’-’,Xn )"
I.
il,...,in E Z+
ii,...,I n 6Z
with zero 0
v(b)
Then
1.
Let S be a closed connected submonoid of
dim S=I.
il ,...,ain )laK}"
V where V
{(col(a)
,COn(a)) v(col(a),...,con(a))
u(col,...,con) v(col,. ,con)"
for all c
COROLLARY 2.8.
0
u(b)
Conversely suppose
u(col(a)
Thus
A, (S)
with
S.
for all c
,n ).
projection of S into K.
i-,mi Let V
(1,...,l)0 (0,...,0) S 1.
Then 1
S
with zero
Km}.
{(col(a),...,con(a))la
PROOF.
(@,*)
l and 2"
THEOREM 2.7.
(S)
Let
category of finitely generated, commutative, idempotent free, totally
cancellative semigroups with morphisms being semigroup homomorphisms.
0
1.
+
(Kn, ")
with zero
such that S
S defined as above is a
(0,...,0)
and dim S
i.
681
LINEAR ALGEBRAIC SEMIGROUPS
{CmlCa),...,mnCa))la E Km)
V
aEK),
SI=V-’.
such that V
SlCS_
[5;
By
s=V.
v
Let
T(Ca,...,a),...,nCa,...,a)
Sl=l. So S=S1.
i
i
{(a I ,...,a n)laEK}. Define e:K/S
Then
I
it is easy to see that
.2].
and
e
Proposition
d
So there exist i 1,...,inE Z
as
i
(a I ,...,a
e(a)
is a finite morphism in the usual sense of
.2], e(K)
[5;
n).
Then
Section
S.
Let S be a connected monoid with zero, dim S
THEOREM 2.9.
i
+
I.
Then S is
*-isomorphic to a semigroup of the type given in Corollary 2.8.
By Corollary 1.5, S is a d-semigroup.
PROOF.
and Corollary
2.8.
Let S be a connected semigroup, e, fEE(S), e >f.
THEOREM 2.10.
exists a closed connected subsemigroup S
f is the zero of S
and dim S
1
1
1
Then there
of S such that e is the identity of S l,
1.
We can assume that e is the identity element of S (otherwise we work
PROOF.
eSe).
with
We are now done by Lemma 2.2
By Lemma i.i we are reduced to the case when f is the zero of S.
By
Corollary 1.5, we are reduced to the case when S is also a d-semigroup.
A(S)
{All prime ideals of S} U {}.
(s)
(s\l
(S)
Maximal semilattice image of S.
A(S)}.
It is easy to see that (A(S),
finitely generated, then
THEOREM 3.1.
as
S(W)
(ala
E S,
biJections and S
PROOF.
Clearly
some eEE(S).
e
#_
(m(I))
f.
E
(a)
m
-1
u,
So u(I)
Clearly I C_(m(I)).
a E
(A((S)), )
is finite and so
is a complete lattice.
(A(S), C)
@(S), x(a)
0 for all a E I).
0 for all X E W).
Moreover m(A(S))
Then
E
@(S), (e)
0}.
We claim that I= (a(1)).
such that a
I.
E
Define
m:I(S)
S:P(@(S))
/
/
I(S)
m,S are inclusion reversing
Let lEA(S).
Then l=eS for
It follows that u(I)EA(@(S)).
Suppose not.
Let a Hf, f E E(S).
By Lemma 2.1(2), there exists
Define
A(@(S)).
are inclusion reversing.
{I
If S is
is a finite lattice.
Let S be a connected d-samigroup with zero.
(I
P(@(S)) as re(I)
(S)
)
@(S) such
Then
that
Then there exists
fI, fE(u(I)).
(f) =l, (e)
=0.
So
So
M.S. PUTCHA
682
e re(I)
8((I)),
and f
for all I e
A(S), a(I) e A(@(S))
(S).
# (S).
So assume P
m(8(P))=P. By Lemma 2.1, this
(S)\P
Then F
(12)
I
is a subsemigroup of
(Kn, "),
By Lemma 2.2 we can assume that S is a closed submonoid of some
#(S).
0=
8(m(I))
and
We claim that 8(P) e A(S) and
Let P e A(@(S)).
is true for P
So
a contradiction.
