Selected Papers on Algebra and Topology by Garrett Birkhoff

The present volume of reprints are what I consider to be my most interesting and influential papers on algebra and topology. To tie them together, and to place them in context, I have supplemented them by a series of brief essays sketching their historieal background (as I see it). In addition to these I have listed some subsequent papers by others which have further developed some of my key ideas. The papers on universal algebra, lattice theory, and general topology collected in the present volume concern ideas which have become familiar to all working mathematicians. It may be helpful to make them readily accessible in one volume. I have tried in the introduction to each part to state the most significant features of ea ch paper reprinted there, and to indieate later developments. The background that shaped and stimulated my early work on universal algebra, lattice theory, and topology may be of some interest. As a Harvard undergraduate in 1928-32, I was encouraged to do independent reading and to write an original thesis. My tutorial reading included de la Vallee-Poussin's beautiful Cours d'Analyse Infinitesimale, Hausdorff's Grundzüge der Mengenlehre, and Frechet's Espaces Abstraits. In addition, I discovered Caratheodory's 1912 paper "Vber das lineare Mass von Punktmengen" and Hausdorff's 1919 paper on "Dimension und Ausseres Mass," and derived much inspiration from them. A fragment of my thesis, analyzing axiom systems for separable metrizable spaces, was later published [2]. This background led to the work summarized in Part IV.

Contenu
I. Lattices.- [3] On the Combination of Subalgebras.- [3a] Note on the Paper On the Combination of Subalgebras.- [4] Applications of Lattice Algebra.- [6] On the Lattice Theory of Ideals.- [7] Ideals in Algebraic Rings.- [12] Combinatorial Relations in Projective Geometries.- [16] Abstract Linear Dependence and Lattices.- [21] The Logic of Quantum Mechanics (with J. von Neumann).- [36] A Characterization of Boolean Algebras (with M. Ward).- [40] Neutral Elements in General Lattices.- [51] Distributive Postulates for Systems Like Boolean Algebras (with G. D. Birkhoff).- [54] A Ternary Operation in Distributive Lattices (with S. A. Kiss).- II. Universal Algebra.- [15] On the Structure of Abstract Algebras.- [47] Subdirect Unions in Universal Algebra.- [49] Universal Algebra.- [154] Heterogeneous Algebras (with J. D. Lipson).- [174] Universal Algebra and Automata (with J. D. Lipson).- [179] Note on Universal Topological Algebra.- III. Topology.- [8] The Topology of Transformation-Sets.- [24] Moore-Smith Convergence in General Topology.- [25] The Meaning of Completeness.- [28] An Extended Arithmetic.- [29] Rings of Sets.- [43] Generalized Arithmetic.- [58] Representations of Lattices by Sets (with O. Frink, Jr.).- [118] A New Interval Topology for Dually Directed Sets.- IV. Lie Groups and Lie Algebras.- [19] On the Order of Groups of Automorphisms (with P. Hall).- [20] A Note on Topological Groups.- [22] Lie Groups Simply Isomorphic with No Linear Group.- [26] Continuous Groups and Linear Spaces.- [27] Representability of Lie Algebras and Lie Groups by Matrices.- [32] Analytical Groups.- [60] Representation of Jordan and Lie Algebras (with P. M. Whitman).- V. Lattice-Ordered Algebraic Structures.- [33] Dependent Probabilities and Spaces (L).- [42] Lattice-OrderedGroups.- [48] Lattice-Ordered Lie Groups.- [91] Lattice-Ordered Rings (with R. S. Pierce).- VI. History of Algebra.- [160] Current Trends in Algebra.- [169] The Role of Modern Algebra in Computing.- [182a] The Rise of Modern Algebra to 1936.- [182b] The Rise of Modern Algebra, 1936 to 1950.- Permissions.