Unary FA-presentable semigroups

Alan James Cain, Nik Ruskuc, R.M. Thomas

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Automatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups.
We prove the following: Every unary FA-presentable structure admits an injective
unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.
Original languageEnglish
Article number1250038
Number of pages29
JournalInternational Journal of Algebra and Computation
Issue number4
Publication statusPublished - 8 Jun 2012


  • Automatic presentations
  • Semigroups
  • Regular languages


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