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Summary: REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES IN
IDEAL LATTICES OF VARIOUS ALGEBRAIC STRUCTURES
K.R. GOODEARL AND F. WEHRUNG
Abstract. We study the relationships among existing results about repre
sentations of distributive semilattices by ideals in dimension groups, von Neu
mann 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.
Introduction
Many algebraic theories afford a notion of ideal, and the collection of all ideals
of a given object typically forms a complete lattice with respect to inclusion. It is
natural to ask which lattices can be represented as a lattice of ideals for a given type
of object. Often, the lattice of ideals of an object is algebraic, in which case this
lattice is isomorphic to the lattice of ideals of the (join) subsemilattice of compact
elements. For instance, this holds for lattices of ideals of rings, monoids, and par
tially ordered abelian groups. Hence, lattice representation problems often reduce
to corresponding representation problems for (join) semilattices. For example, to
prove that a given algebraic lattice L occurs as the lattice of ideals of a ring of some
type, it suffices to show that the semilattice of compact elements of L occurs as the
semilattice of finitely generated ideals of a suitable ring.
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