(0,...,0)
E S and that
S into K, i=l,...,n.
where e =l if
i
xi6A,
#(S)
<
Xi q
eo =0 if
i
by Lemma 2.3, there exist u, v 6
u(e) # v(e)
v(e)
Clearly
0.
u(e) 2
Since
u(e)
u(x 1,...,Xn)
We claim that e 6 S.
A.
n)
F(XI,
and
Let
Then<A>= F.
{XilXiEF).
Let A
where X is the i
i
>
X1,...,Xn
X
v(e) 2
projection of
e=(el,...,en)
Suppose not.
u(a)=v(a)
such that
Then
for all a6Sand
v(e) we can assume that u(e)
Xn).
v(x I,
th
i,
u(e)=l, u(X1,...,xn)
Since
By Lemma 2.2 and Theorem 2.7, we can assume that S is as in Theorem 2.7, with
(I,...,I),
e
S
V
I
=(0,...,0).
Let V
mn(a,
(ml(a,...,a),
connected.
I
{(I (a’’’’’a)’’’’’n(a’’’’’a))la6K}’
SI= i, SIC S.
Then e, f6S I, dim
I.
e(a)
f
,a)).
Define e:K
Then
e
/
If X C]R
as
So S
is a *-homomorphism.
1
is
This proves the theorem.
n
then we let
C(X)
denote the convex hull of X
n
vex hull of a finite set in ]R is called a polytope
[h].
to be a face of P
[h; p. 35]
and only if a, b
E X.
(X(P),C_)
if for all a, b
X(P))- I.
[h; p. 3].
Two polytopes
X(P2) (see [h; p. 38]).
[: P. 9].
Then dimension of P
PI’ P2 have the
If u
1
Then
(length of any
same combinatorial
t_
By [h; p. 2hh], every polytope of dimension
(l’’’’’an),
v
n
I a-8 i
e (0,i), ma + (l-m)b e X
if
[4; p. 21],
Dimension of P is defined to be the dimension of
same combinatorial type as some rational polytope.
general
a
The con-
If the vertices of P
Let X(P) denote the set of all faces of P.
is a finite lattice.
the affine hull of P
e P,
(see [4])
If X CP, then X is said
are rational, then P is said to be a rational polyope.
i= 1
I
POLYTOPES
3
in
S
denote the inner product of u and v.
maximal chain
if
X(PI)
<--3
has the
However this is not true in
(81,...,8n)
n then let u
v
LINEAR ALGEBRAIC SEMIGROUPS
Let S be a semigroup.
An ideal I of S is said to be s_emiprime, if for all
a6S, a 6I implies aI.
b 6 I.
I is
Prime
if for all a, bS, abI implies aI or
Let
[(S)
{All ideals of S}
A(S)
{All principal ideals of S}
F(S)
{All semiprime ideals of S} U {@}
involves only those
v(X1,...,Xn)
Xl’S
with
{XIX
e #(S), x(e):0}
m(8(P))
u(x1,...,Xn)6F. Since v(e)=0,
X.l with Xi @ F. So v(x1,...,Xn) 6P. This
F.
6
Xi
involves at least one
tradiction shows that e 6 S.
P
683
a(eS)
P.
for all P
So
Clearly
a(eS).
x(e)
1 for X 6 F,
By
(12), 8(P)
6
A(S)
x(e)
6
0 for
eS e
8(a(eS))
P.
con-
Hence
A(S),
So
e A((S)), 8(P)
and
a(8(P))
P
(13)
Clearly
a(I 1
U
I2) a(Ii) N (I2)
WI, W2 e r((s)). Then
8(WI)8(W2) C__B(W10 W2).
Let
Then there exist
e
W2,
8(W n
W2)
I
A(S)
Ii,...
I
k
e A(S).
a(I)
S(a(I))
Let a
8(a) # 0.
12
8(Wi)C_8(WINW2) i i, 2. So
6 8(W
W2). Suppose a @ 8(Wi), i=l,
1
such that el(a) # 0, i=l, 2. So
a
8(W1 (] W2), a contradiction. Thus
So
8(Wl) U 8(W2)
for all
Let I e [(S).
is finite.
