Neighbouringly normal archimedean ordered semigroups

Acta Mathematica Academiae Seientiarum Hungaricae Tomus 20 (1--2) , (1969), pp. 105~-110.

NEIGHBOURINGLY NORMAL ARCHIMEDEAN ORDERED SEMIGROUPS

By T. SAITO (Clayton)

KOWALSKI [5] proved the following

THeOReM. Let S be a nonperiodic archimedean ordered semigroup. Then there exists an o-homomorphism of S into the additive semigroup of positive numbers such that two distinct elements of S have the same image i f and only i f they form an anomalous pair.

The purpose of this note is to give a theorem (Theorem 1) which generalizes the Kowalski's Theorem such as includes a certain kind of periodic archimedean ordered semigroups.

By an ordered semigroup, we mean a semigroup S with a simple order which is compatible with the semigroup operation:

a ,b, c E S and a<=b imply ac<=be and ca<=eb.

An element a of an ordered semigroup S is called positive if a 2 >a and is called nonnegative if a z >-_ a. An element a of S is called positive (nonnegative) in the strict sense if ax >a and xa >a ( ax >-a and xa ~=a) for every x E S. An ordered semigroup Sis called positively (nonnegatively) ordered (in the strict sense), if every element of S is positive (nonnegative) (in the strict sense). The number of distinct powers of an element a of S is called the order of a. A nonnegatively ordered semigroup S is called archimedean if, for each pair of elements a, b of S, there exists a natural number n such that a <= b n.

REMagK. There are some slight differences between the above terminologies and those in FucI~s [2]. An ordered semigroup is a fully ordered semigroup in Fuchs' sense. An element is nonnegative in the strict sense when it is positive in Fuchs' sense. A nonnegatively ordered semigroup is archimedean when it is archimedean without identity in Fuchs' sense. We treat exclusively nonnegatively ordered semigroups, and so we say simply an archimedean ordered semigroup in the place of an archimedean nonnegatively ordered semigroup.

The importance of archimedean ordered semigroups was pointed out in our previous paper [6], from which now we give some lemmas.

LEMMA 1 ([6] Lemma 2. 2). An archimedean ordered semigroup is nonnegatively ordered in the strict sense.

LEMMA 2 ([6] Lemma 2. 3). For an archimedean ordered semigroup S, the following conditions are equivalent:

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(1) S contains an idempotent; (2) S has the greatest element; (3) S has the zero element; (4) Every element of S is an element of finite order; (5) S contains an element of finite order.

Moreover, under these conditions, an idempotent of S is the greatest element and also the zero element o f S.

LEMMA 3 ([6] Corollary 2.4). An archimedean ordered semigroup contains at most one idempotent.

I f an archimedean ordered semigroup S satisfies any one of the conditions in Lemma 2, then S is called a periodie archimedean ordered semigroup. Otherwise S is called a nonperiodie archimedean ordered semigroup.

Following ALIMOV [1], tWO distinct elements a, b of a nonnegatively ordered semigroup S are said to form an anomalous pair if a" < b "+ ~ and b " < a "+ 1 for every natural number n. Now we say that two distinct elements a, b of S form a neighbouring pair if they satisfy the following two conditions:

(1) I f Xk<=al # a 1+t and 1/k<n/m for some x E S and some natural numbers k , / , m, n, then x m <= b";

(2) I f xk<=bl#b l+~ and l / k < n / m for some x E S and some natural numbers k, l, m, n, then xm<=a ".

LEMMA 4. Let a and b be two distinct elements of an arehimedean ordered semigroup S. I f they form an anomalous pair, then they form a neighbouring pair. I f S is nonperiodic, then the converse holds also.

PROOF. First we suppose that a and b form an anomalous pair and that x k < a t # a l+ 1 and l / k< n/m. Then

xkm <=atm <btm+ l <=b kn and so xm <b n.

We can similarly prove that, if xk<:bt#b ~+1 and l / k<n/m, then x;~<a ". Next we suppose that S is nonperiodic and that a and b form a neighbouring pair. Then, putting x, k, l, m, n by a, 2n, 2n, 2n, 2n + 1, respectively, in (1) of the definition of a neighbouring pair, we have aZ"<=b 2"+1, and also b 2"+1 < b 2n+2, since S is nonperiodic. Therefore a" < b "+ i. We can prove b" < a "+ ~ in a similar way.

