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/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

In this paper, we consider the group    Aut  (  ℚ  ,  ⩽  )    of order‐automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of    Aut  (  ℚ  ,  ⩽  )    is contained in an   N  ‐generated subgroup of    Aut  (  ℚ  ,  ⩽  )    for some fixed    N  ∈  ℕ   . We show that the least such   N   is 2. Moreover, for every countable subset of    Aut  (  ℚ  ,  ⩽  )   , we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that   a   and   b   freely generate the free semigroup{  a  ,  b  }   +consisting of the non‐empty words over   a   and   b  . Then we show that there exists a sequence of wordsw  1   ,   w  2   ,  …    over    {  a  ,  b  }    such that for every sequencef  1   ,   f  2   ,  …  ∈  Aut  (  ℚ  ,  ⩽  )    there is a homomorphism    ϕ  :    {  a  ,  b  }   +   →  Aut  (  ℚ  ,  ⩽  )    where    (   w  i   )  ϕ  =   f  ifor every   i  .  As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of    Aut  (  ℚ  ,  ⩽  )    is uncountable, or equivalently that    Aut  (  ℚ  ,  ⩽  )    has uncountable cofinality and Bergman's property.

Universal sequences for the order‐automorphisms of the rationals