6. Linearly Ordered Semimodules Over Real Numbers

6.

L i n e a r l y O r d e r e d S e m i m o d u l e s O v e r Real N u m b e r s

Let (R,+,=,Z) denote an arbitrary subring of the linearly ordered field of real numbers with zero O E R and unity 1 E R . Let (H,*,z) be a linearly ordered commutative monoid. In this chapter (R+,+,.,l;O;H,*,() dered) semimodule.

As

will always be a (linearly or-

multiplication is commutative the semi-

module is commutative, too. Further (R+,+,z) is a d-monoid as for u,B E R + with (4.17)

a

5 B we know B

- a €

R+. From the proof of

we conclude that R+ is either equal to Z+ or R + is a

dense subset of IR+. In any case, z+c_R+ as { 0 , 1 1

c_

R+.

Therefore the properties of the d i s c r e t e semimodule (Z+ ,+,

,f;O;H,

*,z) play

an Important role.

We do not assume in the following that the semimodule considered I s ordered. Therefore the monotonicity conditions (cf. 5.10) may be invalid. Nevertheless, for discrete semimodules we find: (6.1)

e C a O a

and

a o b c e

f o r all a E Z Z + x f 0 ) and all a , b E H with e

(6.2)

a l e

*

a o a f B o a ,

and

C

a, b

C

e,

uob,Bob

--1

for all a , B E Z+ and all a,b E H with e 5 a , b 5 e l and

for all a E Z +

and all a , b E H . We remark that (6.2) and (6.3)

for all a , B E R + and all a,b E H are equivalent to (5.10). We will show how these properties are related and that they hold In all r a t i o n a l semimodules (R+ 91

g+).

96

Ordered Algebraic Stntctures

(6.4) Proposition L e t H be a l i n e a r l y o r d e r e d commutative monoid which i s a

semimodule o v e r R+.

*

a ( b

(11

If

a o a z a o b

VaER+

V a,bEH

then t h e following p r o p e r t i e s are s a t i s f i e d :

v

< a o a

(2)

e

(2')

a o a < e

(3)

e

(3')

a o a

(4)

a

5 B

9

a o a

5

(4.')

a

5 8

*

aOa

2

5

aER+L{O)

V a E H

with e C a,

V aER+\{O}

V aEH

with a < e ,

v

aER+

V aEH

with e

5

a,

V aER+

V a € H

with a

5

e,

Boa

V a,BER+

V aEH

with e

5

a,

Boa

V a,BER+

V aEH

with a

5 e.

a o a

5

e

Furthermore

(2),

( 3 ) and

as w e l l as

(4)

(Z'),

(3')

and

(4')

are

equivalent. Proof.

(1)

*

( 3 ) . L e t a = e . Then a

same way b = e s h o w s of

(21,

( 3 ) and

(4).

(1)

*

(3'1.

0

a = e leads t o

(3). In the

We p r o v e o n l y t h e e q u i v a l e n c e

The e q u i v a l e n c e o f

(Z'),

( 3 ' ) and

(4')

follows similarly. (2)

*

( 3 ) i s obvious.

(3)

*

( 2 ) . L e t a E R + L { O j and l e t e < a E H .

Now 1 5 n a

Suppose e = a O a .

f o r some n E N . T h e n

contrary t o e < a. (4)

*

( 3 ) . Let a = 0 i n

(3)

*

(4).

e

5 (B

- a)

This leads t o

(4).

Let a,BER+ with a 0

a. T h e r e f o r e

(a

0

5 B a)

*e

and l e t e

5

(a

0

a)

(3).

5

*

aEH. [ (8

-

a)

Then 0

a]. m

Linearly Ordered Semimodules over Real Numbers

99

(6.5) T h e o r e m Let H be a linearly ordered commutative monoid which is a semimodule over R + .

If R + C_ Q + then H is a linearly ordered semi-

module. Proof. Due tc proposition ( 6 . 4 )

it suffices to prove ( 6 . 4 . 1 ) .

Let a E R + and let a , b E H with a

1. b.

a

Ua

a =

5

a

If a = b or a = 0 then

O b . Now assume a > 0 and a < b. A s R + C _ , Q +

we find

n/m f o r some m , n E W with greatest common divisor 1. There-

fore there exist p , q E Z with p n + q m = 1. This implies that

Let a' = (l/m) O a and b' = (l/m) O a . Now a < b implies a' Cb'.. The latter yields (at)mn 5 (b')mn, i.e.

a O a f aob. rn

In the general case R + may contain irrational or, in particular, transcendental numbers. Then even in the case (H,*,z) rn

( m , +-, < )

the monotonicity conditions may be invalid. This is shown by the following example.

