Dedekind cuts of Archimedean complete ordered abelian groups

Algebra univers. 37 (1997) 223–234 0002–5240/97/020223–12 $ 1.50+ 0.20/0 © Birkhaüser Verlag, Basel, 1997

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 m a \ b and n b \ 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

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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]), Ho¨lder ([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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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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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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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 n a B 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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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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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).

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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%


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 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

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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

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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 ]. REFERENCES [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10]

[11] [12]

[13] [14] [15] [16] [17]

BAER, R., Zur Topologie der Gruppen, J. Reine Angew. Math. 160 (1929), 208 – 226. BAER, R., Dichte Archimedizita¨t und Starrheit geordneter Ko¨rper, Math. Ann. 188 (1970), 165 – 205. ¨ ber die Ver6ollsta¨ndigung geordneter Gruppen, Math. Nachr. 16 (1957), 51 – 71. BANASCHEWSKI, B., U BROUWER, L. E. J., On the Foundations of Mathematics, L. E. J. Brouwer Collected Works I: Philosophy and Foundations of Mathematics, A. Heyting, Ed., North-Holland Publishing Company, Amsterdam, 1975, pp. 13–126. (Translation of: O6er De Grondslagen Der Wiskunde, Maas & Van Suchtelen, Amsterdam, 1907.) CLIFFORD, A. H., Note on Hahn’s theorem on ordered Abelian groups, Proc. Amer. Math. Soc. 5 (1954), 860–863. COHEN, L. W. and GOFFMAN, C., A theory of transfinite con6ergence, Trans. Amer. Math. Soc. 66 (1949), 65–74. COHEN, L. W. and GOFFMAN, C., The topology of ordered Abelian groups, Trans. Amer. Math. Soc. 67 (1949), 310–319. COHEN, L. W. and GOFFMAN, C., On completeness in the sense of Archimedes, Amer. J. Math. 72 (1950), 747–751. CONRAD, P., Embedding theorems for Abelian groups with 6aluations, Amer. J. Math. 75 (1953), 1–29. ENRIQUES, F., Prinzipien der Geometrie, Encyklopa¨die der Mathematischen Wissenschaften III, Erster Teil, Erste Ha¨lfte: Geometrie, Verlag und Druck Von B.G. Teubner, Leipzig, 1907, pp. 1–129. EHRLICH, P., General introduction, in Real Numbers, Generalizations of the Reals, and Theories of Continua (Edited by P. Ehrlich), Kluwer Academic Publishers. Dordrecht, Holland, 1994. ¨ ber die nichtarchimedischen Gro¨ssensysteme and the Development of the EHRLICH, P., Hahn’s U Modern Theory of Magnitudes and Numbers to Measure Them, in From Dedekind to Go¨del: Essays on the De6elopment of the Foundations of Mathematics, edited by Jaakko Hintikka, Kluwer Academic Publishers, Dordrecht, Holland, 1996, pp. 165 – 213. EHRLICH, P., From Completeness to Archimedean Completeness: An Essay in the Foundations of Euclidean Geometry, Synthese 110 (1997). FUCHS, L., Partially Ordered Algebraic Systems. Pergamon Press, 1963. GOFFMAN, C., Completeness of the real numbers, Math. Mag. January – February (1974), 1 – 8. GRAVETT, K., Ordered Abelian groups, Quart. J. Math. Oxford 7 (1956), 57 – 63. ¨ ber die nichtarchimedischen Gro¨ssensysteme, Sitzungsberichte der Kaiserlichen Akademie HAHN, H., U der Wissenschaften, Wien, Mathematisch – Naturwissenschaftliche Klasse 116 (Abteilung IIa) (1907), 601–655.

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