Intrinsic metrics preserving maps on Abelian lattice-ordered groups

1992 ◽  
Vol 29 (3) ◽  
pp. 338-345 ◽  
Author(s):  
Endre Pap
Keyword(s):  
Intrinsic Metrics ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice

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.

scholarly journals Free abelian lattice-ordered groups

2005 ◽  
Vol 134 (2-3) ◽  
pp. 265-283 ◽  
Author(s):  
A.M.W. Glass ◽  
Angus Macintyre ◽  
Françoise Point
Keyword(s):  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice

Weak relatively uniform convergences on abelian lattice ordered groups

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Štefan Černák ◽  
Ján Jakubík
Keyword(s):  
Uniform Convergence ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice Ordered Group ◽  
Relative Uniform Convergence ◽  
Brouwerian Lattice ◽  
Abelian Lattice ◽  
Relatively Uniform Convergence

AbstractThe notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G +, then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex ℓ-subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.

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