English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We investigate the structure of lattice-preserving homomorphisms of free lattice-ordered Abelian groups to the ordered group of integers. For any lattice-ordered group, a choice of generators induces on such homomorphisms a partial commutative monoid canonically embedded into a direct product of the group of integers. Free lattice-ordered Abelian groups can be characterised in terms of this dual object and its embedding. For finite sets of generators, we obtain the stronger result: a lattice-ordered Abelian group is free on a finite generating set if and only if the generators make ℤ-valued homomorphisms a free Abelian group of finite rank. One of the main points of the paper is that all results are proved in an entirely elementary and self-contained manner. To achieve this end, we give a short new proof of the standard result of Weinberg that free lattice-ordered Abelian groups have enough ℤ-valued homomorphisms. The argument uses the ultrasimplicial property of ordered Abelian groups, first established by Elliott in a different connection. The paper is made self-contained by a new proof of Elliott's result.

Weinberg's theorem, Elliott's ultrasimplicial property, and a characterisation of free lattice-ordered Abelian groups