An archimedean ordered semigroup S is called neighbouringly normal or n-normal, if, for the nonidempotent element a of S, the set of natural numbers k for which there exist x E S and a natural number l such that x k <: a t # a t+ 1 is unbounded above.

LEMMA 5. The above definition of neighbouring normality is determined irrespective of the choice of an element a in an arehimedean ordered semigroup S. S is n-normal i f and only i f either

(1) S is nonperiodic, or (2) S is periodic and, for every a E S and every natural number k, there exists

an element x E S such that x ~ <= a. The condition (2) can be replaced by (2') S is periodic and, for every a E S, there exists an element x E S such that

X 2 ~ a .

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NEIGHBOURINGLY NORMAL ARCHIMEDEAN ORDERED SEMIGROUPS 107

PROOF. First we suppose that S is an n-normalperiodic archimedean ordered semigroup and that a is the element satisfying the condition in the definition of the neighbouring normality. We denote by e the greatest element of S. We take b E S with b < e and a natural number k arbitrarily. Since S is archimedean, there exists a natural number m such that b m =e . Also by the definition of n-normality, there exist x E S and natural numbers 1 and n such that x" =< a*< e and mk <-n. Then we have

xmk<=x"~=aZ<e=b " and so xk<b.

Thus we have proved that, if S is n-normal, then S satisfies (1) or (2). I t is easily verified that, if S satisfies (1) or (2), then S is n-normal. This proves the second assertion, f rom which the first assertion is trivial. (2) implies (2") trivially. Finally we assume the condition (2"). We take a E S and a natural number k arbitrarily. Choosing a natural number n such that k ~= 2", we can find, by (2"), a sequence of elements x l , ..., x, such that

xl z -<= a, xz 2 ~ x l , �9 .., xZ, = x , - 1 . <

Then we have x, k ~ x 2" <= a. Thus (2) holds. A nonnegatively ordered semigroup S is called naturallj~ ordered, if a, b E S

and a < b imply the existence of c, dE S such that ae=b and da=b.

L~MA 6. An arehimedean naturally ordered semigroup S is either cyclic (that is, S is generated by a single element) or n-normal.

PROOF. By Lemma 5, we suppose that S is periodic. First we suppose that S has the least element a. For every x E S, there exists a natural number n such that a"<=x~=a "+l. I f S were not cyclic, then, for some x E S , we have a " < x < a "+~ and so there exists c E S such that a"e=x. Hence we have a"+l<=a"e=x-<a "+~, which is a contradiction. Next we suppose that S has not the least element. We take an arbitrary element a E S. Then we can find an element x E S such that x-< a and so there exists y E S such that x y = a . Putting z = m i n {x, y}, we have zZ<=a and so, by Lemma 5, S is n-normal.

LEMMA 7. Let S be an n-normal periodic archimedean ordered semigroup with the greatest element e, and let xP ~ y q, y~=x s, xP<e, yr <e for some x, y E S and some natural numbers p, q, r, s. Then r/q ~=s/p.

PROOF. We choose the natural numbers h and k arbitrarily. Since S is n-normal, we can find u, y E S such that u~<=x and v~<=y. We put z = m i n { u , v}. Since S is archimedean and x < e , y < e , we have Z I ~ X < Z I+1, z m ~ y < z m+l for some l ~ h and m --~k. Hence

zpl ~ X p ~ yq ~ zq(m + 1), zVm ~ y~ ~ x S ~= zS(l + i ).

Since xP~e , we have zpt<e and so, since zpZ<=zq(m+l), we have pl~=q(m+l). Similarly rm<=s(l+l). Therefore pr /qs~( (m+l ) /m) ( ( l+ l ) / l ) . The numbers 1 and m become sufficiently large with h and k, and so, f rom the above inequality, we obtain pr/qs ~ 1 and r/q ~ sip.

Let S and T be two ordered semigroups. A mapping co of a subset S* of S into T is called a partial o-homomorphism, if it satisfies the following two conditions:

(1) for every s l , s2 E S* such that s is z E S* in S, we have co(sls2)=co(sl)co(s2); (2) for every s l , SzES* such that sl<=sz, we have co(si)~=o~(sz).

