scholarly journals a∗-closures of completely distributive lattice-ordered groups

1975 ◽  
Vol 59 (1) ◽  
pp. 43-67 ◽  
Author(s):  
Andrew Glass ◽  
W. Charles Holland ◽  
Stephen McCleary
Keyword(s):  
Distributive Lattice ◽  
Completely Distributive ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Completely Distributive Lattice

Torsion Radicals and Torsion Classes of Cyclically Ordered Groups

2015 ◽  
Vol 65 (2) ◽  
Author(s):  
Jan Jákubík ◽  
Judita Lihoyá
Keyword(s):  
Proper Class ◽  
Ordered Sets ◽  
Ordered Group ◽  
Completely Distributive ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Complete Completely Distributive Lattice ◽  
Cyclically Ordered Group ◽  
Completely Distributive Lattice ◽  
Torsion Radical

AbstractThe notion of torsion radical of cyclically ordered groups is defined analogously as in the case of lattice ordered groups. We denote by T the collection of all torsion radicals of cyclically ordered groups. For τ1, τ2 ∈ T, we put τ1 ∈ τ2 if τ1(G) □ τ2(G) for each cyclically ordered group G. We show that T is a proper class; nevertheless, we apply for T the usual terminology of the theory of partially ordered sets. We prove that T is a complete completely distributive lattice. The analogous result fails to be valid for torsion radicals of lattice ordered groups. Further, we deal with products of torsion classes of cyclically ordered groups.

Cevian properties in ideal lattices of Abelian ℓ-groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Miroslav Ploščica
Keyword(s):  
Distributive Lattice ◽  
Recent Result ◽  
Ideal Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Spectral Spaces ◽  
Abelian Lattice

Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

Metrizability Conditions for Completely Distributive Lattices

1983 ◽  
Vol 26 (4) ◽  
pp. 446-453
Author(s):  
G. Gierz ◽  
J. D. Lawson ◽  
A. R. Stralka
Keyword(s):  
Distributive Lattice ◽  
Essential Extension ◽  
Distributive Lattices ◽  
Interval Topology ◽  
Completely Distributive ◽  
Countable Chain ◽  
Completely Distributive Lattice

AbstractA lattice is said to be essentially metrizable if it is an essential extension of a countable lattice. The main result of this paper is that for a completely distributive lattice the following conditions are equivalent: (1) the interval topology on L is metrizable, (2) L is essentially metrizable, (3) L has a doubly ordergenerating sublattice, (4) L is an essential extension of a countable chain.

scholarly journals Essential completions of distributive lattices

1985 ◽  
Vol 32 (3) ◽  
pp. 361-374 ◽  
Author(s):  
Gerhard Gierz ◽  
Albert R. Stralka
Keyword(s):  
Distributive Lattice ◽  
Salient Feature ◽  
Distributive Lattices ◽  
Compact Topological Space ◽  
Completely Distributive ◽  
Completion Process ◽  
The One ◽  
One Point Compactification ◽  
Essential Completion ◽  
Completely Distributive Lattice

The salient feature of the essential completion process is that for most common distributive lattices it will yield a completely distributive lattice. In this note it is shown that for those distributive lattices which have at least one completely distributive essential extension the essential completion is minimal among the completions by infinitely distributive lattices. Thus in its setting the essential completion of a distributive lattice behaves in much the some way as the one-point compactification of locally compact topological space does in its setting.

On a type of distributivity of lattice ordered groups

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Ján Jakubík
Keyword(s):  
Distributive Lattice ◽  
Boolean Algebras ◽  
Radical Class ◽  
Infinite Cardinal ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Homogeneity Condition ◽  
Lattice Ordered Groups

AbstractLet m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.

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