† In the sense of Waerden's, van derModerne Algebra, 2 vols. (Berlin, 1930–1931)CrossRefGoogle Scholar. A technical definition is given in section 2, which includes groups, rings, the synthetic algebras of modern writers, Boolean algebra, and the topology of Frechet (the operation being the limit point of a sequence).

‡ Cf. Klein's, F. “Abstrakte Verknüpfungen”, Math. Ann. 105 (1931), 308CrossRefGoogle Scholar, and his “Zerlegungssatz bei abstrakte Verknüpfungen”, Math. Ann. 106 (1932), 114. These will be cited in later references as K 1 and K 2. The following papers by Remak, R., Journal für Math. 162 (1930), 1–16Google Scholar; 163 (1930), 1–44; 164 (1931), 197–242; 165 (1931), 159–79; 166 (1932), 65–100; 167 (1932), 360–78, have been very suggestive to the author.

* Klein calls a finite lattice a “Verband” in K 2; a sublattice an “Unterverband”. He calls a finite C-lattice (which we shall define later) an “A-Menge” in K 1.

† Waerden, van der (Moderne Algebra, 1, p. 37Google Scholar) calls a system of double composition. He (Ibid. p. 60) inverts the symbols (,) and ∩. Our usage of (,) for common part is, however, more widely accepted.

* Theorem 6·1 and Corollary 6.4 are proved in K 2.

* By Theorem 4.2 and Corollary 6.4, as has been remarked before [cf. K 2], VI is equivalent to a pair of one-sided inequalities, and V to a single one. But formally, V and VI are simpler, so we use them. Klein calls a C-lattice an “A-menge” in K 1.

† Cf. a similar theorem by Quine, W. V.: Journal London Math. Soc. 8 (1933), 89.CrossRefGoogle Scholar

* For, by Theorem 16.3, it is the join of certain meets, and any meets are in ʂ_ρ−1 at which point we can use induction and

* This definition corresponds to the conventional one of direct product for groups, rings, and topological manifolds, and direct sum of linear algebras. The theorems of § 19 are related to the Zerlegungssatz of K 2.

† By we mean that there exists a (1, 1) correspondence between the elements and operators of and which preserves and the formation of elements by operations. This yields (1, 1) isomorphy in group and ring theory, and homeomorphy in topology.

* Strictly, we should show in detail that, if then the sublattices (C ₁, M) and (M, C ₂) correspond to the factors which are discussed in the preceding paragraph.

* Cf. Whitehead, and Russell, , Principia mathematica, vol. 1, p. 218.Google Scholar

* We are in the statement of Theorem 25.2 assuming that any class can be well-ordered. Cf. van der Waerden, , Moderne Algebra, 1, p. 194.Google Scholar

† In Theorems 26.1–26.2 there is no restriction on finiteness.

* “Über Untergruppen direkter Produkte von drei Faktoren”, Journal für Math. 166 (1932), 100.Google Scholar

* By the projection of two such linear manifolds, we mean the set of vectors linearly dependent upon them.