Actas de las X jornadas by Antonio Monteiro; Oscar VarsavskyThe Journal of Symbolic Logic, Vol. 29, No. 1 (Mar., 1964), pp. 54-55Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2269786 .

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54 REVIEWS

ANTONIO MONTEIRO and OSCAR VARSAVSKY. A Igebras de Heyting monddicas. Actas de las X jornadas, Uni6n Matematica Argentina, Instituto de Matema6ticas, Universidad Nacional del Sur, Bahia Blanca 1957, pp. 52-62.

This paper deals with a generalization of the monadic Boolean algebras of Halmos (XXIII 219) suggested by intuitionistic logic. The paper consists mostly of definitions of various concepts pertaining to these generalized algebras and of a discussion of the complexity of the representation problem for these algebras.

A Heyting algebra (hereafter called an algebra) (H, A, v, -?, 0) where H is a set, o e H, and A, v, -* are binary operations on H is defined by means of six axioms. The pseudo-complement -a of a e H is defined to be a -*0, in particular 1 = -0. The concepts of homomorphism and filter are as in the theory of Boolean algebras. The radical RH of H is defined to be the intersection of all ultrafilters of H; it consists of all dense (--a = 1) elements a of H. The quotient algebra H/RH is a Boolean algebra and is isomorphic to the sub-algebra of regular (--a = a) elements a of H. The algebra H is semisimple iff (i.e., if and only if )RH = {1}. The only simple H is the Boolean algebra {0, 1}.

A Heyting monadic algebra (hereafter called a monadic algebra) is a sequence (H. A, v, -A, O. V, A) such that (H, A, v, -A-, 0) is an algebra and V and A are unary operations on H satisfying a list of eleven axioms. The operators V and A play re- spectively the roles of associated existential and universal quantifiers. In general we do not have A -V- nor A(a v Ab) Aa v Ab as is the case in a Boolean monadic algebra. Examples of monadic algebras are: (1) The functional monadic algebra B' where B is a complete algebra and I a non-empty set, in which V and A are defined by extrema as in the Boolean case; (2) the algebra of all open sets in a topological space, in which the quantifiers are defined by means of an open equivalence relation on that space; (3) the monadic algebra arising from a monadic intuitionistic calculus.

Two questions are raised: (i) Is every monadic algebra isomorphic to a subalgebra of a functional algebra? (ii) Is every monadic algebra isomorphic to a topological algebra as in example (2)?

It is stated in a footnote that the first author has shown that the answer to (ii) is positive. It is asserted that the answer to (i) is unknown. Two facts are mentioned which illustrate the complexity of problem (i): (a) If B and I are finite, the functional alge- bras B' need not be isomorphic to a functionally rich functional algebra; (b) there exists an algebra of three elements which is representable only by means of an infinite B and an infinite I.

The author introduces the concept of the monadic radical RmH and that of the free radical RlH for a monadic algebra H. Monadic algebras need not be semi-simple; the only simple monadic algebra is {O, 1}.

Correction: page 53, line 12, replace the two occurrences of "-a" by "--a AUBERT DAIGNEAULT

ANTONIO MONTEIRO. Normalidad en las dlgebras de Heyting monddicas. Ibid., pp. 50-51.

We use the notation and terminology of the preceding review. Let A = B' be a functional monadic algebra, I containing at least two elements. A filter J of B' is said to be free if Vf = 1 for all / c J. Contrary to what happens in the Boolean case (XXIV 252), a quotient A/J when J is maximal free, need not be isomorphic to B. The following result is announced.

THEOREM. In order for the algebra A/J to be isomorphic to B for any maximal free filter J of A, it is necessary and sufficient that B satisfies the following condition of normality: for all a, b e B there are a', Y' e B such that a' A b' 0 and a v Y bva' = 1.

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REVIEWS 55

Mention is made of the fact that if B is the algebra of all open sets of a topological space X, then B satisfies the above normality condition iff X is normal.

Correction: page 50, line 5, in the displayed formulas replace "X" by "x". AUBERT DAIGNEAULT

GONZALO ZUBIETA R. Clases aritmdticas definidas sin igualdad. Boletin de la Sociedad Matemdtica Mexicana, second series, vol. 2 (1957), pp. 45-53.

This paper is connected with XXI 405. (Not too closely in the terminology, however; "semejante," for example, is not Tarski's "similar.")

<r,, . . .r, n> being an n-tuple of positive integers, a structure of species <ri, . n> is a relational system W- <Ko, Wl . 9 .....i n>. Given two structures of same species, W and A, g is a similarity transformation of W onto Z, and we write W g Z, if

(1) g is a mapping of to onto Zo; (2) for every i, 1 < i < n, and every <cl . Cr> E tort, <cl. CrC> E Wi if

and only if <g(ci), . . ., g(cr,)> E Zi - The structures W and Z are similar (a Q Z) if there exist a structure E and similarity

transformations g and h such that W g C and Z 'h E. We write

C( = {e: W - $I

and, K being a class of structures of same species,

C(K) = U{C(W): e K}.

Let F be a fragment of the functional calculus of first order with identity and Xl, X2, ... formulas of F. M(XI, X2, ...) will be the class of structures that are models of each of the formulas Xl, X2, A class K of structures is axiomatizable in F if there exists a set {Xl, X2, .. .} of formulas of F such that K =M(X,

X2, ...). F* is the part of F obtained when all formulas containing the sign = are omitted. The principal result of the paper can now be stated:

K is axiomatizable in F* if and only if K is axiomatizable in F and C(K) C K. The proof of this theorem requires a number of lemmas that have interest in them-

selves. Let us now write two more definitions: K is finitely axiomatizable in a calculus if there exists a finite set X , Xn of

formulas of the calculus such that K = M(X1, ... , Xn). K is u-axiomatizable in a calculus if there exist closed universal formulas X1, X2,

of the calculus such that K = M(X1, X2, ...).

We then have the theorems: K is finitely axiomatizable in F* if and only if K is finitely axiomatizable in F and

C(K) C K. K is u-axiomatizable in F* if and only if K is u-axiomatizable in F and C(K) c K. Let S(W) be the class of the substructures of W. We write

S(K) = U{S(W) : e K, Sco(K) ={W WE S(K), to finite},

and, for n 2 2,

Sn(K) = {W E 9 e S(K), cardinal number of 9to < n}.

Then we have: K is u-axiomatizable in F if and only if (1) S(K) C K, (2) I(K) a K,

(3) for every W, S(v(W) C K implies W E K, where I(K) is the class of the structures that are isomorphic to structures of K, and

K is finitely u-axiomatizable in F if and only if (1) S(K) c K, (2) I(K) C K, (3) there exists an n such that, for every W, Sn(M) c K implies W E K.

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