Proprietà delle Funzioni Simmetriche Elementari Nel Algebre di Boole e Nel Algebre deiLivelli. by Giulio AndreoliReview by: Ruggero FerroThe Journal of Symbolic Logic, Vol. 38, No. 1 (Mar., 1973), pp. 153-154Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271751 .

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REVIEWS 153

Hajnal, and Rado, Partition relations for cardinal numbers, Acta mathematica, vol. 16 (1965), pp. 93-196.

The authors conclude II with several problems concerning the classes E = K -* (K)2 f} and

Echo where Ea = {K: K-- (ac)2' W}. These classes of Erdds cardinals are of foundational signifi- cance. W. N. REINHARDT

L. RIEGER. On the consistency of the generalized continuum hypothesis. Rozprawy mate- matyczne no. 31. Pan'stwowe Wydawnictwo Naukowe, Warsaw 1963, 45 pp.

This paper gives a reformulation of Gddel's consistency proof for the generalized continuum hypothesis (G.C.H.) (VI 112). The changes are not very far-reaching; the eight fundamental operations which Gbdel used are changed slightly by removing the relativizing factor from F4 - F8; and some changes are made in the construction of L and the proof of the Godel axioms, the axiom of choice, and the G.C.H. in L. But the proofs are still very close in spirit to the Godel proofs; thus no attempt is made to give a proof as indicated in Gbdel's 1939 paper V 117, and nothing is proved beyond what is proved in his monograph VI 112.

Some insight is given into the meaning of V = L, in terms of the "progenitors" of a set (i.e., the arguments of the fundamental operation used to construct the set); and some of the proper- ties of the segments of the constructible hierarchy (in Godel's terms, FPa for regular cardinals a) are pointed out. F. R. DRAKE

VLAD BoIcEscu. Sur la representation des alg~bres de Lukasiewicz n-valentes. Comptes rendus hebdomadaires des seances de l'Acadimie des Sciences, s6r. A t. 270 (1970), p. 4-7.

L'auteur consider les algbbres L de Lukasiewicz n-valentes (n > 3), dans la terminologie de Moisil (XIII 50) (voir 'a ce propos Cignoli, Moisil algebras, Notas de l6gica matemAtica, no 27, Bahia Blanca 1970).

Une algebre de Lukasiewicz n-valente (n entier, n > 2) est d'apres Moisil, un system (L, 1, O, r', U, N, al , a2 ,, a,-i) former par un ensemble non vide L, deux elements 0, 1 E L,

deux operations binaires r, u d6finies sur L et n operations unaires N, a, , a2, - - - , an-1

appeles negation et endomorphismes chrysippiens, respectivement, tels que: (I) (L, 1, 0, r, u) soit un reticule distributif avec premier (0) et dernier (1) 6l6ments. (II) N(x n y) = Nx U Ny; N(xuy) = NxrnNy;NNx = x. (III) aiO = 0;ail = 1;a1(x Uy) = aix Uaiy; aj(x ( y) = aix r aiy; aix U Nax= 1; aix r Naix = 0, pour tout i = 1, 2, *-* *, n- 1; ajajX = ajx; aiNx = Nan- x; ax _ a2X _*--_ a-jx, pour tout x eL; si ajx = ajy pour tout i = 1, 2, *-* *, n-1 alors x = y.

Pour chaque i tel que 1 _ i _ [n/2], Boicescu considere la relation binaire xRiy definie par les deux conditions aix = ajy et an-ix = an,-y. Il montre que: (P1) Ri est compatible avec ri, us, N, at , et an- i, et que (P2) l'algebre quotient Li = L/Rj est une algebre de Lukasiewicz trivalente, donc il en est de meme pour P = L1 x ... x Lrn/21. Il cherche alors a definir sur P des operateurs a', 1 _ i _ n - 1, de facon A obtenir ainsi une algebre L' de Lukasiewicz n-valente. Mais les operateurs ai ne sont pas bien d6finis. En effet soit L = {0, a, b, 1} l'algebre de Luka- siewicz tetravalente, ofi 0 < a < b < 1. Les relations R1 et R2 sont donnees par les partitions de L: R1 = {{O}, {a, b}, {1}}, R2 = {{O, a}, {b, 1}}. Soit p = ({a, b}, {O, a}) E P, alors

a30(R2) = {O, a} et a3a(R2) = {b, 1}

(ot A(R2) indique la classe d'equivalence qui contient l'element x, par rapport a la partition R2) donc a3 n'est pas compatible avec R2 et (d'apres la definition de l'auteur) a3' n'est pas bien definie. Dans une chatne avec cinq elements on trouve la meme difficulte, donc les propositions 4 et 5 de Boicescu ne sont pas valables. Luiz MONTEIRO

GIULMo ANDREOLI. Propriety delle funzioni simmetriche elementary nelle algebre di Boole e nelle algebre dei livelli. La ricerca (Naples), vol. 10 ser. 2 (Oct.-Dec. 1959), pp. 1-10.

