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Dedekind cuts of Archimedean complete ordered abelian groups
P. EHRLICH1
0. Introduction
A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty
subsets of G where Y = G −X and every member of X precedes every member of Y.
A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has
a least member, but not both; if every Dedekind cut of G is a continuous cut, G is
said to be (Dedekind) continuous. The ordered abelian group R of real numbers is,
of course, up to isomorphism the unique (Dedekind) continuous ordered abelian
group. R is also up to isomorphism the unique Archimedean complete, Archimedean
ordered abelian group. The idea of an Archimedean complete ordered abelian
group was introduced by Hans Hahn [17] as a generalization of Hilbert’s [19, 20]
classical continuity condition which characterizes R as an Archimedean ordered
field which admits no proper extension to an Archimedean ordered field.
DEFINITION 1. An ordered abelian group G is said to be Archimedean
complete if it admits no proper Archimedean extension to an ordered abelian group,
that is, if there is no G%³ G such that for each a G%−{0} there are positive
integers m and n and some b G− {0} for which ma\ b and nb\ a.
Hahn [17; also see 5, 9, 14, 16] showed that the Archimedean complete ordered
abelian groups coincide to within isomorphism with the so-called Hahn groups (see
§3); and L. W. Cohen and Casper Goffman [8; also see 15] later provided an
alternative characterization of these distinguished structures (see §1). In §2 of this
paper we introduce the idea of a Dedekind cut (X, Y) of an ordered abelian group
that is B(X, Y)-continuous, where B(X, Y) is the breadth of (X, Y) in the sense of
Kijma and Nishi [29: Definition 1.1, p. 89] and, relying heavily upon the aforementioned theorem of Cohen and Goffman, prove the following generalization of the
classical relationship that exists between ordered abelian groups that are continuous
in the alternative classical senses of Hilbert and Dedekind:
Presented by M. Henriksen.
Received December 18, 1995; accepted in final form October 21, 1996.
1
Research supported in part by NSF (Scholars Award cSBR-9223839).
223
224
P. EHRLICH
ALGEBRA UNIVERS.
THEOREM 1. An ordered abelian group G is Archimedean complete if and only
if e6ery Dedekind cut (X, Y) of G is B(X, Y)-continuous.
In §1, which is historico-expository in nature, we begin by drawing attention to
the apparently little-known origin of a now familiar generalization of Dedekind
continuity which plays a crucial role in Cohen and Goffman’s analysis as well as in
our own. This is followed by a brief discussion of the Cohen–Goffman Theorem,
which, as we already noted, is employed in §2 in our proof of Theorem 1. Finally,
in §3, we establish three theorems which collectively constitute a constructive proof
that Dedekind cuts of Hahn groups are B(X, Y)-continuous and thereby add flesh
to the abstract content of Theorem 1. Being essentially a reformulation of their
classical theorem, Theorem 1 is respectfully dedicated to Cohen and Goffman.
1. Veronese continuity and the Cohen–Goffman Theorem
An ordered group G is said to be discrete if it contains a smallest positive
member. If G is nondiscrete, then G is densely ordered, that is, for all x, y G where
x By there is a z G such that xBz By.
DEFINITION 2. We will say that a Dedekind cut (X, Y) of an ordered group
G is a Veronese cut of G, if for each positive d G there are x X and y Y for
which y −x B d; if G is nondiscrete and every Veronese cut of G is a continuous cut
we will say that G is Veronese continuous.
Although these notions are well known to contemporary mathematicians under
a variety of other names (cf. [1], [2], [3], [7] p. 312, [15], [18], [23] p. 71, [24], [25],
[26], [27], [28], [29] p. 96 (Remark 2.9), [30], [31], [32], [36], [37], [38] p. 66, [39], [43],
[51] p. 219), we have chosen these appellations to draw attention to the apparently
little-known fact that both conceptions were introduced (in 1889!) by Giuseppe
Veronese [45, Princ. IV, p. 612], who made extensive use of them in his nonstandard
theory of rectilinear continua and in his pioneering work on non-Archimedean
geometry more generally ([45], [46], [47], [48] pp. 39–51, [49], also see [11] pp.
xvii – xxi).
During the decades bracketing the turn of the twentieth century, Veronese’s
continuity condition was discussed by numerous authors including Levi-Civita ([34],
[35] also see [33]), Hölder ([21] pp. 10–11, [22] p. 89), Schoenflies ([40] p. 205, [41]
p. 27, [42] pp. 58 – 64), Brouwer ([4] pp. 49–50), Vahlen [44], Vitali ([50] pp.
133 – 134), Enriques ([10] pp. 37–38), and Hahn ([17] p. 603), the latter of whom was
aware that for ordered abelian groups, Archimedean completeness implies, but is not
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Dedekind cuts of Archimedean complete ordered abelian groups
225
implied by, Veronese continuity ([17] p. 603, also see [7] p. 316 and [15] p. 7). The
precise relation between the two types of structures, however, was first revealed by
Cohen and Goffman in their characterization theorem for Archimedean complete
ordered abelian groups alluded to above. To state the relation we require the
following familiar notions.
A subset G% of an ordered set G is said to be a con6ex subset of G, if every
member of G that lies between some pair of members of G% is also a member of G%.
If G% and G are also ordered groups, then G% is said to be a con6ex subgroup of G.
For each convex subgroup G% of G, let G/G%= {x+ G%: x G} denote the ordered
factor group of G modulo G%.
COHEN – GOFFMAN THEOREM. An ordered abelian group G is Archimedean complete if and only if for e6ery proper con6ex subgroup G% of G the ordered
factor group G/G% is Veronese continuous.2
2. Further definitions and proof of Theorem 1
If (X, Y) is a Veronese cut in G, then G%= {0} is the largest convex subgroup of
G for which x + g% X for all x X and all g% G%. Extending this idea to arbitrary
Dedekind cuts of G leads to
DEFINITION 3. The breadth of a Dedekind cut (X, Y) of G, written B(X, Y),
is the largest convex subgroup G% of G for which x+ g% X for all x X and all
g% G%.
LEMMA 1. If G% is a proper con6ex subgroup of an ordered abelian group G,
then (X ={x G : ×g G% x 5g}, Y= G−X) is a Dedekind cut of G for which
B(X, Y) = G%.
If (X, Y) is a Dedekind cut of G and G% = B(X, Y), then for each positive
a G% − G, there is an x X such that x+a Q X; moreover, the breadth of a
Dedekind cut of a densely ordered abelian group is {0} if and only if the cut is a
Veronese cut. Accordingly, if {u+G%: u U¤G}¤G/G% is denoted by U+ G%, one
can readily prove
2
Cohen and Goffman employ the terminology ‘‘nondiscrete and topologically complete’’ in place of
‘‘Veronese continuous’’.
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P. EHRLICH
ALGEBRA UNIVERS.
LEMMA 2. If (X, Y) is a Dedekind cut of G and G/B(X, Y) is densely ordered,
then (X +B(X, Y), Y +B(X, Y)) is a Veronese cut in G/B(X, Y).
Every continuous cut (X, Y) of G is a Veronese cut; and, so, B(X, Y)= {0} if
(X, Y) is continuous. Moreover, it is evident that a Dedekind cut (X, Y) of G is
continuous if and only if (X +{0}, Y+ {0}) is a continuous cut in G= G/{0}.
Extrapolating these ideas to Dedekind cuts having arbitrary breadths leads to
DEFINITION 4. A Dedekind cut (X, Y) of G is B(X, Y)-continuous if
(X +B(X, Y), Y +B(X, Y)) is a continuous cut in G/B(X, Y).
Proof of Theorem 1. Let G be an ordered abelian group. In virtue of the
Cohen – Goffman Theorem, it suffices to prove the equivalence of the following two
propositions:
(a) Every Dedekind cut (X, Y) of G is B(X, Y)-continuous.
(b) For every proper convex subgroup G% of G the ordered factor group G/G%
is Veronese continuous.
Suppose (a) and let G% be a proper convex subgroup of G. By Lemma 1,
(X = {x G : ×g G% x 5g}, Y =G− X) is a Dedekind cut of G for which B(X, Y)=
G% and, so, by the hypothesis, (X+ G%, Y+G%) is a continuous cut in G/G%.
Accordingly, since every member of a discrete ordered group has an immediate
successor and an immediate predecessor, G/G% must be densely ordered. Now
suppose (C%, D%) is a Veronese cut in G/G% and let p: G“ G/G% be the mapping
defined by the condition p(g)= g+ G% for all g G. Then (C= p−1(C%), D=
p−1(D%)) is a Dedekind cut in G and (C+ G%, D+ G%)= (C%, D%). Moreover, it is not
difficult to see that B(C, D)= G%. Indeed, since the convex subgroups of G are
totally ordered by proper inclusion, if B(C, D)" G%, then B(C, D)¦ G% or G% ¦
B(C, D); but if B(C, D) ¦G%, then c+ g% D for some c C and some g% G% −
B(C, D), and so c+ G% D +G%, which is impossible since (C+ G%, D+ G%) is a
Dedekind cut in G/G%; on the other hand, if G%¦ B(C, D), there is a positive
x B(C, D) −G% such that (i) x+ G% is a positive member of G/G% and (ii) c+ x C
for all c C; but, in virtue of (ii), (c+ x)+ G% C+ G% for all c C, which together
with (i) contradicts the fact that (C +G%, D+ G%) is a Veronese cut in G/G%.
