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Kn(w) Kn(an)(W + o(1)) n n - n ~~~~~+o(1). (14) Then relations (14), (10), and the continuity of 9Z(w) yield relation (4), which com- pletes the proof. Given an additive function f(m), define its strongly additive contraction by f(m) = E f(p). p/m One can easily prove THEOREM B. Let f(m) be an additive arithmetic function and f(m) its strongly ad- ditive contraction, and define AZ, B, as in equations (3). Then, assuming relation (5), if one of f(m) - A f(m) - An Bn Bn, has a continuous distribution function F(w), the other also has F(w) as its distribution function. Combining Theorems A and B yields THEOREM C. Relations (1), (5), and (7) imply relation (4). * This work was done under the sponsorship of the U.S. Office of Naval Research under Con- tract N6ori-201, T.O. No. 1. Reproduction in whole or in part is permitted for any purpose of the United States Government. 1 P. Erdos and M. Kac, "The Gaussian Law of Errors in the Theory of Additive Number Theo- retic Functions," Am. J. Math., 62, 738-742, 1940. 2 H. Halberstam, "On the Distribution of Additive Number Theoretic Functions. II," J. Lon- don Math. Soc., 31, 1-14, 1956.
THE REES-SUSCHKETVITSCH STRUCTURE THEOREM FOR COMPACT SIMPLE SEMIGROUPS BY A. D. WALLACE DEPARTMENT OF MATHEMATICS, TULANE UNIVERSITY OF LOUISIANA Communicated by S. Lefschetz, May 2, 1956 The structure theorem for completely simple semigroups was first proved by Suschkewitschl in a special case and later by Rees2 in the general case. Here a semigroup S is simple if S = SaS for each a e S. For our purpose it is enough to say that S is completely simple if also it contains both a minimal left and a minimal right ideal (see A. H. Clifford3). Our purpose is to outline a proof of the Rees-Suschkewitsch theorem when S is compact. The essential difficulty, of course, is that of selecting the various func- tions so that they are continuous. Our procedure is so formulated that if compact- ness is not available as a hypothesis, then it will suffice to check the continuity of the functions r and 0 (defined below) in order to verify the continuity of all the other functions involved. Except as noted, we use the language and symbolism of an earlier paper, and, in Downloaded by guest on September 30, 2021 VOL. 42, 1956 MATHEMATICS: A. D. WALLACE 431
particular, it is alwvays assumed that S is Hausdorff. If E is the set of idempotents of S and if e e E, then He will denote the maximal subgroup of S which contains e. Let H= U{HeIEEI, and note that He, n He, id 0 implies He, = He,. If x e H, let v(x) be the unit of the group which contains x and let 0(x) be the inverse of x in this group. LEMMA 1.5 If S is compact, then H is closed and tq and 0 are continuous. Let S' be a subsemigroup of S, and let X and Y be two Hausdorff spaces in which a multiplication is defined by xx' = x' for all x, x' E X and yy' = y for all y, y' E Y. Let (p: X X Y -i S' be a continuous function, and denote by [S', X, Y] the space S' X X X Y with the multiplication (t, x, y)(t', x', y') = (t
T X (PtE- n K XXp K is analytic, and, if S is compact, then r is continuous. The proofs of Lemmas 1-4 rely mainly on set-theoretic topology. On the other hand, the proof of Lemma 5 is largely computational and is quite lengthy. LEMMA 5. Assume that T and 0 are continuous, and let Lo E £, Ro e 61, and eo = T(Lo,Ro). Define a: St-Ebya(R) = r(Lo,R),/A: - Eby f(L) = r(L,Ro), and (p: £X a- HEfbysp(L,R) = (L)*a(R). Then#: [He, £, a]I K,definedby t(t, L, R) = a (R)t (3(L), maps [He., £, R1] topologically and isomorphically onto K. THEOREM. Let S be compact, let eo eE n K, and let £ and 61 be the spaces of minimal left and right ideals. The Rees product [He., 2, 6(R] is topologically isomorphic with K. We now give some applications. Let us note that if £ and (R are degenerate, then cardinal (E n K) = 1 and K is a group. If cardinal (H,,) = 1 (eo e E n K), then cardinal (He) = 1 for each e e E n K; if also cardinal (S) = 1, then K c E, and K Downloaded by guest on September 30, 2021 432 MATHEMATICS: A. D. WALLACE PROC. N. A. S.
