Definition: a Semiring S Is Said to Be Semi-Isomorphic to the Semiring The

118 MA THEMA TICS: S. BO URNE PROC. N. A. S.

that the dodecahedra are too small to accommodate the chlorine molecules. Moreover, the tetrakaidecahedra are barely long enough to hold the chlorine molecules along the polar axis; the cos2 a weighting of the density of the spherical shells representing the centers of the chlorine atoms.thus appears to be justified. On the basis of this structure for chlorine hydrate, the empirical formula is 6Cl2. 46H20, or C12. 72/3H20. This is in fair agreement with the gener- ally accepted formula C12. 8H20, for which Harris7 has recently provided further support. For molecules slightly smaller than chlorine, which could occupy the dodecahedra also, the predicted formula is M * 53/4H20.

* Contribution No. 1652. 1 Davy, H., Phil. Trans. Roy. Soc., 101, 155 (1811). 2 Faraday, M., Quart. J. Sci., 15, 71 (1823). 3 Fiat Review of German Science, Vol. 26, Part IV. 4 v. Stackelberg, M., Gotzen, O., Pietuchovsky, J., Witscher, O., Fruhbuss, H., and Meinhold, W., Fortschr. Mineral., 26, 122 (1947); cf. Chem. Abs., 44, 9846 (1950). 6 Claussen, W. F., J. Chem. Phys., 19, 259, 662 (1951). 6 v. Stackelberg, M., and Muller, H. R., Ibid., 19, 1319 (1951). 7 In June, 1951, a brief description of the present structure was communicated by letter to Prof. W. H. Rodebush and Dr. W. F. Claussen. The structure was then in- dependently constructed by Dr. Claussen, who has published a note on it (J. Chem. Phys., 19, 1425 (1951)). Dr. Claussen has kindly informed us that the structure has also been discovered by H. R. Muller and M. v. Stackelberg, 8 Harris, I., Nature, 151, 309 (1943).

ON THE HOMOMORPHISM THEOREM FOR SEMIRINGS By SAMUEL BOURNE DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF CONNECTICUT, STORRS, CONNECTICUT Communicated by F. D. Murnaghan, December 10, 1951

In a recent paper,1 we have given a statement of the homomorphism theorem for semirings,2 which is in part incorrect. We have noted this in these PROCEEDINGS, 37, 461 (1951). At present, it is our purpose to give and prove a corrected statement of this theorem. Definition: A semiring S is said to be semi-isomorphic to the semiring S', if S is homomorphic to S' and the kernel of this homomorphism is (0). THEOREM. If the semiring S is homomorphic to the semiring 5', then the difference semiring S - I is semi-isomorphic to S', where I is the ideal of elements mapped onto O'. Downloaded by guest on September 24, 2021 VOL. 38, 1952 MATHEMATICS: BARRATTANDPAECHTER1 119

Proof: The correspondence s s', where s' is the image of an element of the coset s, is a semi-isomorphism of S = S - I into S'. The correspondence is certainly a homomorphism, for if s = (sI, s2, ...), then s, + il =s2+i2,i,andi2in l,and(s,+il)' = si' = (s2+i2)' = S2'. s+ t s' + t' and st -- s't' follow from coset addition and multiplication. Ifs -O', then s, + il = s2 + i2,iiand i2 inI, and sl' = s2' = O'. Thus s, and s2 are both contained in I. This implies that the kernel of the homomorphism s s' is 0 and this homomorphism is a semi-isomorphism. Previously,' we have proved the converse of the above theorem. Hence, we may state the homomorphism theorem for semirings as THEOREM: If I is an ideal of S, then S is homomorphic to the difference semiring S - I. Conversely, if the semiring S is homomorphic to the semiring S', then the difference semiring S - I is semi-isomorphic to S', where I is the ideal of elements mapped onto 0'. 1 Bourne, S., "The Jacobson Radical of a Semiring," PROC. NATL. ACAD. Scr., 37, 163 (1951). 2 Vandiver, H. S., "Note on a Simple Type of Algebra in Which the Cancellation Law of Addition Does Not Hold," Bull. Am. Math. Soc., 40, 920 (1934).

A NOTE ONTr(Vn m) By M. G. BARRATT AND G. F. PAECHTER MAGDALEN COLLEGE, OXFORD Communicated by S. Lefschetz, November 28, 1951 Introduction.-Let k ) 3. We shall prove THEOREM 1.1. 7rk+3(Sk) has an element of order four. Let Vk+m, m be the Stiefel Manifold of all orthogonal m-frames in real Euclidean (k + m)-space. THEOREM 1.2. The groups 7rk+2( Vk+m, m) are given by the following table, in which Zp, Z,, are cyclic groups of order p, co, respectively.

r2k. m m = 1 m = 2 m = 3 m 2 4 k 1 0 2X, Z. +Z.. Z. k =4s-2 Z2 Z2 +22 Z2 0 k =4s Z2 Z2 + 22 Z2 + 22 Z2 + 22 k =4s-1 22 Z4 Z2 +2. 22 k =4s + 1 Z2 24 Z4 +2Z 28 Let yn+1 be the (n - 1)-fold suspension of the real projective plane, so that yn+1 consists of an n-sphere Sn and an (n + 1)-cell en+l attached to Sn by a map of degree 2. We prove THEOREM 1.3 7rn+2(Yn+l) = Z4ifn 3 Downloaded by guest on September 24, 2021