(6.6) S e m i m o d u l e c o n t a i n i n g t r a n s c e n d e n t a l n u m b e r Let a € (0,l) be a transcendental number and consider the subring R generated by

ZL

U{a).

Then the elements of R have the

form (cf. 4 . 2 0 ) r j 1 P, a j=o with p tion

j

0:

€ 2 3 , j = O,l,...,r.

R x l R +lR

by

Now we define an external composi-

100

Ordered Algebraic Structures

for all a E I R . Then lR is a module over R. B u t now

shows that ( 6 . 4 . 3 )

is invalid. Therefore the monotonicity con-

ditions (5.10) are invalid. On the other hand we will show that for an important class of monoids all monotonicity properties hold.

( 6 . 7 ) Proposition Let H be a d-monoid which is a semimodule over R+. Then ( 6 . 4 . 1 1 , (6.4.21,

(6.4.31,

and ( 6 . 4 . 4 )

and ( 6 . 4 . 4 ' ) .

(6.4.3'1,

Proof. Due to ( 6 . 4 ) (6.4.1).

are equivalent and imply ( 6 . 4 . 2 ' 1 ,

it suffices to show that (6.4.2) implies

Let a , b E H , u E R + and assume ( 6 4 . 2 ) .

If a = b or a = O

then a O a = a a b . Otherwise a * c = b for some c E H with c > e. Then e <

a

a c which implies a m a 5 (a a a

*

(a 0 c ) = a o b . m

Proposition ( 6 . 7 )

shows that a d-monoid is a linearly ordered

commutative semimodule over R + if it i s a semimodule over R + and the external composition restricted to R + x H + has only values in H+. Unfortunately, example ( 6 . 6 )

shows that even in

the case of the additive group of real numbers we can construct a

semimodule in which this condition is invalid.

101

Linearly Ordered Semimodules over Real Numbers

(6.8) T h e o r e m Let H be a positively and linearly ordered commutative monoid which is a semimodule over R+. Then and ( 6 . 4 . 4 )

hold,

(1)

(6.4.21,

(2)

if H is a d-monoid then H is a linearly ordered semi-

(6.4.3),

module over R + . Proof. A s H is positively ordered we know H = H+. Therefore (6.4.3)

is satisfied. Now ( 1 ) follows from ( 6 . 4 )

and ( 2 ) follows

from ( 6 . 7 ) .

In chapter 4 we discussed monoids which have an ordinal decomposition. If a monoid H is the ordinal sum of a family (HX'

hEh)

of linearly ordered commutative semigroups and if H is a semimodule over R + then this semimodule is called ordinal. The

i n d e x X(a) for a E H is defined with respect to the ordinal decomposition by A(a):=

p if a € Hu

.

(6.9) P r o p o s i t i o n Let H be an ordinal semimodule over R+. Then

for all a E R +

{ O } and all a - E H.

Proof. Let aER+\{O}

and let a E H . Then h ( ( n + a ) m a ) =max(X(a),

A(a0a)) for all nEIN. Therefore it is sufficient to consider a € ( o , ~ ) . NOW X ( a o a ) < max{A(aoa),X((I-a)

oa)} = X(a). On

the other hand let n €IN with 1 5 n a . Then X(a = max{X(a),X((na-l)

0

a) = A ( (na) 0 a)

oa)l 2 A(a). w

102

Ordered Algebraic Structures

Proposition (6.9) shows that the external composition decomfor h E A .

poses into external compositions on R+'{O}xHh-rHh

From (4.9) we know that H h is positively ordered with the possible exception of h = min A .

Therefore we find the

following result.

(6.10) P r o p o s i t i o n Let H be an ordinal semimodule over R+. Then H h U { e ) semimodule over R+ with (6.4.2),

(6.4.3).

is a

and (6.4.4) for all

X E A ~ m i nA .

We remark that if A = min A then H A is a semimodule over R +

but the monotonicity properties (6.4.2),

(6.4.3), and (6.4.4)

may be invalid.

A

d-monold H has an ordinal decomposition ( H A ; h E A ) .

positively ordered then H

=

{el for .A

= min A .

