Intrinsic metrics preserving maps on Abelian lattice-ordered groups

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Abstract

Swamy studied the natural "metric" |x-y| on Abelian lattice-ordered groups G, and he proved that the stable isometries which preserve this metric have to be automorphisms of G. Holland proved that the only intrinsic metrics on lattice-ordered groups, i.e., invariant and symmetric metrics, are the multiples n|x -y| for some integer n. We show that if f is an arbitrary surjection from an Abelian lattice-ordered group G1 onto an Archimedean Abelian lattice-ordered group G2 such that f(0)]0 and, for some non-zero intrinsic metrics D and d, D(f(x),f(y)) depends functionally on d(x,y), then f is a homomorphism of G1 onto G2.

Original languageEnglish
Pages (from-to)338-345
Number of pages8
JournalAlgebra Universalis
Volume29
Issue number3
DOIs
Publication statusPublished - Sep 1992

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Lattice-ordered Group
Abelian group
Metric
Surjection
Isometry
Homomorphism
Automorphisms
Integer
Invariant
Arbitrary

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Intrinsic metrics preserving maps on Abelian lattice-ordered groups. / Pap, E.

In: Algebra Universalis, Vol. 29, No. 3, 09.1992, p. 338-345.

Research output: Contribution to journalArticle

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