The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of new terms, and extend the meaning of some accepted ones.
† By a “sequence” we mean a “well-ordered set”. We can use the locutions “finite sequence” and “enumerated sequence” to express that the ordinal number of the set is finite, or that of the ordered positive integers.
‡ The first statement is known; the second will be proved in § 15.
† These facts were proved by the author in “On the combination of subalgebras”, Proc. Cambridge Phil. Soc. 29 (1933), 441–64: A ∩ B is the set of elements common to the sub-algebras A and B, A ∪ B is the meet of the subalgebras containing both A and B. In later citations, the above paper will be referred to for short as “Subalgebras”.
‡ Cf. van der Waerden's, B. L.Moderne Algebra, 1 (Berlin, 1930–1931), 28–32.
§ The notion of inclusion in an abstract lattice is naturally defined by writing a ⊂ b if and only if a ∪ b = b.
† Steinitz, E., Algebraische Theorie der Körper (Berlin, 1930), p. 143.
‡ Pontrjagin, L., “Theory of topological commutative groups”, Ann. of Math. 35 (1934), 361–88, Theorems 2 and 4. If X is continuous, we admit only closed subgroups. (5·1) was added in revision; the surprising thing is that it has not been explicitly stated before.
§ This representation dates back at least to Vogt, H., Résolution algébrique des équations (Paris, 1895), p. 91.
† Moore, E. H., “Tactical memoranda I–III”, Am. Jour. Math. 18 (1896), 264. The definition is too long to repeat.
‡ This definition includes the standard definitions of direct products of groups and topological manifolds, and of the direct sum of linear algebras (of hypercomplex numbers).
† By A ˜ B (A and B any algebras), we denote “A and B are isomorphic”.
‡ This definition generalizes a usage in group theory started by Remak, R., Journal für Math. 163 (1930), 6.
§ For a counter-example, cf. 6 of the author's paper “Group synthesis”, now in the hands of the editors of the Trans. Amer. Math. Soc.
† In case B is the set of equations G 2–G 4 of § 3 on algebras of species (2, 1), we have the so-called “free” groups. Another connection with standard usage is made by calling F (B, m) the “calculus” on m symbols defined by the laws of B.
† Gesammelte Werke, 1, p. 121. Cf. also a paper by O. Ore on modular lattices, for which he prefers the term Dedekind structures, Annals of Math. 36 (1935), 406–37.
† “Subalgebras”, Theorems 26·1, 27·1, and (essentially) §28. On the other hand, the closed (normal) subgroups of the translations of the line do not satisfy L 5. To show this, let r and s be any two incommensurables, and take the subgroups generated by the translations x → x + r, x → x + s, and x → x + 2r.
‡ Birkhoff, G., “Combinatorial relations in projective geometries”, Annals of Math. 36 (1935), 743–8.
§ “Hausdorffgroupoids”, Annals of Math. 35 (1934), 360.
† Hausdorff, F., Mengenlehre (Berlin, 1927), Chapter v.
‡ “Subalgebras”, Theorem 25·2. Other instances of distributive lattices are listed in Klein's, F. “Einige distributive Systeme in Mathematik und Logik”, Jahr. d. D.M.V. 38 (1929), 35–40.
§ Proved by Stone, M. H., “Boolean algebras and their application to topology”, Proc. U.S.A. Acad. 20 (1934), 197–202. The family of Boolean algebras is hence generated by any Boolean algebra containing more than one element. It is a corollary that any family of equations between functions of species (2, 2, 1) which contains an equation not derivable from L 2–L 7 contains the equation x = y. Cf. Lukasiewicz, J., “Ein Vollständigkeitsbeweis des zweiwertigen Aussagenkalkuls”, Comptes Rendus de Varsovie, 24 (1932), 153.
∥ Added in revision.
† Fund. Math. 24 (1935), 177–98.
‡ Or equivalently satisfying one of the chain of equations between functions of species (2, 1):
(14·1) is a consequence of a result in Burnside's Theory of groups of finite order, 2nd ed. (Cambridge, 1911), p. 163; (14·2) follows from Speiser's Gruppentheorie, 2nd ed., Theorem 80.
† Hasse, , Höhere Algebra, 1 (1927), 17. What Hasse (and I) call an “equivalence relation”, Carnap calls an “equality relation”, and P. A. Macmahon would call a “distribution of n(C) objects of type (1n(C)) into classes of type (m)”.
‡ “Abstraction class” according to Carnap (Logische Aufbau der Welt (Berlin, 1928), p. 102).
† Aitken, A. C., Edin. Math. Notes, 28 (1933), xviii–xxiii.
‡ x ⌢ y is merely the conventional “logical product” of the relations x and y. The operation of join is, however, new.
† The first inequality is a consequence of the previous statement; the second requires Theorems 19 and 9·2 of “Subalgebras”.
‡ H. Hasse, op. cit. p. 60; the proof is, from the standpoint of group theory, elementary.
* The technique consists in passing from G x to G x ⌣ ν through a finite or transfinite sequence of such intransitive groups, and showing that G x ⌣ G ν cannot fail to contain any first one of them.
† This section was added in revision.
‡ We need k i to be finite to ensure that n should be finite.
§ "Subalgebras”, Theorems 26·1 and 27·1, and Speiser, op. cit. Theorem 23. By an abstraction of the same method, we can show that of an algeora A is such that to any homomorphic equivalence a (x ⌣ y) b corresponds an element c such that axc and cyb, then the lattice of the homomorphisms of A is a modular lattice. It is the lattice H (A) which describes the “structure” of A; hence we may call it the “structure lattice” of A.
∥ This result was implicitly announced by Ore in a lecture at Harvard University, and will presumably appear in his paper already cited.
† This is the accepted usage for both groups and linear algebras.
‡ Klein, F., “Beiträge zur Theorie der Verbände”, Math. Zeitschrift, 39 (1934), 227–239.
† “Zur topologische Algebra. I. Komplettierungstheorie”, Math. Annalen, 107 (1933), 587–626.
† As in “Subalgebras”, §2.
‡ Stated in “Subalgebras”, §3.
§ “Die Deformationssätze der einfach zusammenhangenden Flächen”, Math. Zeits. 25 (1926), 362.
∥ Carathéodory, C., Vorlesungen über reelle Funktionen (Berlin, 1927), 2nd ed. p. 238.
† Proved by Daniell, P. J., “The modular difference of classes”, Bull. Amer. Math. Soc. 23 (1916), 446–50.
‡ In the sense of van Dantzig, op. cit.
§ Dedekind, R., Ges. Werke, 2, 147, states that D (1) = 1, D (2) = 4, D (3) = 18, D (4) = 166.
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