has the multiplication gyy' = !I for each y, t/' e K. Ili this case each element of K is a right zero. COROLLARY 1 (see Faucett6). If ' is a continutitm and if K has a cutpoint, then each element of K is a left zero or each element of K is a right zero. Proof: If eo e E n K, then K is topologically He, X £ X (R. But no Cartesian product of nondegenerate continua has a cutpoint, and no compact connected group has a cutpoint. Hence, say, He, and 2 are degenerate, and the result follows from the preceding remarks. COROLLARY 2.7 Let Sbe compact, and letP = { xS = S} El. I e E n P. then P is topologically isomorphic with Heo X U, where U = E n P is the set of left units for S. Proof: This follows easily, since P is right simple, i.e., xP = P for each x E P. We say that S is a clan if it is compact and connected and has a unit. As is well known, no n-sphere is the Cartesian product of nondegenerate spaces. COROLLARY 3. If S is a clan whose minimal ideal is topologically an n-sphere, then n = 1 or 3. Proof: Let eo e E n K, so that two of the three sets He,, £, and 61 have cardinal one. If £ and (R have cardinal one, then He. is an n-sphere, and, by a well-known result of E. Cartan, we have n = 1 or 3. If H,0 is degenerate, then Ha(S) t H'(eoSeo)7 and He, = eoSeo, so that Ha(S) = 0. But also7 Ha(S) Cs H'(K), so that H'(K) = 0, contrary to the fact that K is an n-sphere. For suitable coefficient domains it is known how to construct the cohomology ring H(K) from the rings H(He,), H(5C), and H(61). If S is a continuum with left unit, then H(S) q H(K).7 In certain cases we know that H(S) A H(HeO) 0 H(£) 0) H(6R), and useful consequences may be adduced from this fact, particularly if the coefficient domain is the rational numbers or a finite field. Additional information concerning the structure of K can be obtained in case S is one of the simpler Euclidean forms. Suppose that S has the property that any map of S into a proper part of S has a fixed point. If e E E, then the function x -f exe is a retraction. If eSe = S, then S has a unit, and if S is a manifold, then S is a Lie group and K = S. If eSe 5 S, then K 5 S, so that, if we take e e K, it follows that He = {e} and hence K c E. It follows then that K is topologically isomor- phic with £ X 6R, and the multiplication is given by (L1, R,) * (L2, R2) = (L2, R1). The determination of K in this case overlaps with some recent results of P. E. Con- ner and Edwin Hewitt, as Professor Hewitt kindly informed me. Let us note finally that the measure-theoretic aspects of S are largely governed by the structure theorem. I am grateful to the National Science Foundation for supporting this work. 1 A. Suschkewitsch, "UIber die endlichen Gruppen," Math. Ann., 99, 30-50, 1928. 2 D. Rees, "On Semi-groups," Proc. Cambridge Phil. Soc., 36, 387-400, 1940. 3 A. H. Clifford, "Semigroups Containing Minimal Ideals," Am. J. Math., 70, 521-526, 1948. 4 A. D. Wallace, "The Structure of Topological Semigroups," Bull. Am. Math. Soc., 61, 95-112, 1955. 6 A. D. Wallace, "A Note on Mobs. II," Anais Aead. Brasil. Ci., 24, 329-334, 1952; 25, 335- 336, 1953. ' W. M. Faucett, "Topological Semigroups and Continua with Cutpoints," Proc. Amer. Math. Soc., 6, 748-756, 1955. 7 A. D. Wallace, "Cohomology, Dimension and Mobs," Summa Brasil. Math., 3, 43-54, 1953. Downloaded by guest on September 30, 2021