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THEOREM 1. Let S be an n-normal archimedean ordered semigroup and let S* be the set of non-idempotents o f S. (Thus, if S is nonperiodic, then S* = S, and, if S is periodic with the greatest element e, then S* = S\e.) Then there exists a partial o-homomorphism of S* into the additive semigroup P of positive real numbers such that two distinct elements o f S* have the same image if and only i f they form a neighbouring pair.

PROOF. In the case when S is nonperiodic, the notion of a neighbouring pair is equivalent to that of an anomalous pair. Thus the assertion is nothing but the Kowalski's Theorem. Hence, in the rest of the proof, we assume that S is periodic with the greatest element e. We fix an element a of S*. For b E S*, we define co(b) as the infimum of quotients Ilk of two natural numbers l and k such that x k_-< a and b <= x~ for some x E S*.

1 ~ For every bE S*, we have co(b)>0. In fact, for an arbitrarily chosen real number e >0 , there exist x E S* and na tu ra l

numbers k and l such that

x k <= a, b ~ x ~, co(b) <= l /k< co(b) + 5.

We put c = m i n {a, b}. Since S is archimedean, there exists a natural number n such that a <= c". Hence

X k-<a<zt'n c<=b<=x t, c - < e , x k < e .

Therefore, by Lemma 7, we have 1/n <= l / k< co(b) + e. Hence we have co(b) _--> 1/n > O. 2 ~ I f b, c, bc E S*, then co(bc) <~ co(b) + co(c). In fact, for an arbitrarily chosen real number 5>0 , there exist x, y E S* and

natural numbers k, l, m, n such that

x~<=a, b<=x t, ym<=a, c<=y n, co(b)<=I/k<co(b)+5, co(c)<~n/m<co(c)+5.

We take a natural number p such that lip < 5 and then, since S is n-normal, take z l , z2 of S such that v-< z 1 = x and z~<=y. We put z - - ra in {z 1, Za}. Since S is archimedean and x < e , there exists a natural number q such that zq<=x<z~ +~. Then, since ~p < ~p < " we have p <= q and also

Similarly, for some p <= r, we have

Z mr ~ a , c ~ z n(r + 1)

Therefore zmax{qk, mr} ~ a , b e ~ z l(q+ 1)+n(r+ 1).

Hence, by definition, we have

co(bc) <= (l(q + 1) + n(r + 1))/max {qk, mr} <= (l(q + 1)/qk) + (n(r + 1)~mr) =

= ((l/k) (1 + (l/q))) + ((n/m) (1 + (l/r))) < (co(b) + 5) (1 + 5) + (co(c) + e) (1 + 5).

Hence we have co(bc) <- co(b) + co(c). 3 ~ I f b, c, bc E S*, then co(b) + co(c) ~_ co(bc).

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NEIGHBOURINGLY NORMAL ARCHIMEDEAN ORDERED SEMIGROUPS 1.09

In fact, for an arbitrarily chosen real number ~ > 0 , there exist x E S* and natural numbers k and l such that

xk<=a, be<=x l, co(bc)<=l/k<co(bc)+e.

We take a natural number p such that lip < E. Then, since S is n-normal, there exists y 6 S* such that yP =< x. Since S is archimedean, there exist natural numbers p ' , q, r, s such that

yZ/<=x<yv'+1, yq<=b<yq+l, y,'<=e<y~+l, yS<=a<y~+ L

Then we have p =<p' and so, since yv <=yp" <=x<=x~,~a, we obtain p _-<-s. Now we get

xk <=a < yS+ *, yq+r.<=bc<=xZ and so, by Lemma 7,

(q + r)/(s + 1) ---<_ l / k< co(be) + e. By definition, we have

co(b)<=(q+ l)/s, co(c)<=(r + O/s and so

co(b) +co(c) <=(q+r+2)/s<=((q+r)/(s+ 1))(1 + (l/s)) + 2 / s <

< (co(bc) +0(1 + 0 +2~. Hence co(b) + co(c) ~ co(bc).