Ii,
clearly
i=l, 2
Wi,
i
W
I
e18 2
Clearly
8
for all
By (12)
Wl,
Then I
8((I ))
I
r
r
r((s))
W
2
I
1
1
2.
U
12
U...
.,k.
(5)
U
I
k
By (lh)
for some
(15)
a(Ik)
B(a(II) )U... U 8((Ik)
a(Ii)...
IlU
N
uI
k
I
So
8(a(I))
Since
@(S)
I for all I e I(S)
is finitely generated,
p. 125,Exercise
9], W=W1
(]
A((S))
(16)
is finite.
...(] W for some W
k
1
Let
,Wk
W e F(@(S)).
A((S)).
By [2;
Then by
(13)
684
M.S. Pb’CHA
m(8(Wr ))=Wr
Then by
for r=l,...,k.
(lh), (15),
sCwi) u... u(wk)
(s(wi)) n... n(s(wk)
s(w)
(s(w))
WIN
NW
k
W
So
(s(w))
By (16) and
(17),
REMARK.
Wor iwe v(#(s))
-i=8.
By
(17)
(12), (13), e(A(S))fA(#(S)).
This proves the theorem.
The classical Hilbert’s Nullstellensatz yields a 1- 1 correspondence
n
between the closed subsets of K and the radical ideals of
K[X1,... ,Xn].
Moreover
this restricts to a l-1 correspondence between the closed irreducible subsets of
Kn and the prime ideals of
K[Xl,...,Xn].
Analagously,
Theorem 3.1 yields a l-1
correspondence between the ideals of a connected d-semigroup with zero S and the
__
semiprime ideals of its character semigroup
#(S).
Moreover this correspondence
restricts to a correspondence between the principal ideals of S and the prime
ideals of
@(S).
THEOREM 3.2.
(u(s), i)
Let S be a connected d-semigroup with zero.
(E(S), <_)
(((S)), _).
Clearly
(A(S), C_)_(U(S), <_)
PROOF.
is anti-isomorphic to
(X(@(S)), c_).
Let
Qmn
FACT 3 3
.
i
A
denote the set of all
Let A
0 implies
PROOF.
(A(@(S)), C_)
(A(S), )
is anti-isomorphic to
This proves the theorem.
Qmn
(81,...,8m) 6 Qm
exists v
Clearly
Theorem 3.i,
mn
matrices over Q.
The following result is
However, we include a proof here for the convenience of the reader.
well known.
B i>O,
(A(@(S)), c_).
(E(S), <_). By
Then
Let N
u
(al""’am) 6Bm
such that vA
such that uA
0 and for
0.
i=l,...,m,mi>
Then there
0 implies
8i 0
{XIX e]RTM,
omn, there exist ul,...,ut
XA
Qm
0)denote the left null space of A.
such that
Ul,...,ut
is a basis of N.
Since
So
LINEAR ALGEBRAIC SEMIGROUPS
t
7. juj
J=l
U
685
+
for some
en
t
For Z
v
j=l
laj-gjl
u-vl < "
small enough,
For
g
small enough,
the conclusion of the lemma clearly holds.
COROLLARY 3. h.
Let
m
n
;.
such that
.u.
m
-
i
By Fact 3 3 we can choose ml,
PROOF.
n
alUi
Z 8jvj.
j =l
i,... ,n.
Clearly
Then for some
m
THEOREM 3.5.
S
,am,
e Z+
sm
Si
i’
Z
8jvj.
{X(S)
IS
m.u.
J=l
The classes
n
polytope in ]R for some n E Z
PROOF
Let C
+)
8j
e Z+
,m
i=l
{X(P)
P is a rational
are identical to within lattice isomorphisms.
By Fact 3.3,
In-
then
assume that u
l(u-
u.1
Qd,
0 for i
If
0C. So C-C
<_ I u-
i,... ,n.
aES, then
z+ and set a
mk
s
By Theorem A we can assume that S
C(Ul,...,Un).
small enough, v.
ml
8’:j
such that
is a finitely generated, commutative,
there exists u 6R d such that u. a> 0 for all a6C.
d,
such
Let S be a finitely generated, commutative, idempotent-free, totally
cancellative semigroup.