If H is

Otherwise

hO

the structure o f H i

is described in theorem (4.10). 0

( 6 . 1 1 ) Theorem Let H be a d-monoid which I s a semimodule over R+. If the over R+ satisfies the monotonicity rule (6.4.3)

semimodule H xO

then H is a linearly ordered semimodule over R+. Proof. Due to (6.7), (6.8), and (6.10) H ordered semimodule over R + for all h over R

. The

HA

is a semimodule

monotonicity rule (6.4.3) together with propo0

-

ho.

U {el Is a linearly

0

sition (6.7) shows that H A over R +

i

is a linearly ordered semimodule

103

Linearly Ordered Semimodules over Real Numbers

In ( 4 . 1 8 ) we proved that dom(a) and pos(a) for a E H are sub-d-monoids of the d-monoid H. A similar result holds with respect to the corresponding semimodules. For convenience we define R + o A : =

{ a ma1

aER+

,

a E A )

for

A

C_ H.

(6.12) Proposition Let H be a d-monoid which is a semimodule over R + . (1)

R,

12 f

H is a linearly ordered semimodule over R+ iff R

Proof. -

odom(a) = dom(a)

Then

+ Opos(a)

=

for all a E H ,

pos(a)

( 1 ) Let b E dom(a).

for all a € H.

Then bn E dom(a) for all n El".

fore it suffices to consider a E R + =

(aob)* (anal

*

n

(0,l). Now ( a

( ( 1 - a ) o a ) = ( a 0 (a*b))

*

0

b)

There-

*a

=

( ( 1 - a ) m a ) = a.

( 2 ) Assume that H is a linearly ordered semimodule over R+.

Again it suffices to consider a E R +

b E dom(a) then a

0

n

(0,1)

and bEpos(a1. If

b E dom(a) C_ pos(a) follows from ( 1 ) . Other-

wise e < b and therefore e < ( a o b ) as ( 6 . 4 . 2 ) Then a f a

*

is valid.

( a o b ) . For the reverse implication it suffices

to show that ( 6 . 4 . 3 )

holds in H over R+. For the special choice

of a = e we find R+UH+ = H+

In proposition ( 4 . 1 4 )

,

i.e.

(6.4.3).

cancellation rules and related proper-

ties in weakly cancellative d-monoids are stated. ( 4 . 1 9 . 1 ) implies the following cancellation rule.

Ordered Algebraic Structures

I04

(6.13) P r o p o s i t i o n Let H be a weakly cancellative d-monoid which is a linearly ordered semimodule over R+. If a O a

*

6 O a or X(a)

5 A(b)

then

*

(aoa) * b = (Boa)*c f o r all a , b , c E H and all a , 6 E R +

b = [ ( B - a ) ma1 * c

with a

2 6.

Proof. The case a = e is trivial. If a < e then the inverse element a

-1

> e exists. Composition with u o a

b = (Boa)* ( a o a plies b =

[ ( f ?-

-1

)

* c . Then 6 O a = “ 6

-1

- a ) Oa]

leads to

*

( a O a ) im-

a ) o a ] * c . N o w let e < a. At first we assume

a o a 8 8 m a . Then a O a <

tla and thus a < 6 .

> A(b). Then B o a > a o a = $ O a

*c

Suppose A(a)

leads to a contradic-

tion. Therefore A(a) 5 X(b). In the second case X(a)

1. A ( b )

is valid, too. Therefore in both cases A(u o a ) 5 X(b) and A(aoa) < A ( ( ( B - a ) o a ) *c).

(4.19.1)

implies b = [ ( B - a ) m a ] * c . rn

Weakly cancellative d-monoids which are semimodules over R + play an important role in the consideration of optimization problems in part 11. The ordi.na1 decomposition of such a monoid H may contain H A with IH

xI

=

1. Such t r i v i a l subsemigroups with

an idempotent element (a + e for X * A o )

lead to difficulties in

the formulation and validation of algorithms. We avoid these difficulties by a certain extension of the underlying semimodule. Let H be a weakly cancellative d-monoid with ordinal decomposition ( U x :

A E A ) and assume that H is a linearly ordered semi-

module over R

+.

Let H

P

=

{a] for some v E A . Then it is con-

venient to extend the trivial semigroup H

P

to

105

Linearly Ordered Semiwwdules over Real Numbers

in the case p > X (6.14.2)

= min h and to

~ [ a , r I)

N

H":=

in the case p = Xo.