Therefore, since B(C, D) = G%, it follows from the hypothesis that (C+ G%, D+ G%)
is a continuous cut in G/G%. Consequently, G/G% is Veronese continuous. Now
suppose (b) and let (X, Y) be a Dedekind cut of G. Then G/B(X, Y) is densely
ordered and, hence by Lemma 2, (X +B(X, Y), Y+ B(X, Y)) is a Veronese cut in
G/B(X, Y). Therefore, by the Cohen–Goffman Theorem, (X+ B(X, Y), Y+
B(X, Y)) is a continuous cut in G/B(X, Y) and, so, (X, Y) is B(X, Y)-continuous.
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Dedekind cuts of Archimedean complete ordered abelian groups
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For a geometrical application of the above theorem, see [13].
3. Cuts of Hahn groups
As we mentioned in the Introduction, Hahn [17] showed that an ordered abelian
group is Archimedean complete if and only if it is isomorphic to a Hahn group. For
our purpose, it is convenient to define the latter structures as follows.
DEFINITION 5. Let R[G] be the set of all formal series of the form
% r a v ya
aBb
where b is an ordinal, {ya :aB b} is a descending sequence of elements of an
ordered set G, and ra R − {0} for each aB b where R is the ordered group of
reals. The unique such series for which b=0 – the empty series – is the 0 of the
Hahn group (with exponents in G) that arises by defining addition and order
according to the rules
% ay v y + % by v y = % (ay + by )v y,
yG
yG
yG
y G ay v y By G by v y if at the first y% G such that ay% " by% , ay% B by% , it being
understood that terms with zeros for coefficients are inserted and deleted as needed.
In the degenerate cases where G is empty or G is a singleton, R[G] is isomorphic
to {0} and R, respectively; in all other cases R[G] is non-Archimedean, as is evident
from the lexicographic ordering of R[G]. For the historical development of the
theory of Hahn groups, the reader may consult [12].
DEFINITION 6. x will be said to be a truncation of a B b ra v ya R[G] if and
only if x = a B s ra v ya for some s 5 b; if x is a truncation of a and x" a, then x
will be said to be a proper truncation of a.
Plainly, a truncation of a member of R[G] is itself a member of R[G], 0 is a
truncation of every member of R[G], and every member of R[G] is a truncation
of itself; moreover, if a B b ra v ya + a"0, then a B b ra v ya is a truncation of
a B b ra v ya +a if and only if a v ya for all aB b, where for any elements a and
b of an ordered (additive) group, a b if and only if naB b for all members
n of the set Z+ of positive integers. Using these elementary results together with the
definition of R[G], it is a simple matter to prove
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P. EHRLICH
ALGEBRA UNIVERS.
LEMMA 3. If G " ¥, then for each a B b ra v ya R[G]− {0},
!
z R[G]: % ra v ya is a truncation of % ra v ya + z
aBb
"
aBb
={z R[G]: z v ya for all aB b}= R[{x G: xBya for all aBb}]
={z R[G]: z v ys }= R[{x G: xB ys }]
if b= s +1.
To complete our preparations we require the following definitions and collateral
lemmas which combine the idea of a truncation of a member of R[G] with that of
a Dedekind cut of R[G].
DEFINITION 7. If (X, Y) is a Dedekind cut of R[G], then by T(X, Y) we
mean the set of all z such that z is a truncation of some x X and z is a truncation
of some y Y.
LEMMA 4. For each Dedekind cut (X, Y) of R[G], T(X, Y) is a nonempty set
that is closed under truncation and well ordered by the proper truncation relation.
Proof. Since the result holds vacuously when G=¥, we may suppose G" ¥.