]R
Q+
,8n
81
idempotent-free, totally cancellative smeigroup) and
v
+
n
7.
i=l
0S.
Z
+
J=l
m
J
81,...,8n
]R
Z 8jvj.
c,,.u.
i=l
i=l
8n E
n
7.
that
81
,mm,
Then there exist al,...,am,
8jvj.
i=l
,VnE Qd al
’urn’ Vl
Ul,
lal
By [h; p. ii
],
So u. u. > 0, i=l,...,n
i
II II
so
fo=
If
II u- vll
So, without loss of generality, we can
e(a)
let
+
.
=<Ul,...,Un>C (zd,+),
+
kak"
e
Qd.
u-a
Let al,...,ak6S,
Then
k
e()
8ie(ai)
k
[
i=l
C(e(a].),...,e(.)),
(.a.
8i
i where
i i
a’u
(8)
u
> 0,
i
l,...,k.
M.S. PUTCHA
686
C(e(X)) c p.
(X)
let
C(e(Ul),..., 8(Un))
C(e(S))
So P
X(S).
Let X
exist
al,...,ap,
Y1
’Yr
Eel
ESJ EYk
+
ZSjbj
3.4
there exist
y 6
((X)).
mp +
I
el,...,mp
Z
+
Then
By (18)
So
(X) e X(P)
and
@(F)
X(P).
(X)
So there exist
Em.a.X.
i I
a
X.
Clearly X
and
C_ @((X)).
al,...,ap
@((X))
(19)
X
e(a)+ (l-)e(b) e F
e(a), e(b)eF. Hence a, b6(F)
Let a, bS such
and
@(F)
X(S).
6
for some
Clearly
e (0,i).
((F))CF.
Let
k
(S).
Then xeP
such that
7.i
i.
Then
((F)). So (#(F))
for all
Since
such that
Hence
7. ie(ai
So x
al,...,akeS, l,...,ake(Ql)
for some
i=l
x e
+
such that
e(ap))
or a subsemigroup of S.
is
By (18), e(a+b)
x e F.
]R
By Corollary 3 h, there exist
Za.a..
i 1
such that m a
Let F)P).
There
ZYke(Ck ),
z
C(e(al),
Since x
Hence
C(e(x)).
e(a)
such that ma
that
So
(X).
x, y
XeX(S),
(X).
X
ZYkCk
for all
z 6
81"’’’ 8 q’ YI’’’’’Yr
al’’’’’p’
ap, bl,...,bqX.
e(bq)),
zsje(bj),
y
ml,...,mp, 81,...,8 q, Yl,...,Yr 6
Zgjb
X(S), aI,
C(e(bl)
Let a
a
+
(l-a)y
S.
ZYkC k
1
r.a.a
zeie(ai),
that x
So there exist
I.
such that ax +
(0,1)
then
{ala6S, e(a)F}
(F)
then let
e X(S),
Cl,...,Cr X, al,-..,ap, 81,..-,8 q,
bl,...,bqS,
Z.a.1
Since X
X(P),
Let x, yP, a
(0,i) such
By Corollary
If F 6
If X
is a rational polytope.
,
(X(S), c..c..)
e(a i)
F.
6
F, i
i,
,k
So
al,...,ake(F)
and
Hence
FeX(P), (F) e X(S)
and
((F))
(20)
F
are clearly inclusion preserving, it follows from
(19), (20)
that
(X(p),c_).
c_m be a rational polytope. Then P C(al,...,an) for some
If a 6 Z +, then clearly (X(P), _c)
(X(aP), _c). So we can assume
=
zm" Let u. (ai,1), i=l,...,n, d m+l. Then Pl=C(u1,...,un)=
Conversely let P
al"’"an 6 Qm.
that a1,...,an
LINEAR ALGEBRAIC SE.flGROUPS
C__ m d
P
{i}
<
u]_,...,u
is a rational polytope and
>C Z
d.
nen 0S
(X(P), C__)
687
(X(pl) C__).
and S is a finitely generated,
cancellative, idempotent-free semigroup.
Let u
(0,i) 6 Z d.
Let S
commutativetotally
Then u
u.l
i,
U.
