## The additive group of the rationals does not have an automatic presentation.(English)Zbl 1247.03070

Summary: We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are $$p$$-divisible for infinitely many primes $$p$$, or groups of the form $$\oplus _{p \in I} \mathbf Z(p^{\infty })$$, where $$I$$ is an infinite set of primes.

### MSC:

 03D05 Automata and formal grammars in connection with logical questions 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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### References:

 [1] Logical Methods in Computer Science 3 pp 18– (2007) [2] Bulletin of the European Association for Theoretical Computer Science 94 pp 181– (2008) [3] Word processing in groups (1992) [4] Additive combinatorics 105 (2006) · Zbl 1127.11002 [5] Nachala strukturnoi teorii slozheniya mnozhestv (1966) [6] Logic Colloquium 2007 35 pp 132– (2010)
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