Semimodular Lattices

The theory of lattices, initiated by Dedekind in the past centu ry, and revived in the thirties by Garrett Birkhoff, F. Klein-Barmen, ore, and von Neumann, is only in our time coming into its own. The fledgling theory was handicapped by a contingent historical circumstance. The peculiarities of mathematical personality of the founders made lattice theory less welcome to the mathematical public of the time than it otherwise might have been. Thus Dedekind was wi dely thought in his time to be far too abstract for his own good, and some of his peers, notably Kronecker, did not hesitate to state their loud and clear disapproval. Later on, the tempers of Garrett Birkhoff and John von Neumann clashed with those of some of the "mainstream"' mathematicians of their time. Norman Levinson once related to me the following anecdote about von Neumann. Invited to deliver the weekly mathematics colloquium at Harvard sometime in the thirties, he chose the subject of his current interest, namely, continuous geometries. At the end of the lecture, as the public was streaming out, G. H. Hardy, who was at the time visiting Cambridge, was overheard whispering to G. D. Birkhoff (Gar rett's father): "He is quite clearly a very brilliant man, but why does he waste his time on this stuff?" I myself, when still an assistant professor, was once stopped in the hall of M. I. T.

I. Semimodular Lattices. - II. Strong, Balanced, and Consistent Lattices. - III. Strong Semimodular Lattices. - IV. Lattices of Infinite Length with Covering Properties. - V. Appendix. - List of symbols and notations.