Some results on abstract commutative ideal theory

Conditions are given for a multiplicative lattice to be a finite Boolean algebra. Multiplicative lattices in which semiprimary elements are primary or in which prime elements are weak meet principal are studied. The lattice of filters of a bounded commutative semilattice are investigated. Finally, we study compactly packed lattices.     

A fixed point theorem with applications to truncated lattices

We prove a fixed point theorem related to the set P2 of [17]. The result gives access to nontrivial infinite ordered sets with the fixed point property. We also show how the result can be used to provide an elementary proof of part of Baclawski and Björner’s results on truncated lattices.Dedicated to the memory of Ivan RivalReceived December 1, 2002; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Topological representation of posets

There is a canonical imbedding of a poset into a complete Boolean lattice and hence into a Boolean lattice. This gives it a representation as a collection of clopen sets of a Boolean space. There are reflective functions from a category of distributive posets to the subcategories of distributive and Boolean lattices and consequently a topological dual equivalence that extends the Stone duality of Boolean lattices.Presented by B. Jonsson.     

s-Prime elements in multiplicative lattices

Let be aC-lattice which is strong join principally generated. In this paper, we consider prime elements of for which every semiprimary element is primary. We show, for example, that a compact nonmaximal primep with this property is principal. We also show that if every primepm has this property, then is either a one dimensional domain or a primary lattice. It follows that if every primep satisfies the property, and if there are only a finite number of minimal primes in , then is the finite direct product of one-dimensional domains and primary lattices.     

Pseudo-trees and Boolean algebras

We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras.     

Der Verband der zusammenhängenden Untergruppen einer kommutativen algebraischen Gruppe

We classify the connected commutative algebraic groups having as lattice of connected closed subgroups a projective space and determine the coordinate fields of these projective spaces. Furthermore we determine the not necessarily commutative connected algebraic groups with a distributive or boolean lattice of connected closed subgroups.

Received: 3 December 2004     

Six problems of Gian-Carlo Rota in lattice theory and universal algebra

No Abstract. .The algebraic structure sooner or later comes to dominate, whether or not it is recognized when a subject is born. Algebra dictates the analysis. Gian-Carlo Rota [33]     

Lattices and infinite-dimensional forms. ‘The Lattice Method’

Hermitean vector spaces E of infinite dimensions are considered. Let G be a subgroup of the orthogonal group of E acting on a set M. The Lattice Method is a technique for classifying the orbits in M under G. We discuss the method in abstract terms and we illustrate it by means of three classification results showing that it is decisive to do a considerable amount of explicit calculations with vector subspace lattices.     

CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES

Abstract

The notion of Cauchy point (= regular Cauchy filter) and the corresponding Cauchy spectrum, for a nearness frame (= uniform without the star-refinement condition) are investigated in various directions, including basic motivation, several functorial aspects, and the recognition of the Cauchy spectrum as the ordinary spectrum of the completion, after the unique existence of the latter is obtained as a central new result in this context.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.