In this paper the author considers the elementary symmetric functions (e.s.f.) on a Boolean algebra: a= a, + a2 + *- - + an; a2 = aja2 + ala3 + - - - + a2a3 + * - - + an-,an; a3 = a1a2a3 + *- + an-2an1an;. *.*-; an = a1a2a3 * *an. For these we have the following theo- rems: (1) The e.s.f. constitute a descending chain of inclusions, a, 2 a2 2 a3 2... 2 an . (2) If the a's form a chain of inclusions then the a's form the same chain, i.e., if ai1 O ai2 v

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154 REVIEWS

a13 ... *- an then a, = ail, a2 = aj2 ,.* , an = ain . It follows that if the A's are the e.s.f. obtained replacing the a's for the a's then af = aj . (3) If the a's can be decomposed into one or

more chains of inclusions, then the a's can be decomposed similarly; i.e., if a, D a2 D ... ** am

and a,+, l a,+2 * * a,+, and v = min(m, s), v + r = max(m, s), then a, - a, + a,+l, a2 D a2 + a,+2, D*Dav D av + a+V ,av+i D ao , - * Dav+, T ap+; where p=v + I if m > s,

p = r + v +l1 if s > m. (4) Since a*(a) = a,,-r(a*), if r of the a's equal I then ar = o2=

* * * = a, = 1 and vice versa. The dual is also true. (5) A generalization of (4). If r of the a's are

a and s of the a's are a* then a,, a2, * * *, ar D a and an-S+l , aUns+ 2, ' ** * an C a* and , * * , ap = 1; an -+ I, * * *, an = 0 where p = min(r, s) and X- = max(r, s). (6) Fundamental theorem. Given two sets of values al, *. , a'; a' , * * *, a'; if a' ' a' (v = 1,* , r), a'+j, a'+ a'

Xi=1,* * * , n -r) then av av.

Next the author considers an algebra in which each atom can have h levels, where a set is a totality in which a level is associated to each atom. The sum (product) of two such sets is the set in which each atom is at the higher (lower) level of those at which it appears in the given sets. The e.s.f. can be introduced also in this algebra, and the following results hold: (1) (a) In a, each atom takes the highest level; (b) in a2 each atom takes the second highest level; (c) in a3

each atom takes the third highest level; and so on. (2) The a's can be considered as a system of n-ary commutative and associative operations. These are also mutually distributive, i.e., if c, = a,(a, b, c,.. , 1) and a = ap(a, , *, * , A) then a,(ap(a, fi, y,* , A), b, c, * * *, 1) =

,al(ar(as b, c, * * * 1), qr(,83 b, C, *3 * 1), * * * 3, a,(A, b, C, * * * , 1)). RUGGERO FERRO

DIETER KLAUA. Konstruktive Analysis. Mathematische Forschungsberichte no. 11, VEB Deutscher Verlag der Wissenschaften, Berlin 1961, VIII + 160 pp.

The three chapters of this book give a systematic presentation of the theory of recursiveness in the domains of the integers, the rationals, and the reals, respectively. As classical logic is used throughout, the book sheds no light on the philosophical problem of the foundations of mathe- matics. Like XXXVI 148(3), it is a mathematical study of a particular subsystem of classical analysis, which may be of importance in computer science. This is in contrast to the constructiv- ist view of the foundational problem as developed by Markov (e.g. XXXI 258) and ganin. This book may be useful as a reference for detailed proofs of elementary results in recursive analysis, but it gives a misleading impression of its likeness to classical analysis.

In the chapter on recursiveness in the domain of the integers, the recursive functions and func- tionals are defined, and various standard results are derived for later use. The next chapter con- sists of three sections. The first gives three definitions of a recursive sequence of rationals and shows they are equivalent. The second gives three definitions of a recursive function with rational values and arguments. It proves that they are equivalent and that the defined class includes many rational functions and relations that would naturally be considered recursive. The third section gives three definitions of a recursive functional with rational values and rational functions as arguments. It shows that they are equivalent and give rise to a class of functionals that is closed under such operations as composition. The author does not fail to develop the analogous results for rational relations and relationals.

We are now fully prepared for the final chapter on the recursive real numbers, which takes up more than half the book. It begins with a proof of the equivalence of seven definitions, given by constructivizing various classical definitions of the real numbers. The closure of the recursive reals under the arithmetic operations and the theory of sequences present no difficulty. How- ever, Cantor's diagonal argument gives something new (p. 90): Every recursive sequence of reals omits some recursive real number. For the definition of the recursive functions on the real numbers we use the rational functionals of the previous chapter. The class formed by these functions includes x + y, x - y, x * y, x . y, xn, maxlx, y], and enjoys various closure proper- ties. Next comes a section on continuity in which the most important classical theorems are shown to hold for recursive real functions. Constructive proofs of special cases are also given, e.g. Weierstrass's theorem when we have a positive lower bound on the distance between the roots of the function (p. 115). This is followed by a short section on differentiation and inte- gration that culminates in a recursive version of the fundamental theorem for calculus (p. 140). In the final section the theory of recursive series of reals is developed. Its most important

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