Lattices

Abstract

In a number of important considerations in the theory of groups and of rings one is concerned primarily with certain distinguished subsets (invariant subgroups, ideals) of these systems rather than with the elements themselves. This is particularly true of the Jordan-Hölder-Schreier theory. Here the arguments concern the system of M-subgroups and the compositions in this system of intersection and group generated. Similarly, parts of the theory of rings are concerned with the systems of ideals (left, right, two-sided) of a ring and the compositions of intersection and sum in these systems. One is therefore led to the definition of an abstract system—called a lattice—that includes these two as instances. The concept of a lattice was first defined by Dedekind, but it attracted very little attention until quite recently (around 1930). Besides the applications to algebra many applications to the foundations of geometry and to other fields have been discovered. It should be noted also that prior to Dedekind’s work a special class of lattices, Boolean algebras, had been introduced by Boole.