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Discussiones Mathematicae General Algebra and Applications 25(1) (2005) 119-134
DOI: https://doi.org/10.7151/dmgaa.1095

REPRESENTATIONS OF A FREE GROUP OF RANK TWO BY TIME-VARYING MEALY AUTOMATA

Adam Woryna

Institute of Mathematics
Silesian University of Technology
Kaszubska 23, 44-100 Gliwice, Poland
e-mail: Adam.Woryna@polsl.pl

Abstract

In the group theory various representations of free groups are used. A representation of a free group of rank two by the so-calledtime-varying Mealy automata over the changing alphabet is given.Two different constructions of such automata are presented.

Key words and phrases: changing alphabet, Mealy automaton, time-varying automaton, group generated by time-varying automaton, free group.

2000 Mathematics Subject Classification: 20E05, 20E08, 20F65, 68R15, 68Q70.

References

[1]	S.V. Aleshin, A free group of finite automata, (Russian), Vestnik Moskov. Univ. Ser. I. Mat. Mekh. 38 (1983), 12-14 (English Transl.: Moscow Univ. Math. Bull. 38 (1983), No. 4, 10-13).
[2]	L. Bartholdi, R.I. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, p. 25-118 in: "Fractals in Graz 2001", Birkhäuser, Basel 2003.
[3]	R.I. Grigorchuk, V. V. Nekrashevich and V.I. Sushchanskii, Automata, Dynamical Systems and Groups, Proc. Steklov Inst. Math. 231 (2000), 128-203.
[4]	R.I. Grigorchuk and A. Żuk, Lectures on Automata Groups, Dynamics on Tress, and L2-invariants, in: ``Advanced Course on Automata Groups" (July 5-16, 2004 at CRM), Centre de Recerca Matemàtica, Universitat Autonòma de Barcelona, Bellaterra, Spain, (preprint, 2004).
[5]	A. Olijnyk, Free products of C2 as groups of finitely automatic permutations, (in Russian), Voprosy Algebry 14 (1999), 158-165.
[6]	A.S. Olijnyk and V.I. Sushchanskii, Free Groups of Infinitely Unitriangular Matrices, Math. Notes 67 (2000), 320-324.
[7]	V.I. Sushchanskii, Group of Automatic Permutations, (Ukrainian), Dopov. Nats. Akad. Ukr. Mat. Prirodozn. Tekh. Nauki 1998, no. 6, 47-51 .
[8]	V.I. Sushchanskii, Group of Finite Automatic Permutations, (Ukrainian), Dopov. Nats. Akad. Ukr. Mat. Prirodozn. Tekh. Nauki 1999, no. 2, 48-52.
[9]	A. Woryna, On transformations given by time-varying Mealy automata, (Polish), Zeszyty Nauk. Politech. Śląskiej, no. 1581, Ser. Automatyka 138 (2003), 201-215 .
[10]	A. Woryna, On the group permutations generated by time-varying Mealy automata, Publ. Math. Debrecen, 67 (2005), 115-130.
[11]	A. Woryna, On representation of a semidirect product of cyclic groups by a 2-state time-varying Mealy automaton, Zeszyty Nauk. Politech. Śląskiej, no. 1652, Ser. Math. 91 (2004), 343-355.

Received 13 May 2005
Revised 19 July 2005