Internat. J. Math. & Math. Sci.
VOL. ii NO.
(1988)81-86
81
THE SEMIGROUP OF NONEMPTY FINITE
SUBSETS OF RATIONALS
REUBEN SPAKE
Department of 4athematics
University of California
Davis, California 95616
(Received December 8, 1986)
ABSTRACT.
Let
Q
additive semigroup of all nonempty finite subsets of
<X
to be the basis of
greatest
min(X)>
,
In this paper we
X.
N
if and only if
AX
Q.
M(X)
e
denote
Ay
AX
define
In thc
the archimedeun
and determine
examine the structure of
M(X)
non-singleton, then the idempotent-free
be the
X>.
<max(X)
the basis of
let
X
For
In particular, we show that for
and B
By. Furthermore, if
X
its greatest semilattice decomposition.
(Y)
BX
and
decomposition of
semilattice
component containing
(X)
,
be the additive group of rational numbers and let
X,Y e N,
X
is a
is isomorphic to the direct product of
Q.
a power joined subsemigroup and the group
KEY WORD AND PHRASES. Greatest semilattice decomposition, archimedean components.
19gO AMS SUBJECT CLASSIFICATION CODES. 20M14, IOLIO, 05A99.
I.
INTRODUCTION.
In [I] we determined the structure of the semigroup of nonempty finite subsets of
In this paper we extend the results of [i] for the semigroup of nonempty
integers.
finite subsets of rationals.
In particular, we give a complete description of its
We also propose an isomorphism problem. It is
lattice
decomposition.
greatest semi
assumed the reader is familiar with the basic notions on commutative semigroups and
greatest semilattice decompositions- otherwise refer to Clifford and Preston
[2]
and
[3]. We begin with some notation and several definitions.
Let Q be the group o rational numbers, Z the group of integers, and define
Petrich
.
to be the semigroup consisting of a11 nonempty finite subsets of
Q
with the
opera t on
A + B
A singleton element of
The semigroup
N
{a
N
+ b
a e
A, b e B}
will be identified with the rational number it contains.
is a commutative countable semigroup with identity element
{al/b a2/b2,...,an/bn}
N,
b
X
1,
are relatively prime integers,
(if
as
x/l).
max(X)
Let
Define
A, B e N
min(X)
al/b I,
e
X
where
al/b <---< an/bn
contains an integer
an/bn,
and let
x,
if(X)
O.
and each
then express
a.,1
x
denote the least
R. SPAKE
82
b I, b
(positive) common multiple of the integers
consists only of
gcd(X) to be the grea,est (non-negative) common
where gcd(O)
0 and g,t(X O {0})
gcd(X). Let
diviso," of
then define
integers,
X,
the integers in
,
a,b e Z
with
a
the set U, and for
m e
b.
For
Z+,
r e Q
U e
Z
e
Z+
be the
a
x
b}
<U>
denote the semigroup generated by
U},
and
let
define
U +---+ U, rU
mU
{x
[a,b]
set of positive integers and define the integer interval
if
X
If
bn.
2
{ru
u e
Zm
Z/<-m,m>.
,
denote the
let
In the greatest semilattice decomposition of
(X)
on the (lower)
Define the partial order
archimedean component containing X.
and n e
scmilattice as"
(X)
(Y) if and only if nX Y + W for some W e
(equivalently" U + V e (X) for some (all) U e (X) and V e M(Y)). Note that
Z+
(0)
co,,sists
evidently
(0) g Q. Moreover, since 0 is
(X) is idempotent-free if and only
We will show later
in Theorem 2.1 that there are in fact
clearly the only idempotent in
X
if
,
and
of all the singletons in
is a non-singleton.
inlinitely ny archimedean components in the greatest semilattice decomposition of
For
X e
AX
define
X>.
<max(X)
fd(X)
,
Also,
N
BX
and
X
elements, respectively (if
introduction with an example.
determine
AX and Bx.
First,
X
max(X)
Consequently,
AX
1/30
have at most
Ax
is a singleton then
Let
X
{-3/10,
(X)
30, so
the basis of
min(Ax\{O})
id(X)
x e X.
(Bx) for all
+ (Ax)id(X
(X-x)
(Ax)
BX
and
min(X)>
is a non-singleton define
Note that
min(Bx\{O}).
a non-singleton,
X
if
<X
to be the sis of
When
We
11/6, 2}.
{-9,-6,2,55,60}.
1/30
1/30
{0,3,33,64,69} and
min(X)
1/30
X
{0,5,36,66,69}.
{0, 1/10, 32/15} and BX
{0,3,64}
is
We close this
{0}).
-I/5, 4/5,
X
X
I+(Bx)fd(X
and
Bx
and
wish to
Thus
{0,5,36,69}
1/30
{0,1/6,6/5,23/10}.
2.
STRUCTURE OF
R.
In this section we
examine
by determining its greatest
the structure of
semilattice decomposition and describing the structure of its archimedean components.
The first result gives a necessary and sufficient condition for two elements of
to be in the same archimedean component.
THEOREM 2.1.
For X,Y e
,
M(X)
M(Y)
if and
onlK
if
Ay
Ax
and
By.
Bx
PROOF. Let X,Y e N and without loss of generality assume min(X)
O.
min(Y)
U and V be such that U
(X)X and V (Y)Y. Note that U and V are
finite sets of integers. Suppose A
By. Since min(X) min(Y)
X Ay and BX
Let
O,
[I],
this implies
(U)
(V)
(X)
(AX)
(Ay)
(Y).
and therefore it follows that
Conversely, suppose
(X)
(Y).
Hence
AU
(X)
(Y).
There exist
n, m e
AV
Z+
and
and
BU
S, T e
BV.
N
By
such
SEMIGROUP OF NONEMPTY FINITE SUBSETS OF RATIONALS
83
that
nX
Since necessarily
min(S)
Y + S and mY
O,
min(T)
Ay
AX _C
and likewise
Ay.
AX
<Ay>.
Similarly
_c Y _c Y
By
+ S
<Ax>
Consequently,
BX
X + T
evidently
_c
<Ax>
<Ay>
and by definition this implies
and this completes the proof.
_
<Ax>
and
By
C
<Bx>-
AX+Y
A
X
and
BX+Y
BX.
Using the above theorem we can determine when two archimedean components are
related with respect to the order on the semilattice.
THEOREI 2.2.
The
followin ar___e equivalent.
(i) (x) <_ (Y).
(ii)
(iii)
Ay
PROOF.
Suppose
d(X)
d(Y).
n(X
Since
Y
min(X))
U e
and
Z+
n e
such that
min(Y) + U.
O.
min(U)
Ay
Similarly
There exist
<Bx>.
By
Y
Suppose
min(Y) + U
Y
min(Y)
Y
Then
<Ax>
<Ay>
rain(Y)
<Ax>
(ii) holds.
next that assertion
and thus
AX
Hence
AX+
AX.
Y
X + Y e d(X)"
Since
d(X)
that is,
Ay
BX+Y
Likewise
and
By
<Ax>.
X + Y- min(X + Y)
BX.
Finally, if (iii) holds, then by Theorem 2.1
d(Y) and the proof
are finite sets,
it
is complete.
is relatively easy to determine when
{-I0/7,-8/7, 22/7, 33/7,
(X)
d(Y) using Theorem 2.2(ii). For example, let W
5}, X {1/7, 5/21, 29/21, 68/21, 23/7}, and Y {-15, -13, 8, 28, 30}. Then fi(W)
7, (X)
21, and (V)
I. Thus
1/7
W min(W)
{0,2,13,43,45}
{0,2,32,43,45}, max(W) W 1/7
1/21
X min(X)
(O,l,dO.6d,66},
(0,2,26,65,66), max(X) X 1/21
Y min(Y)
{0,2,23,d3,45}, and max(Y) Y {0,2,22,43,5}.