~ E R )

Then let

(6.15)

(a,r)

(a,r'):=

for all (a,r), (a,r') E';i

(a, r + r'),

u and all bEH\{a).

The external com-

position is defined by if a = 0, (6.16)

a

0

( a , r ): =

i f a > 0.

(a, a r )

We identify a

t)

(a,l) if l~ > X N

*he order relation on H

v

and a

c--)

(a,O) if p = X

0

.

is the usual one with respect to the N

second component. In this way H

p

replaces H

p

in the ordinal

decomposition. If this is done for all trivial semigroups the new semimodule

is called e x t e n d e d . This is again a linearly

ordered semimodule over R+. We remark that therefore an extended semimodule is always linearly ordered by definition.

linear-

A

ly ordered semimodule without trivial semigroups in its ordinal N

decomposition is clearly extended. In particular, H is a weakly cancellative d-monoid. The ordinal decomposition of an extended semimodule H has w.1.o.g.

(cf. remarks after 4 . 1 3 )

the following form. H A

=: G I

0

0

is a nontrivial linearly ordered commutative group with neutral element e which is also the neutral element of H. For X > X o we find that H A is the strict positive cone of a nontrivial

Ordered Algebraic Structures

106

linearly ordered commutative group G A . We may identify the neutral element of G A with e. Then G The order relation on G tive cone H A u {el

A = {a-ll a E H A l U {el U H A .

is completely determined by the posi-

(cf. remarks after 2.7). Using (5.12) to

continue the external composition on R x G A

+

G A for A > A.

find that G A is a module over the ring R for all A € A .

we

We

remark that it is not useful to consider the ordinal sum of the groups

GA.

> 1 then such an ordinal sum is not an or-

If I h l

dered semigroup. Several cancellation properties in extended semimodules will be helpful in part 11.

(6.17) Proposition Let H be a weakly cancellative d-monoid the ordinal decomposition of which contains no trivial subsemigroups. Then

-

for all a , b E H . If H is an extended semimodule over R+ then (4 1

a z B

(5)

a ( B

c,

a a a ~ B o a , a o b z B o b ,

for all a , 6 E R + and all a , b E H with b < e < a and (6)

a z b

for all a E R + , then

c.

u o a z a o b

a > 0 and all a , b E H .

If a < 6 or A(a)

5 A(b)

107

Linearly Ordered Semimodules over Real Numbers

for all a , b , c E H and all a , B E R + with a Clearly, a < a

Proof. ( 1 )

*b

5 8.

implies e < b and X (a)

the reverse implication assume e < b and A ( a ) A(a)

5

A(b).

For

5 A(b). If

< X(b) then a < a * b follows from the definition of ordi-

nal sums. If X(a) = X ( b ) then a < a * b follows from ( 4 . 1 9 . 4 ) applied to an extended semimodule. (2)

and ( 3 ) are proved similarly.

( 4 ) As

(5.10)

implies a a

0

a =

B

0

is satisfied it suffices to prove that a n a

5 B if e

C

a. Suppose

5 B Ua

a. If a o a < B O a then a < 8 . Now assume

B <

Then [ ( a

a.

together with ( 2 ) imply ( a

-

- B)

0

a1

*

(B Da)

=

( B ma)

8 ) O a = e contrary to ( 6 . 4 . 2 )

which

holds in a linearly ordered semimodule. ( 5 ) and

( 6 ) are proved similarly.

(7) If X ( a ) 5 X(b) then (7) follows from ( 6 . 1 3 ) . Now assume h ( a ) > X(b) and a < 6 .

Then h(a) > X o and therefore a > e . In

an extended semimodule this implies a o a

C

B O a . Again

(6.13)

shows (7). 8

From ( 1 )

-

(3)

in proposition ( 6 . 1 7 ) we conclude the following

corollary which describes the sets corresponding to the different definitions of positivity and negativity considered in the discussion of d-monoids in chapter 4 .

(6.18)

Corollary

Let H be a weakly cancellative d-monoid the ordinal decomposition of which contains no trivial subsemigroups. (1)

The set H+ of all positive elements is equal to {a1 e

5 a);

strictly positive elements exist only if A has a maximum

Ordered Algebraic Structures

and then the set of all strictly positive elements is (a1 e < a , i(a) = max A } .

