The Many Lives of Lattice Theory

Introduction Never in the history of mathematics has a mathematical theory been the object of such vociferous vituperation as lattice theory. Dedekind, Jónsson, Kurosh, Malcev, Ore, von Neumann, Tarski, and most prominently Garrett Birkhoff have contributed a new vision of mathematics, a vision that has been cursed by a conjunction of misunderstandings, resentment, and raw prejudice. The hostility towards lattice theory began when Dedekind published the two fundamental papers that brought the theory to life well over one hundred years ago. Kronecker in one of his letters accused Dedekind of “losing his mind in abstractions,” or something to that effect. I took a course in lattice theory from Oystein Ore while a graduate student at Yale in the fall of 1954. The lectures were scheduled at 8 a.m., and only one other student attended besides me—María Wonenburger. It is the only course I have ever attended that met at 8 o’clock in the morning. The first lecture was somewhat of a letdown, beginning with the words: “I think lattice theory is played out” (Ore’s words have remained imprinted in my mind). For some years I did not come back to lattice theory. In 1963, when I taught my first course in combinatorics, I was amazed to find that lattice theory fit combinatorics like a shoe. The temptation is strong to spend the next fifty minutes on the mutual stimulation of lattice theory and combinatorics of the last thirty-five years. I will, however, deal with other aspects of lattice theory, those that were dear to Garrett Birkhoff and which bring together ideas from different areas of mathematics. Lattices are partially ordered sets in which least upper bounds and greatest lower bounds of any two elements exist. Dedekind discovered that this property may be axiomatized by identities. A lattice is a set on which two operations are defined, called join and meet and denoted by ∨ and ∧ , which satisfy the idempotent, commutative and associative laws, as well as the absorption laws: