130 MATHEMATICS: P. R. HALMOS PRoc. N. A. S. Hence Rao*-a* = 2 (ga*g-^* + ga6*g,,*) (6) where K = (2/n)pY*; a*. Setting pass* = (K/2)ga,* (the solution of which is as- sured) in equation (6), the converse is immediate. In fact, - log g = gys R*a*. = ( 2) Kgas* = (n + 1)par*. COROLLARY 1. A manifold of "constant" holomorphic curvature can always be mapped projectively (with the geodesics preserved) on a space of the same type. COROLLARY 2. A Gsp-flat compact manifold is projectively flat. COROLLARY 3. A Gsp-flat manifold is conformal with some Kaehler (Einstein) space. Remark: The tensors G,,6*ya* and Ha{R,*,* in Bochner's paper2 are related: H,,8Y* = 1/2(G.,*7y* + Gya*aj,*). Hence they need not be treated separately. In fact, Ha4764*,* = 0 if and only if Gaj*76* = 0. Moreover, Theorem 6 in the same paper need only be stated for the tensor a, the statement for the tensor H being redundant. I S. Bochner, "Curvature in Hermitian Metric," Bull. Am. Math. Soc., 53, 179-195, 1947. 2 S. Bochner, "Curvature and Betti Numbers. II," Ann. Math., 50 (No. 1), 77-93, 1949. 3 S. I. Goldberg, "Tensorfields and Curvature in Hermitian Manifolds with Torsion," Ann. Math. (to appear). 4 J. Igusa, "On the Structure of a Certain Class of Kaehler Varieties," Am. J. Math., 76 (No. 3), 669-677, 1954. 6 H. Weyl, "Zur Infinitesimalgeometrie: Einordnung der projectiven und der konformen Auffassung," G6ttinger Nachr., pp. 99-112, 1921. PREDICATES, TERMS<, OPERATIONS, AND EQUALITY IN POLYADIC BOOLEAN ALGEBRAS 1BY PAUL R. HALMOS DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO Communicated by A. A. Albert, January 6, 1956 The theory of Boolean algebras is an algebraic counterpart of the logical theory of the propositional calculus; similarly, the theory of polyadic (Boolean) algebras' is an algebraic way of studying the first-order functional calculus. The Godel completeness theorem, for instance, can be formulated in algebraic language as a representation theorem for a large class of simple polyadic algebras, together with the statement that every polyadic algebra is semisimple.2 The next desideratum is an algebraic study of the celebrated Godel incompleteness theorem. Before that can be achieved, it is necessary to investigate the algebraic counterparts of some fundamental logical concepts (such as the ones mentioned in the title above). The purpose of this rather technical note is to report (without proofs) the results of such an investigation; the details are being published elsewhere.3 Downloaded by guest on October 5, 2021 VOL. 42, 1956 MATHEMATICS: P. R. HALMOS 131 1. Basic Concepts.-For convenience of reference, this section contains the defi- nitions of the (already known) basic concepts of algebraic logic; all further work is based on these concepts and on their elementary properties. The following notation will be used for every Boolean algebra A: the supremum of two elements p and q of A is p v q, the infimum of p and q is p A q, the comple- ment of p is p', the zero element of A is 0, and the unit element of A is 1. Some- times it is convenient to write p q instead of p' v q. The natural order relation is denoted by <, so that p < q means that p v q = q (or, equivalently, that p A q = p or that p -- q = 1). The simple Boolean algebra { 0, 1 is denoted by 0. A quantifier (more precisely, an existential quantifier) on a Boolean algebra A is a mapping 3 of A into itself such that (i) 30 = 0, (ii) p < 3p, and (iii) 3(p A 3q) = 3p A 3q, for all p and q in A. Suppose that A is a Boolean algebra, I is a set, S is a mapping that associates a Boolean endomorphism S(T) of A with every transforma- tion r from I into I, and 3 is a mapping that associates a quantifier 3(J) on A with every subset J of 1. The quadruple (A, I, S, 3) is a polyadic algebra if (i) 3(0) is the identity mapping on A (here 0 is the empty set); (ii) 3(J u K) = 3(J) 3(K) whenever J and K are subsets of