The dimension monoid of a lattice
The paper is one more step toward solving R. P. Dilworth's longstanding problem of the representation of a distributive algebraic lattice as a congruence lattice \(\text{Con }L\) for some lattice \(L\). In this work, the dimension monoid \(\text{Dim }L\) is associated with any lattice \(L\). To any pair \(a,b \in L\) an element \(\Delta (a,b)\) is given which may be viewed as a monoid-valued ``distance'' between \(a\) and \(b\) and \(\text{Dim }L\) is generated by these elements \(\Delta (a,b)\). The relations defining the \(\Delta \) function are satisfied by the mapping \(\theta \) that with any \(a,b \in L\) associates the principal congruence \(\theta (a,b)\). In particular, \(\text{Dim }L\) is a precursor of \(\text{Con }L\) and, moreover, the semilattice of all compact congruences of \(L\) is isomorphic to the maximal semilattice quotient of \(\text{Dim }L\). The dimension monoid gives much more information about \(L\) than \(\text{Con }L\) does. This dimension theory provides a generalization to all lattices of the von Neumann dimension theory for continuous geometries. In particular, if \(L\) is an irreducible continuous geometry then \(\text{Dim }L\) is either isomorphic to \(Z^+\) or to \(R^+\). If \(L\) has no infinite bounded chains then \(\text{Dim }L\) embeds into a power of \(Z^+\cup \{\infty \}\). If \(L\) is modular or \(L\) has no infinite bounded chains then \(\text{Dim }L\) is a refinement monoid. If \(R\) is a von Neumann ring and \(L\) its lattice of principal right ideals of the matrix ring \(M_2(R)\) then \(\text{Dim }L\) is isomorphic to the monoid of isomorphism classes of finitely generated projective right \(R\)-modules. Hence, the dimension theory of lattices provides a wide lattice-theoretical generalization of \(K\)-theory of regular rings. This 164 pages long paper is in fact a monograph divided into 13 chapters and including an Introduction, Contents, Index and a list of 69 references. It contains also 14 open problems concerning such properties of \(\text{Dim }L\) as decidability of words, dimension-preserving extensions, representability as a refinement monoid, etc. The paper is a continuation of the author's series of long papers concerning refinement monoids. The importance of it is formally expressed also by the fact that the whole No. 3 of Vol. 40 of Algebra Universalis is devoted only to this paper.
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