### 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 G_{1} onto an Archimedean Abelian lattice-ordered group G_{2} 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 G_{1} onto G_{2}.

Original language | English |
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Pages (from-to) | 338-345 |

Number of pages | 8 |

Journal | Algebra Universalis |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1992 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

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

Research output: Contribution to journal › Article

*Algebra Universalis*, vol. 29, no. 3, pp. 338-345. https://doi.org/10.1007/BF01212436

}

TY - JOUR

T1 - Intrinsic metrics preserving maps on Abelian lattice-ordered groups

AU - Pap, E.

PY - 1992/9

Y1 - 1992/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0011618912&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011618912&partnerID=8YFLogxK

U2 - 10.1007/BF01212436

DO - 10.1007/BF01212436

M3 - Article

AN - SCOPUS:0011618912

VL - 29

SP - 338

EP - 345

JO - Algebra Universalis

JF - Algebra Universalis

SN - 0002-5270

IS - 3

ER -