I; (iii) S(5) is the identity mapping on A (here 5 is the identity transformation on I); (iv) S(aU) = S(u) S(r) whenever a and r are transformations on I; (v) S(a) 3(J) = S(r) 3(J) whenever ai = ri for all i in I - J; and (vi) 3(J) S(r) = S(-r) 3(r-1J) whenever T is a transformation that never maps two distinct elements of I onto the same element of the set J. If I is a set, X is a non-empty set, and B is a Boolean algebra, the set of all func- tions from the Cartesian product X' into B is a Boolean algebra with respect to the obvious pointwise operations. If 'r is a transformation on 1, and if x and y are in X', write r*x = y whenever y1 = xri for all i in I. (The value of a function x from I into X, i.e., of an element x of X', at an element i of I will always be denoted by Xi.) If p is a function from X' into B, write S(r)p for the function defined by (S(T)p) (x) = p(r*x). If J is a subset of I and if x and y are in X', write x J* y whenever xi = yi for all i in I - J. If p is a function from X' into B, and if the set {p(y): x J* y} has a supremum. in B for each x in X', write 3(J)p for the function whose value at x is that supremum. A functional polyadic algebra is a Boolean subalgebra A of the algebra of all functions from X' into B, such that S(r)p e A whenever p e A and r is a transformation on I, and such that 3(J)p exists and belongs to A whenever p e A and J is a subset of I. The set X is called the domain of this functional polyadic algebra. It is convenient to be slightly elliptical and, instead of saying that (A, I, S, 1) is a polyadic algebra, to say that A is a polyadic algebra, or, alternatively, to say that A is an I-algebra. An element of I is called a variable of the algebra A. The degree of A is the cardinal number of the set of its variables. An element p of A is independent of a subset J of I if 3(J)p = p; the set J is a support of p if p is independent of I - J. The algebra A is locally finite if each of its elements has a finite support. Some of the results that follow are true for arbitrary polyadic algebras, but most are not. To simplify the statements, it will be assumed through- out the sequel that (A, I, S, 3) is a fixed, locally finite polyadic algebra of infinite degree. Associated with every subset J of I there is a polyadic algebra (A-, I-, S-, 3-) that is said to be obtained from A by fixing the variables of J. The set A- is the Downloaded by guest on October 5, 2021 132 MATHEMATICS: P. R. HALMOS PROc. N. A. S. same as the set A, and I- = I - J. If -r is a transformation on I-, let T be its canonical extension to I (i.e., r is the extension of T- to I such that ri = i whenever i-e J), and write S-(T-)p = S(T)p for every p in A. If J- is a subset of I-, then J- is also a subset of I; write 3-(J-)p = 3(J-)p for every p in A. Suppose that c is a mapping that associates a Boolean endomorphism of A with every subset K of I; denote the value of c at K by S(K/c). The mapping c is a constant of A if (i) S(0/c) is the identity mapping on A; (ii) S(H U K/c) = S(H/c) S(K/c), (iii) S(H/c) 3(K) = 3(K) S(H - K/c), and (iv) 3(H) S(K/c) = S(K/c) 3(H - K), whenever H and K are subsets of I; and (v) S(K/c) S(T) = S(r) S(Tr-K/c) whenever K is a subset of I and T is a transformation on I. (The notation here is different from the one used before for constants; the innovation has a beneficial unifying effect.) 2. Predicates.-An n-place predicate of A (n = 1, 2, 3, . ) is a function P from I' into A such that if (il,. ., in) e In and if r is a transformation on I, then S(T) P(il, ** , in)i = P(Tri, . rin)r THEOREM 1. If P is an n-place predicate of A and if (i,...., in) E In, then {il. ... in} supports P(i1, . *X in) THEOREM 2. If p e A and ifjl, .. 'in are distinct variables such that {jl, ** nj supports p, then there exists a unique n-place predicate P of A such that P(j,...,j,. ) = p. COROLLARY. Every element of A is in the range of some predicate of A.