The set H- of all negative

elements is nonempty only if l A l = l ; then H-

=

{a1 a z e )

and the set of all strictly negative elements is equal to H - \ {el. The set P -

of all self-negative elements is equal to H'H+;

the only idempotent is e and the set { a / a < a * a ]

self-positive

of all

elements is equal to H + \{el.

Let a E H . The partition N(a) < dom(a) < P(a) of H in strictly negative, dominated and strictly positive elements with respect to a is given by {el <

~+xCel

and is given by N(a) = @ and < X(a)}

< {bl X ( b )

5

X(a))

we remark that any weakly cancellative d-monoid can be embedded into a weakly cancellative d-monoid the ordinal decomposition of which does not contain trivial subsemigroups (via the corresponding discrete semimodule). Finally we discuss the possibility that a given linearly ordered semimodule H over R + is contained in a linearly ordered semimodule H' of R I with H $ H'

sub-semimodule

of H' over R ' .

or

$ R',

R

i.e.

H over R + is a

At first we consider two examples

and show that the semimodules considered are not sub-semimodules of a semimodule with H = H' and R Let H = ZZ

*

C

R'.

be the additive monoid of the nonnegative integers.

109

Linearly Ordered Semimodules over Real Numbers

Then H is a linearly ordered semimodule over Z+ with usual multiplication as external composition. Suppose that H is a sub-semimodule of a linearly ordered semimodule H over R+ with Z+

5

R+. Then R+ is dense in IR+. F o r a E (0,l) we try to

define a o l E Z + . Monotonicity shows 0 < a n 1 fore a 0 1

5 1 and there-

1. For the special choice a € (0,1/21 we find the

=

contradiction 1 = (2a)ol = Z o ( a o 1 ) = 2 0 1 = 1 + 1 = 2 .

Therefore no such semimodule exists. Now let H = Q + be the additive monoid of nonnegative rationals. Then H is a linearly ordered semimodule over.Q+ with usual multiplication as external composition. Suppose that H is a subsemimodule of a linearly ordered semimodule H over IR+ Let B € I R + \ Q + .

Then a

5 B 01 5

y for

.

all a , y E Q + with a < B

C

y.

There exists no rational number B O 1 with this property. Therefore no such semimodule exists. In both examples the semimodules considered are subsemimodules over IR o f the linearly ordered semimodule (IR+,+,z)

with

+

respect to usual multiplication as external composition. Secondl y , the following example shows that a given linearly ordered

*

semimodule H over R+ with R C I R exists which is not a subsemimodule of any linearly ordered semimodule H' over R; and R +

5

with H

C

A'

R;.

Let H be the fundamental monoid ([O,l)],*,c-)

with respect to

the usual order relation and a *b:= min(a+b,l). ordered semimodule over Z

+

= min(za,l).

E is a linearly

with external composition z n a = 'a

Now we try to define a n 1 for a € (0,l).

Odered Algebraic Structures

I10 (a 0 1)

*

( a n 1 ) = a 0 (1

*

1 ) = a 0 1 shows that a 0 1 is an idem-

potent element of H'. Let 1

-<

(nu)

1 = (a

1)

n

=

a

1.

5 n a for some nElN. Then 1 In particular, let a E

(0,

=

1 0 1

1/21

and a E [ 1 / 2 , 11. Then a o l = a O ( a * a ) = a 0 ( 2 o a ) = ( 2 a ) o a

< -

loa = a < 1/2. This contradiction to 1

1. a 0 1

shows that no

such semimodule exists. Thirdly, let H be a submonoid of the linearly ordered commutative monoid H'. Then H and H' are linearly ordered semimodules over 22 +

and H is a sub-semimodule of H'. In general, if H i s

a linearly ordered semimodule over R + then it is not known

whether the external composition on R

0:

R+ x H

-D

H can be continued

x H ' such that H' is a linearly ordered semimodule over

R+.

Nevertheless, we make the following conjecture. Let H be a weakly cancellative d-monoid which is a linearly ordered semimodule over R+. Then this semimodule can be embedded in a linearly ordered semimodule H' over lR+ with weakly cancellative d-monoid H' 2 H. We remark that using the ordinal decomposition of such a semimodule and using the embedding result

of HAHN (cf. theorem 3.5) the cases R + C_ Q, and Q,

it i s possible to provide a proof for

5

R+

.