So e(u
i--1,...,n.
l
i)
half of this theorem,
u
u
u.
(X(S), c__)
i’
(X(PI)
By the proof of the first
..,n.
i=l
C__).
This proves the theorem.
If S is a connected d-semigroup with zero, then "by
dim S
length of any maximal chain in
3.6.
THEOREM
dim S>0) and
{(E(S), <--)
The classes
{(X(P), c)
E(S).
EXAMPLE 3.7.
If
SK1*,.),
(Kn, -),
More generally if S
3.17],
Theorem
By Theorems 2.6, 3.2 and 3.5 we have,
S is a connected d-semigroup with zero,
P is a rational polytope in
identical to within lattice isomorphisms.
[7;
IRn for some n6Z
+}
are
Moreover, for any corresponding S
and
then the corresponding polytope P is a tetrahedron
(n-1)-sim-
then the corrsponding polytope P is the
plex.
EXAMPLE 3.8.
Then by
[7;
Let S
Example
1,.7] S
Moreover dim S= 1. and
{(al,bl,a2,bp,a3,b3)lai,bj6K, aibj=ajb i,
i, J=l,2, 3}.
(K6 ")
is a closed connected d-submonoid with zero, of
IECS)I
22.
The corresponding polytope P can be shown to
be the triangular prism"
EXAMPLE 3.9.
2
a2a 5
2
2
ala4, a2a4
Let S
2
a5a3}.
{(al,a2,a 3,a h,a 5,a
Define
(K6)
Clearly dim S
6 / 6 as
K
223
(xI ,x2 ,x 3 ,x1* ,x ,x6
5
Then
0: K
2
2
6)lal,..-,a6 K, a3aI a5a2,
22
2
2
22
x3x4x 5 x2x3x 5 XlX2X 3 XlX2X4’ XlXhX 5
x
S and so S is a closed connected d-submonoid with zero, of (K
4
and
E(S)I
be the pentagonal pyramid:
24.
6)
6 -)
The corresponding polytope P can be shown to
M.S. PUTCHA
688
the
Let S be a connected semigroup such that U(S) is
COROLLARY B.10.
following lattice
Then n < 2.
S
I
_
be a maximal connected d-submonoid with zero, of S.
E(S I)
N J
lu(s)l
IE(S )I
2(K)
1.8(B)
2. By Theorem 3.6, the polytope P
I
So P is the line
.
IE(Ss.)
(,.)
and
Then by Theorem
and dim S
has dimension 1.
1
EXAMPLE B.ll.
h.
U(S)
for all J
corresponding to S
so
Let
By the proof of Theorem 1.9, we can assume that S is a monoid.
PROOF.
Hence n
<_
2.
show that n can be i or 2 in Corollary B.10.
SEMILATTICES
As usual, by a semilattice, we mean a commutative, idempotent semigroup.
LEMMA h.1.
> i.
as
(F)
sPPI"
If F
Let m be a maximal element of
F N
then
(FI)
fl FI such that fl FI"
So af2P for some
f26F1.
1
So F
contradiction.
F
I
X(I)
F,,
F
G.
I #
,
So G
X(I
I
FI.
sp
Otherwise f
X(),
I
Let F
i"
F
F, G
F I, F
and set
Then f
this is clear.
Then {m}
X().
PI I\FI"
Let
PfPI’
Now we claim that F
I
PPI"
IPI.
U {m}
So assume
:X()/X(I
Define
We claim that
a contradiction.
Then F
.
+
If
Suppose not.
I.
So
SPl C_ PI"
So there exists
X().
Otherwise
FI_FI.
flf2 (fl)f2 F1 and aflf2 (af2)flPl,
X(), (F {}) F Thus is surJective. Let
lu
I.
So
U {}
I
.
11"
x( )l
Then
be a finite semilattice.
We prove by induction on
PROOF.
II
n
Let
X(),
F
I
(F)
U {,}.
F
I
(G), F # G. We can
Since m
F,
F C__ F.
So ,F
a
assume that
I
C_FI.
689
LINEAR ALGEBRAIC SEMIGROUPS
-i(@)=
Clearly
of
into
{0,i}.
