Abstract
We consider biautomatic semigroups. There are two different definitions of a biautomatic structure for a group in the literature; whilst these definitions are not equivalent, the idea of a biautomatic group is well defined, in that a group possesses one type of biautomatic structure if and only if it possesses the other. However the two definitions give rise to different notions of biautomaticity for semigroups and we study these ideas in this paper. In particular, we settle the question as to whether automaticity and biautomaticity are equivalent for semigroups by giving examples of semigroups which are automatic but not biautomatic.
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References
Adjan, S.I.: Defining relations and algorithmic problems for groups and semigroups. Proceedings of the Steklov Institute of Mathematics 85 (1966), American Mathematical Society (1967); translated from Trudy. Mat. Inst. Steklov 85 (1966)
Baumslag, G., Gersten, S.M., Shapiro, M., Short, H.: Automatic groups and amalgams. J. Pure Appl. Algebra 76, 229–316 (1991)
Cain, A.J.: A group-embeddable finitely generated non-automatic semigroup whose universal group is automatic, preprint
Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: Automatic semigroups. Theoret. Comput. 365, 365–391 (2001)
Duncan, A.J., Robertson, E.F., Ruškuc, N.: Automatic monoids and change of generators. Math. Proc. Cambridge Philos. Soc. 127, 403–409 (1999)
Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S., Paterson, M.S., Thurston, W.: Word Processing in Groups (Jones and Barlett 1992)
Gersten, S.M., Short, H.B.: Rational subgroups of biautomatic groups. Annals Math. 134, 125–158 (1991)
Hoffmann, M., Kuske, D., Otto, F., Thomas, R.M.: Some relatives of automatic and hyperbolic groups. In: Gomes, G.M.S., Pin, J.-E., Silva, P.V. (eds.) Semigroups, Algorithms, Automata and Languages, pp. 379–406. World Scientific, Singapore (2002)
Hoffmann, M., Thomas, R.M.: Notions of automaticity in semigroups. Semigroup Forum 66, 337–367 (2003)
Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995)
Jackson, D.A.: Decision and separability properties for Baumslag-Solitar semigroups. Internat. J. Algebra Comput. 12, 33–49 (2002)
Lallement, G.: Semigroups and Combinatorial Applications. John Wiley, Chichester (1979)
Otto, F., Sattler-Klein, A., Madlener, K.: Automatic monoids versus monoids with finite convergent presentations. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 32–46. Springer, Heidelberg (1998)
Short, H.: An introduction to automatic groups. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups. NATO ASI Series, vol. C466, pp. 233–253. Kluwer, Dordrecht (1995)
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Hoffmann, M., Thomas, R.M. (2005). Biautomatic Semigroups. In: Liśkiewicz, M., Reischuk, R. (eds) Fundamentals of Computation Theory. FCT 2005. Lecture Notes in Computer Science, vol 3623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537311_6
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DOI: https://doi.org/10.1007/11537311_6
Publisher Name: Springer, Berlin, Heidelberg
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