FA-presentable groups and rings

Abstract

We consider structures which are FA-presentable. It is known that an FA-presentable finitely generated group is virtually abelian; we strengthen this result by showing that an arbitrary FA-presentable group is locally virtually abelian. As a consequence, we prove that any FA-presentable ring is locally finite; this is a significant restriction and allows us to say a great deal about the structure of FA-presentable rings. In particular, we show that any FA-presentable ring with identity and no zero divisors is finite.

Describing Groups

“…Theorem 3.8 ( [22]). Suppose that (N, +) r is weakly FA-presentable with alphabet Σ and k states; then r ≤ (k + 1) log 2 |Σ|.…”

The additive group of the rationals does not have an automatic presentation

“…Even if one considers simpler algebraic structures such as groups, the definition is still too restrictive: Oliver and Thomas [13] observed, as a consequence of Gromov's theorem about finitely generated groups of polynomial growth and a theorem of Romanovskiȋ classifying the virtually polycyclic groups with decidable first order theory, that a finitely generated group has an automatic presentation (in the sense of Definition 1) iff it is virtually abelian. This was extended by Nies and Thomas [12] who showed that every finitely generated subgroup of an automatically presentable group is virtually abelian.…”

The isomorphism problem for tree-automatic ordinals with addition

“…Finally, we refer the reader to the articles of Blumensath and Grädel [2], Delhommé [6], Khoussainov and Minnes [14], Khoussainov and Nerode [16], Kuske [18] and Rubin [24,25] for the background and open questions in the area of automatic structures. For simple and non-trivial examples of word-automatic structures we refer to the articles of Ishihara, Khoussainov and Rubin [11], Nies [22], Nies and Thomas [23] and Stephan [28].…”