4 ~ I f b, e E S* and b <= e, then co(b) <= co(e). In fact, for an arbitrarily chosen real number e >0 , there exist x E S* and natural

numbers k and 1 such that

x~<-_a, c<=x~, co(e)<=l/k <co(e)+e.

Then, since b<=c<=x ~, we have co(b)<=l/k<co(c)+e, and so a~(b)<=(c). 5 ~ I f b, e E S* and b and c form a neighbouring pair, then co(b) = co(c). In fact, for a sufficiently large arbitrary natural number k, we take x E S* such

that xk<=b. Also we take a natural number k" such that xk'<=b<xk'+L Then k<=k '. Also we have

x1"<=b<e, 1/k '< 1 / ( k ' - 1),

and so, since b and e form a neighbouring pair, we have x k'- 1 ~ c. By 1~ ~

co(x), co(b), co(c) >0, co(b) <= (k" + 1)co(x),

and so (k" - 1)co(x) <= co(O,

co(c)/co(b) >=(k' - O/(k" + 5).

Hence co(c)>-co(b). We can similarly prove that co(b)~co(e). 6 ~ I f b, e E S* and co(b) = co(e), then b and e form a neighbouring pair. In fact, we suppose that xk<=bl<e and l /k<n/m. Then we have kco(x)<=

<=lco(b) and so co(x)/co(e) =co(x)/co(b) <= l / k< n/m.

I f c" = e, then x" -<= e" trivially. Next we suppose that e" < e. Then co(x) < (him)co(e) =

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110 T. SAIT() : NEIGHI~OIJRINGLY NORMAL ARCHIMEDEAN ORDERED SEMIGROUIaS

= (1~re)co(c"), and so, by the definition of e)(x), there exist y E S* and natural numbers p and q such that

y"<=a, x<= y q, og(x)<:q/p<(1/m)co (c").

By the way of contradiction, we suppose c n < x". Then we have yp <= a and c" < x" <=y~', and so, by definition, re(c")<: mq/p. Hence (1~re)co(c")<= q/p, which is a contradiction. Thus x m <: c". Similarly, if x k <: d < e and l / k < n/m, then x" <: b".

This completes the p roof of the theorem. Finally we give an example which shows that Theorem I does not hold in general

without the assumption of n-normality.

S is a system with the multiplication EXAMPLE.

a

b

C

d

f g

h

e

and with the order relation

a b c d f g h e

c d f g h h e e

d f g h h e e e

f g h h e e e e

g h h e e e e e

h h e e e e e e

h e e e e e e e

e e e e e e e e

e e e e e e e e

a < b < c < d < f < g < h < e . Then it is verified that S is a periodic archimedean ordered semigroup with the maximal element e. But there is no partial o-homomorphism of S * = S\e into the additive semigroup of positive real numbers. In fact, by the way of contradiction, if co is a partial o-homomorphism, then, since a ~ = b z : f < e and a 4 = b 3 = h < e , we have 3e)(a)=2~o(b)

�9 and 4co(a)= 3co(b) at the same time, which is absurd.

"Received 23 October 1967)

DEPARTMENT OF MATHEMATICS, MONASH UNIVERSITY, CLAYTON, VICTORIA, AUSTRALIA

References

[1] N. G. ALtMOV, On ordered semigroups, lzv. Akad. Nauk SSSR, Ser. Mat., 14 (1950), pp. 569-- 576 (Russian).

[2] L. FtJcHs, Partially ordered algebraic systems (Pergamon Press, 1963). [3] L. FucHs, Note on fully ordered semigroups, Acta Math. Acad. Sci, Hung., 12 (1961); pp. 255--

259. [4] J, GABOWTS, Ordered semigroups, Interuniv. Sci. Sympos. General Algebra (Tartu. Gos. Univ.

1966), pp. 13--31 (Russian). [5] O. KOWALSKI, On archimedean positively fully ordered semigroups, Ann, Univ. Seient. Budapest,

Sectio Math., 8 (1965), pp. 97 99. ~[6] T. SAIT6, The archimedean property in an ordered semigroup, to appear in d. Austral Math. Soc.

Acre lvlathematica Academlae Sclentittrum Hungariccte zo, z969

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Neighbouringly normal archimedean ordered semigroups