Moreover, since X, Y "¥ and 0 is a truncation of every member of R[G],
T(X, Y) " ¥. Further note, since every truncation of a truncation of x R[G] is
itself a truncation of x, T(X, Y) is closed under truncation. In addition, by virtue
of Definition 6, if T(X, Y) is totally ordered by the proper truncation relation, the
ordering is a well-ordering. Therefore, since the proper truncation relation is
obviously transitive and irreflexive, to complete the proof it only remains to show:
whenever x and y are distinct members of T(X, Y), x is a proper truncation of y or
y is a proper truncation of x. However, if we suppose the contrary, there are
members x =a B r ra v ya +rr v yr, y=a B r ra v yr + r%r v y%r of T(X, Y) where rr ,
r%r "0 and either yr "y%r or rr " r%r . But then, in violation of the lexicographic
ordering of R[G], x and y are each truncations of at least one member of X and of
at least one member of Y.
DEFINITION 8. If (X, Y) is a Dedekind cut of R[G], then by C(X, Y) we
mean the member of R[G] of least length whose truncations include the members
of T(X, Y), whereby the length of a B b ra v ya R[G] we mean the ordinal b.
The existence of C(X, Y) follows from the definitions of R[G] and T(X, Y), and
the following result is a simple consequence of Definition 8 together with Lemma
4 and elementary properties of ordinals.
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Dedekind cuts of Archimedean complete ordered abelian groups
229
LEMMA 5. If (X, Y) is a Dedekind cut of R[G], then T(X, Y) is the set of all
proper truncations of C(X, Y) if the length of C(X, Y) is an infinite limit ordinal, and
T(X, Y) is the set of all truncations of C(X, Y) otherwise.
Since R[G] is densely ordered, a Dedekind cut (X, Y) of R[G] that is not a
continuous cut is a gap, i.e., a Dedekind cut in which X has no greatest member
and Y has no least member. Therefore, to provide a constructive proof that
(X +B(X, Y), Y+ B(X, Y)) is a continuous cut of R[G]/B(X, Y), it suffices to
establish the following three theorems, the proof of the first of which is straightforward.
THEOREM 2. If (X, Y) is a continuous cut of R[G], then B(X, Y)= {0} and, so,
the least upper bound of X +B(X, Y) in R[G]/B(X, Y) is the least upper bound of X
in R[G].
THEOREM 3. If (X, Y) is a gap of R[G] for which C(X, Y)= a B b ra v ya where
b is not an infinite limit ordinal, then Gb = {g G: gB ya for all aB b} (which equals
G if b = 0) contains at least two members and one of the following two cases obtain:
CASE I. For some yb Gb (other than the smallest member of Gb if such a
member exists) and some Dedekind cut (R%, R¦) of R,
A%=
!
% ra v ya +r%v yb +z: r% R% & z v yb
"
is a cofinal subset of X
aBb
and
B%=
!
% ra v ya +r¦v yb + z: r¦ R¦ & z v yb
"
is a coinitial subset of Y.3
aBb
In this case,
B(X, Y) =R[{g Gb : g Byb }]
and the least upper bound of X+ R[{g Gb : gB yb }] in R[G]/R[{g Gb : gB yb }] is
% ra v ya +rv yb +R[{g Gb : gB yb }],
aBb
3
X% is said to be a cofinal (coinitial) subset of an ordered set X if for every x X there is an x% X%
such that x% ]x (x%5 x).
230
P. EHRLICH
ALGEBRA UNIVERS.
where r is the least upper bound of R% in R.
CASE II. For some Dedekind cut (G%, G¦) of Gb , either
(i) A¦ =
!
B¦ =
!
and
"
% ra v ya +nv x%: n Z+ & x% G%
is a cofinal subset of X
aBb
or
(ii) A§ =
and
B§ =
"
1
% ra v ya + v x¦: n Z+ & x¦ G¦
n
aBb
!
!
is a coinitial subset of Y,
"
1
% ra v ya − v x¦: n Z+ & x¦ G¦
n
aBb
"
% ra v ya −nv x%: n Z+ & x% G%
is a cofinal subset of X
is a coinitial subset of Y,
aBb
depending upon whether a B b ra v ya is a member of X or a member of Y, respecti6ely; in either case,
B(X, Y) =R[G%]
and the least upper bound of X+ R[G%] in R[G]/R[G%] is
% ra v ya +R[G%].
aBb
Proof. Assume the hypothesis and let Ib = {x R[G]: a B b ra v ya is a truncation
of x}. Since Ib is a convex subset of R[G] for which Ib SX" ¥ and Ib SY" ¥,
there is a Dedekind cut (X%, Y%) of Ib such that X% is a cofinal subset of X and Y%
is a coinitial subset of Y. Now consider hypothesis
(*): for some yb Gb there are x X% and y Y% having truncations
% ra v ya +av yb and % ra v ya + bv yb, respectively, where a, b"0.
aBb
aBb
Vol. 37, 1997
Dedekind cuts of Archimedean complete ordered abelian groups
231
To complete the proof it suffices to show that CASE I holds when (*) holds and
CASE II holds otherwise.