0
Y ()
n
So F
C
I
ikFl
Then clearly
F()
151
+
Let
+
i.
I=I
+
+
denote the semilattice of all hmomorphisms
V()
(X(9),]). Let V ()
be a finite semilattice.
Y(9)\{I,O}.
V().
may or may not be a subsemilattice of
Y (Y())
Then
is a semilattice
V (V()).
PROOF.
Clearly
Define
e(a)(1)
e:
V(F(I))
/
e(m)(0)
i,
as
0.
We can assue that fl (2_ fl x
0
of
0"
into
e()(f i)
Then
f’z e
F(a).
IF*(V())
So
II.
Hence
h.3.
Let
i’ 2
(X(2)
c_.).
Then
COROLLARY h.h.
Let
h.l,
COROLLARY
(X(I)
lattice.
c__)
Then
PROOF.
((i
F(F*()
Let
i
F (n)"
v(].) ,
.
a
f(a).
O.
Let
denote the i
th
e(8).
e(m)
projection
Then
So =8.
for all i.
is in-
By Lemma
2"
i
Again by Lemma
*()
is a semi-
h.2,
h.2,
(v())
is the maximal semilattice
If S is a finitely generated semigroup and if
n
e
be finite semilattice such that
Then by Lemma
v*(v(v(z)))
is a homomorphism.
We claim that
be a finite semilattice such that
V ().
image of S, then clearly
f.z
such that
f’()z fi (8)
V*(()).
e
Then
F*(F(Q)).
x fl
a,8 e
Let
e(g)(f i) for all i.
e(a) (f)
So e(m) e
Jective.
So
_
I(5)I
I-I(FI)I
Thus
a contradiction.
IX()I
{@, {}}. So
LEMMA h.2.
and
\G.
is a semilattice, then let
If
Then
C
G, G
Since
is finite and
(X(S), c_)
=
(X(n), c_).
By Theorem 3.5,
Lemma h.2, Corollaries h.3, h.h, we have.
THEOREM
4.5. (1)
Let
(L,V ,A)
be a finite lattice.
some rational polytope P if and only if
V (L, A)
Then L
X(P) for
is a semilattice and
is
.
690
S. PUTCHA
isomorphic to the maximal semilattice image of some finitely generated, commutative,
idempotent free, totally cancellative semigroup.
(2)
Let
be a finite semilattice.
Then
is the maximal semilattice image
of some finitely generated, commutaitve, idempotent free, totally cancellative
semigroup if and only if
(X(), c)
is isomorphic to
(X(P), c)
for some rational
polytope P.
If P is a polytope, call
Tarski
(see [4; p. 91]),
is solvable.
p.
92].
THEOREM 4.6.
4.5,
the face lattice of P.
By a theorem of
the enumeration problem for face lattices of polytopes
However, for
By Theorem
X(P),
rational polytopes the problem is not yet solved
[4;
we have,
The enumeration problem for face lattices of rational polytopes
is solvable if and only if the enumeration problem for maximal semilattice
images of finitely generated, commutative, idempotent-free totally cancellative
semigroups, is solvable.
REFERENCES
Linear algebraic groups, W. A. Benjamin, N.
i.
BOREL, A.
2.
CLIFFORD, A. H. and G. B. PRESTON
vol. l, Amer. Math.
So__s_c.
Y., 1969.
The algebraic theory of semigroups,
Providence, R. I., 1961.
3.
GRILLET, P. A. The free envelope of a finitely generated commutative semigroup, Trans. AMS 149(1970) 665-682.
4.
GRNBAUM,
Ltd.,
B.
Convex polytopes, Interscience Publishers, John Wiley & Sons
1967.
5.
HUMPHREYS, J. E.
6.
PUTCHA, M. S.
Linear algebraic groups, Springer-Verlag,
On linear algebraic semigroups, Trans.
(1980) 457-469.
Ame.r.
1975.
M_at.h. So__c. 259
7. PUTCHA, M.S. On linear algebraic semigroups II, Trans. Amer. Math. So___9_c. 25
(1980) 471-491.
8. PUTCHA, M. S. Semilattice decompositions of semigroups, Semigroups
(1973) 12-34.
.Forum
6