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ON GROUPS PRESENTED BY MONADIC REWRITING SYSTEMS WITH GENERATORS OF FINITE ORDER

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  06 March 2015

ADAM PIGGOTT
ADAM PIGGOTT*
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA email [email protected]
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Abstract

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We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman’s conjecture in a special case. We also prove that the finite cyclic groups of order at least three are the only finite groups admitting a presentation by more than one finite convergent monadic rewriting system (up to relabelling), and these admit presentation by exactly two such rewriting systems.

Keywords

groupmonoid presentationrewriting systemmonadic systemplain group

MSC classification

Primary: 20F10: Word problems, other decision problems, connections with logic and automata
Secondary: 20E99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015 , pp. 426 - 434
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Avenhaus, J. and Madlener, K., ‘On groups defined by monadic Thue systems’, in: Algebra, Combinatorics and Logic in Computer Science, Vol. I, II (Győr, 1983), Colloquia Mathematica Societatis János Bolyai, 42 (North-Holland, Amsterdam, 1986), 63–71.Google Scholar
Avenhaus, J., Madlener, K. and Otto, F., ‘Groups presented by finite two-monadic Church–Rosser Thue systems’, Trans. Amer. Math. Soc. 297(2) (1986), 427–443.Google Scholar
Book, R. V. and Otto, F., String-Rewriting Systems, Texts and Monographs in Computer Science (Springer, New York, 1993).Google Scholar
Cochet, Y., ‘Church—Rosser congruences on free semigroups’, in: Algebraic Theory of Semigroups (Proc. Sixth Algebraic Conf., Szeged, 1976), Colloquia Mathematica Societatis János Bolyai, 20 (North-Holland, Amsterdam, 1979), 51–60.Google Scholar
Diekert, V., ‘Some remarks on presentations by finite Church–Rosser Thue systems’, STACS 87, 4th Annual Symposium on Theoretical Aspects of Computer Science, Passau, Germany, 1987, Lecture Notes in Computer Science, 247 (Springer, Berlin, 1987), 272–285.Google Scholar
Gilman, R. H., ‘Computations with rational subsets of confluent groups’, EUROSAM84, International Symposium on Symbolic and Algebraic Computation, Cambridge, UK, 1984, Lecture Notes in Computer Science, 174 (Springer, Berlin, 1984), 207–212.Google Scholar
Haring-Smith, R. H., ‘Groups and simple languages’, Trans. Amer. Math. Soc. 279(1) (1983), 337–356.Google Scholar
Madlener, K. and Otto, F., ‘On groups having finite monadic Church–Rosser presentations’, in: Semigroups, Theory and Applications (Oberwolfach, 1986), Lecture Notes in Mathematics, 1320 (Springer, Berlin, 1988), 218–234.Google Scholar
Madlener, K. and Otto, F., ‘About the descriptive power of certain classes of finite string-rewriting systems’, Theoret. Comput. Sci. 67(2–3) (1989), 143–172.Google Scholar
Muller, D. E. and Schupp, P. E., ‘Groups, the theory of ends, and context-free languages’, J. Comput. System Sci. 26(3) (1983), 295–310.CrossRefGoogle Scholar
Otto, F., ‘Elements of finite order for finite monadic Church–Rosser Thue systems’, Trans. Amer. Math. Soc. 291(2) (1985), 629–637.Google Scholar
Serre, J.-P., Trees, Springer Monographs in Mathematics (Springer, Berlin, 2003).Google Scholar
Shapiro, M., ‘Pascal’s triangles in abelian and hyperbolic groups’, J. Austral. Math. Soc. Ser. A 63(2) (1997), 281–288.CrossRefGoogle Scholar