First, suppose (*). Since a B b ra v ya = C(X, Y) and xB y, aB b. Thus, in virtue
of the lexicographic ordering of R[G], there is a Dedekind cut (R%, R¦) of R where
a R%, b R¦ and where A% is a cofinal subset of X% and B% is a coinitial subset of
Y%. By combining this with appeals to Lemma 3 and the definition of B(X, Y), it is
easy to see that B(X, Y) ={z R[G]: z v yb }= R[{g G: gB yb }]. Furthermore,
since A% is a cofinal subset of X% and, hence, of X, and B% is a coinitial subset of Y%
and, hence, of Y, it follows that {a B b ra v ya + r%v yb + B(X, Y): r% R%} is a cofinal
subset of X + B(X, Y) and {a B b ra v ya + r¦v yb + B(X, Y): r¦ R¦} is a coinitial
subset of Y +B(X, Y). Clearly then, since r is the least upper bound of R% in
R, a B b ra v ya +rv yb +B(X, Y) is the least upper bound of X+ B(X, Y) in
R[G]/B(X, Y). Moreover, B(X, Y)" {0}, for otherwise a B b ra v ya + rv yb would
be the least upper bound of X contrary to the assumption that (X, Y) is a gap.
Accordingly, {g G: g B yb } "¥ which is enough to show that Gb contains at least
two members at least one of which is smaller than yb and, thereby, complete the
proof of Case I.
Now suppose (*) fails to hold. Since (X, Y) is a gap, a B b ra v ya is not a least
upper bound of X. Accordingly, there are pairs x X%, y Y% where x\ a B b ra v ya
when a B b ra v ya X and y B a B b ra v ya when a B b ra v ya Y, and for each such
pair there are a, b R −{0} and s%, s¦ {x G: xB ya for each aB b} where s% Bs¦
and for which a B b ra v ya +av s% and a B b ra v ya + bv s¦ are truncations of x and
y, respectively. By now appealing to the lexicographical ordering of R[G], it is a
simple matter to show that there is a Dedekind cut (G%, G¦) of Gb where s% G%,
s¦ G¦ and either A¦ is a cofinal subset of X and B¦ is a coinitial subset of Y or A§
is a cofinal subset of X and B§ is a coinitial subset of Y, depending upon whether
a B b ra v ya is a member of X or a member of Y, respectively. By combining this
with an appeal to the definition of B(X, Y), it is easy to see that B(X, Y)= R[G%]
and that the least upper bound of X+ R[G%] in R[G]/R[G%] is a B b ra v ya + R[G%].
THEOREM 4. If (X, Y) is a gap of R[G] for which C(X, Y)= a B b ra v ya where
b is an infinite limit ordinal, then Gb = {g G: gB ya for all aB b}" ¥ and, in this
case,
B(X, Y)= R[Gb ]
and the least upper bound of X+ R[Gb ] in R[G]/R[Gb ] is
% ra v ya +R[Gb ].
aBb
232
P. EHRLICH
ALGEBRA UNIVERS.
Proof. Assume the hypothesis and, for each r5 b, let Ir = {x R[G]:
a B r ra v ya is a truncation of x}. Since, for each rB b, Ir is a convex subset of
R[G] for which Ir S X "¥ and Ir SY" ¥, for each rBb there is a Dedekind cut
(Xr , Yr ) of Ir such that Xr is a cofinal subset of X and Yr is a coinitial subset of
Y. Moreover, since C(X, Y)= a B b ra v ya Ib = -r B b Ir and Im ³ In whenever
m B n5 b, it follows that either (i) -r B b Xr = {a B b ra v ya + z: z v ya for all
a B b} is a cofinal subset of X and -r B b Yr = ¥ or (ii) -r B b Xr = ¥ and
-r B b Yr ={a B b ra v ya +z: z v ya for all aB b} is a coinitial subset of Y. In
either case, it is easy to see that B(X, Y)= {z R[G]: z v ya for each aBb}
which, by Lemma 3, equals R[Gb ], and that the least upper bound of X+ R[Gb ] in
R[G]/R[Gb ] is a B b ra v ya + R[Gb ].
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Department of Philosophy
Ohio Uni6ersity
Athens, OH 45701-2979
U.S.A.
e-mail: [email protected]
.