Hence, A
{0, 2/21, 65/21},
w {0, 2/7, 43/7}, A
B
Ay
X
B
{0, 1/21},
X
with
A(X)
d(W)
Next, for
do(X
is a
and
do(X
{0,2,3}.
X e
define
subsemigroup of
@ Q.
Therefore, it follows that
{0,2/7,13/7},
w
d(X)
d(W)
(Y)
and
expressed in the form
M(X)
By
{0,2,23},
d(X).
U + q,
Moreover,
do(X)
where
{Y
e
d(X)
min(Y)
In general, elements of
U e
do(X
we have the following
and
q e Q.
0}.
It
is clear that
d(X)
can be uniquely
Hence
it follows that
84
R. SPAKE
THEOREI 2.3.
non-singleton,
isomorphic
is
Sffo(X
.joined subsemiKroup
rational number.
U
which maps
m e
Z+
the direct product
to
and the rouo
qU.
to
U e
First, if
X
where
is a
o_f_f the idemootent-free power
Q.
X be a non-singleton with min(X)
We will first show that o(X)
Let
PROOF.
(X),
The idempotent-free archimedean component
do(X
0
and let
o(qX)
be a non-zero
q
under the isomorphism
U I, X
then there exist
N
e
and
n,
such that
nX
U + U
X + X
mU
and
Hence
qU + qU
n(qX)
qX + qX
m(qU)
qU e
giving
,
there exists
It suffices therefore
o(qX).
V e do(X
T e N and
e
s,
T
to show that
V + S and tV qX + T.
V qV I, S qS 1. and
X + T 1.
V + S and tV
o(X) and consequently o(X) o(qX)
In particular, do(X
O((X)X). Since (X)X
O((X)X) is power joined. Therefore o(X) is
V
o(qX).
o(qX)
Then there exist
e
qT
T
be such that
sX
Hence
V e
Let
V e
for each
such that
q(sX)
V 1, S 1, and
Let
qV 1.
V
such that
Z+
and
Then
1.
for each non-zero rational
q.
by
[I]
is a set of
integers,
power joined and this completes
the proof.
COROLLARY 2.i.
X e N,
For
(qX)
sff(X)
The following equivalence relation on
[2]
and
[3] for
more on the
X $ Y
However, observe that if
N
Hence
X Y
Therefore, in
N
the
is called the
X Y + U and Y
for some U, V e N.
X + V,
Y + U and Y
if and only if
X
if and only if
-class
of
X
X,Y e
N;
v
X
Y
N
(see
X + V
then
X
if and only if
Using Theorem 2.3 we immediately have
and
Y must
U and V
Y + q for some q e Q.
set of all rational translates of
the
is
X
N.
nX
mY
for some
X
Define an equivalence
by
on
on
that is, evidently
(i.e. elements of the form X + q, q e Q).
Let PO denote the least semilattice congruence on
relation
S-relation
-relation):
necessarily be of the same cardinality since
are singletons.
for each non-zero rational number q.
n, m e
Z+.
SEMIGROUP OF NONEMPTY FINITE SUBSETS OF RATIONALS
THEOREM 2.5.
X
PO
Y
if and only i__f
X0 r
X
YO
Y
4(0)
YI
MX U
Y-min(Y)
Q.
(X).
.
is
XO’ YO
for some
Next we look more deeply into the structure of
clear since
R
Th___e least semilattice congruence on
e
85
v $"
PO
That is,
4(0)
Y e M(X) if and only
C
where
BX U
The structure of
Using the above results, evidently
where
Y1
<MX>
_C
max(Y)-Y
and
Y2
is
if
<Bx>.
Y2
More precisely we have the following direct consequence of Theorem 3.2 from [I].
THEOREM 2.6.
where
e
<Bu>
x
X
(mod b)}
j
Let c
fd(U).
Then
Le___t
Y e (X)
A.
