The arithmetical theory of Birkhoff lattices

Abstract

In the development of lattice theory considerable work has been devoted to the study of the arithmetical properties of modular and distributive lattices. Indeed most of the decomposition theorems of abstract algebra have been extended to these more general domains. Nevertheless, there are lattices with very simple arithmetical properties which come under neither of these classifications. For example, the lattices with unique irreducible decompositions, which were studied by the author in a previous paper [3]1 satisfy the Birkhoff condition2 which is even less restrictive than the modular axiom. Furthermore, there are important algebraic systems which give rise to non-modular, Birkhoff lattices. Thus, since every exchange lattice (Mac Lane [4]) is a Birkhoff lattice, the systems which satisfy Mac Lane’s exchange axiom form lattices of the type in question. In this paper we shall study the arithmetical structure of general Birkhoff lattices and in particular determine necessary and sufficient conditions that certain important arithmetical properties hold.