Representations of distributive semilattices in ideal lattices of various algebraic structures

Abstract

Abstract. We study the relationships among existing results about representations of distributive semilattices by ideals in dimension groups, von Neumann regular rings, C*-algebras, and complemented modular lattices. We prove additional representation results which exhibit further connections with the scattered literature on these different topics.

A survey of recent results on congruence lattices of lattices

“…A survey of the connections between ring-theoretical problems and results and congruence lattice representation problems is presented in K. R. Goodearl and F. Wehrung [8]. In Section 7, we present a very short overview of the subject, as well as a few recent results.…”

Semilattices of finitely generated ideals of exchange rings with finite stable rank

“…; see also [11,18]. An ideal of M is a subset I of M such that 0 ∈ I and x + y ∈ I iff x, y ∈ I, for all x, y ∈ M .…”

“…Furthermore, the lattice of (two-sided) ideals of R is isomorphic to the lattice of ideals of V (R); see [11,Proposition 7.3].…”

“…By contrast, we recall that every countable distributive (∨, 0)-semilattice is isomorphic to the semilattice of all compact ideals of some countable dimensional locally matricial ring (over any given field); see [4,11].…”

“…The present work originates in the representation problem of distributive (∨, 0)-semilattices as semilattices of compact (i.e., finitely generated) ideals of von Neumann regular rings; see Section 13 in the chapter "Recent Developments" in [8], or the survey paper [11]. It is known that the original form of this problem has a negative solution, obtained by the author in [21,23]: There exists a distributive (∨, 0)-semilattice that is not isomorphic to the semilattice of all compact ideals of any von Neumann regular ring.…”