Define
for
{x
[O,a-1],
e
{min(Ai)
max
[O,a-1]}
e
and
V e R
where
d
max
and
n
O
gU,
(mod a)} and B.
x
[0, b-l],
X-min(X)
Z+
b
e
[O,b-1]}.
such that
Y-min(Y)
{min(Bi)
e
and
id{U)
a
{x
2 nO
n
integers
be such that
<Au>
e
j e
if and only if there exist
gY and for all
U
be a non-singleton and
gcd(ff(Ax)Ax)/(Ax).
g
a-1
nV
U
{x
e
A.
x
< c-a}
U [c-a+l, nmax(V) + b-d-l]
i=O
b-1
U U
{nmax(V)
x e B., x
x
< d-b}
i---O
<Au> n
<
(nmax(V)
BU>
Next we reproduce several definitions and facts from Tamura [4] that we will use
Let T be an additively denoted idempotent-free commutative
archimedean semigroup. For fixed b e T, define a congruence
on T by
in the next theorem.
Pb
x
Then
T/p
if and only if
y
Pb
G
b
b
standard element
nb + x
mb + y
for some
T
is a group called the structure group of
b.
Z+.
n,m e
determined by the
<
Also, define a compatible partial order
on
T
by
b
x
T
Then
D
TE,
< y if and only if
b
equivalently
T/Pb
x
nb + y
{TE},
tree without smallest element with respect to
for some
Z+
n e
TE
is a discrete
e Gb,
where each
<
discrete tree, with respect to
(a
b
<
is a lower semilattice such that for any
c
<
d
the set
{x
c
finite
Finally, define a relation
chain).
x q y
The relation
q
if and only if
nb + x
q
on
<
b
b
b
x
< d}
is a
b
T as follows"
nb + y
for some
is the least cancellative congruence on
T.
n e
Let
Z+.
Q+
denote the set
of positive rational numbers.
THEOREM 2.7.
g
gcd((A)A)/(A).
Let
A e
and
0
b_e a_ non-singleton with min(A)
determined by .the standard
o_f 40(A
The structure groul
R. SPAKE
86
element
e
A
where
max(A)/g.
m
max(X)/g m i(mod m)}
d0(A
respect
Zm,
__is
to
<
Furthermore,
Moreover,
A
PROOF.
Q
is
i=O
di
where
di
{X
is a discrete tree without smallest element with
the structure
Kro, ID
A
standard element
m-I
U
do(A
d(A}
of
determined b_v
the
Zm
$
This follows from
[1]
O(A)
since
O(1/gA).
Using Theorem 2.7 we have the immediate
PROPOSITION 2.8.
defined bv
relation
N
A
max(X)
h(X)
o__n
Let
do(A
be a non-singleton.
is the greatest cancellative homomorphism.
do(A
Q+
That is,
the
h
defined bv
X
Y
if and only i__[
is the least cancellative congruence.
X o Y
if and
onlz i_!f min(X)
the least cancellative congruence.
max(Y)
max(X)
Moreove[,
th__e relation
o
o__n
d(A)
defined
min(Y) and
mCY)
aCX)
is
The homomorphism
The semigroups
do(A)/N
and
d(A)/o
are
do(A
we
idempotent-free commutative archimedean cancellative semirouDs.
For
a description of the greatest cancellative homomorphic image of
direct the reader to
[1].
We close this report with
an open isomorphism problem.
Any partial solutions would be appreciated.
For
PROBLEM.
X,Y e N,
under what conditions will
d(Y)? See Theorem 5.5 of [5] for
d(X)
be isomorphic to
some related results and also recall Corollary 2.zl.
1.
The Semigroup of Nonempty Finite Subsets of Integers,
SPAKE, R.
Math. and Math. Sci.
(198), 605-61.
2.
CLIFFORD, A.H. and ON, G.B.
Math. Soc., 1961.
.
PETRICH, M.
3.
5.
TAMURA, T.
Nachr.
Internat.
The Algebraic Theory of Semigroups,
.
Amer.
Introduction t__o Semigroups, Merrill, 1973.
Construction of Trees and Commutative Archimedean Semigroups, Math.
3--6 (1968),
255-287.
SPAKE, R. Idempotent-free Archimedean Components of the Power Semigroup of the
Group of Integers I, Math. Japon. 3--1 (1986), 791-810.