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REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES IN IDEAL LATTICES OF VARIOUS ALGEBRAIC STRUCTURES
 

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.

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics