The dimension monoid of a lattice

Abstract

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, $\Dim$, from L x L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Conc L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to Z^+ or to R^+. (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of Z^+ $\cup$ {$\infty$}. (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid. (4) If L is a simple geometric lattice, then DimL is isomorphic to Z^+, if L is modular, and to the two-element semilattice, otherwise. (5) If L is a countably meet-continuous complemented modular lattice, then both Dim L and the dimension function $\Dim$ satisfy (countable) completeness properties. If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M\_2(R), then Dim L is isomorphic to the monoid V(R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.