Amalgamated free product in terms of automata constructions. (English) Zbl 1505.20024
Summary: Automata over finite alphabet are considered. Initial automata over finite alphabet with \(4\) states generating amalgamated free products of finite number of finite cyclic groups are constructed.
MSC:
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
| 20E26 | Residual properties and generalizations; residually finite groups |
| 20E08 | Groups acting on trees |
| 68Q70 | Algebraic theory of languages and automata |
References:
| [1] | Baumslag, G., On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc, 106, 2, 193-209 (1963) · Zbl 0112.25904 · doi:10.1090/S0002-9947-1963-0144949-8 |
| [2] | Brough, T.; Cain, A. J., Automaton semigroup constructions, Semigroup Forum, 90, 3, 763-774 (2015) · Zbl 1336.20062 · doi:10.1007/s00233-014-9632-x |
| [3] | Brough, T.; Cain, A. J., Automaton semigroups: new constructions results and examples of non-automaton semigroups, Theoret. Comput. Sci, 674, 3, 1-15 (2017) · Zbl 1381.20052 · doi:10.1016/j.tcs.2017.02.003 |
| [4] | Brunner, A. M.; Said, S. N., The generation of \(####\) by finite state automata, Internat. J. Algebra Comput, 8, 1, 127-139 (1998) · Zbl 0923.20023 · doi:10.1142/S0218196798000077 |
| [5] | Dömösi, P.; Nehaniv, C. L., Algebraic Theory of Automata Networks: An Introduction (2005), Philadelphia, PA: SIAM, Philadelphia, PA · Zbl 1070.68095 · doi:10.1137/1.9780898718492 |
| [6] | Gecseg, F.; Peák, I., Algebraic Theory of Automata (1972), Budapest, Hungary: Akad. Kiadó, Budapest, Hungary · Zbl 0246.94029 |
| [7] | Grigorchuk, R. I.; Nekrashevich, V. V.; Sushchanskii, V. I., Automata, dynamical systems and groups, Proc. Steklov Inst. Math, 231, 128-203 (2000) · Zbl 1155.37311 |
| [8] | Higman, G., Amalgams of p-groups, J. Algebra, 1, 3, 301-305 (1964) · Zbl 0246.20015 · doi:10.1016/0021-8693(64)90025-0 |
| [9] | Kim, G.; Lee, Y.; McCarron, J., On amalgamated free products of residually p-finite groups, J. Algebra, 162, 1, 1-11 (1993) · Zbl 0804.20024 · doi:10.1006/jabr.1993.1237 |
| [10] | Kim, G.; Lee, Y.; McCarron, J., Residual p-finiteness of certain generalized free products of nilpotent groups, Kyungpook Math. J, 48, 3, 495-502 (2008) · Zbl 1167.20015 · doi:10.5666/KMJ.2008.48.3.495 |
| [11] | Kim, G.; Lee, Y.; Tang C, Y., On generalized free products of residually finite p-groups, J. Algebra, 201, 1, 317-327 (1998) · Zbl 0918.20009 · doi:10.1006/jabr.1997.7256 |
| [12] | Lavrenyuk, Y.; Mazorchuk, V.; Oliynyk, A.; Sushchansky, V., Faithful group actions on rooted trees induced by actions of quotients, Commun. Algebra, 35, 11, 3759-3775 (2007) · Zbl 1187.20021 · doi:10.1080/00914030701410237 |
| [13] | Lyndon, R. C.; Schupp, P. E., Combinatorial group theory (1977), Berlin/Heidelberg: Springer-Verlag, Berlin/Heidelberg · Zbl 0368.20023 · doi:10.1007/978-3-642-61896-3 |
| [14] | Nekrashevych, V., Self-similar groups (2005), Providence, RI: Bull. Amer. Math. Soc.(N.S, Providence, RI · Zbl 1087.20032 |
| [15] | Oliynyk, A., Finite state wreath powers of transformation semigroups, Semigroup Forum, 82, 3, 423-436 (2011) · Zbl 1229.20069 · doi:10.1007/s00233-011-9292-z |
| [16] | Silva, P. V.; Steinberg, B., On the class of automata groups generalizing lamplighter groups, Int. J. Algebra Comput, 15, 5-6, 1213-1234 (2005) · Zbl 1106.20028 · doi:10.1142/S0218196705002761 |
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