The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova's result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.
@article{ITA_2012__46_1_165_0,
author = {Silva, Pedro V.},
title = {Fixed points of endomorphisms of certain free products},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
volume = {46},
year = {2012},
pages = {165-179},
doi = {10.1051/ita/2011125},
mrnumber = {2904968},
zbl = {1266.20069},
language = {en},
url = {http://dml.mathdoc.fr/item/ITA_2012__46_1_165_0}
}
Silva, Pedro V. Fixed points of endomorphisms of certain free products. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) pp. 165-179. doi : 10.1051/ita/2011125. http://gdmltest.u-ga.fr/item/ITA_2012__46_1_165_0/
[1] , Descendants of regular language in a class of rewriting systems: algorithm and complexity of an automata construction, in Proc. of RTA 87. Lect. Notes Comput. Sci. 256 (1987) 121-132. | MR 903667 | Zbl 0643.68119
[2] , Transductions and Context-free Languages. Teubner, Stuttgart (1979). | MR 549481 | Zbl 0424.68040
[3] and , Train tracks and automorphisms of free groups. Ann. Math. 135 (1992) 1-51. | MR 1147956 | Zbl 0757.57004
[4] , , and , The conjugacy problem is solvable in free-by-cyclic groups. Bull. Lond. Math. Soc. 38 (2006) 787-794. | MR 2268363 | Zbl 1116.20027
[5] and , String-Rewriting Systems. Springer-Verlag, New York (1993). | MR 1215932 | Zbl 0832.68061
[6] and , Infinite words and confluent rewriting systems: endomorphism extensions. Int. J. Algebra Comput. 19 (2009) 443-490. | MR 2536187 | Zbl 1213.68477
[7] and , Infinite periodic points of endomorphisms over special confluent rewriting systems. Ann. Inst. Fourier 59 (2009) 769-810. | Numdam | MR 2521435 | Zbl 1166.68021
[8] and , Efficient representatives for automorphisms of free products. Mich. Math. J. 41 (1994) 443-464. | MR 1297701 | Zbl 0820.20035
[9] , Automorphisms of free groups have finitely generated fixed point sets. J. Algebra 111 (1987) 453-456. | MR 916179 | Zbl 0628.20029
[10] , Fixed points of automorphisms of free groups. Adv. Math. 64 (1987) 51-85. | MR 879856 | Zbl 0616.20014
[11] and , Monomorphisms of finitely generated free groups have finitely generated equalizers. Invent. Math. 82 (1985) 283-289. | MR 809716 | Zbl 0582.20023
[12] and , Fixed subgroups of homomorphisms of free groups. Bull. Lond. Math. Soc. 18 (1986) 468-470. | MR 847985 | Zbl 0576.20016
[13] and , Characterization of finite and one-sided infinite fixed points of morphisms on free monoids. Technical Report CS-99-17 (1999).
[14] , Fixed languages and the adult languages of 0L schemes. Int. J. Comput. Math. 10 (1981) 103-107. | MR 645626 | Zbl 0472.68034
[15] , Semigroups. Fizmatgiz. Moscow (1960). English translation by Am. Math. Soc. (1974). | Zbl 0100.02301
[16] , The fixed point group of a free group automorphism. Algebra i Logika 42 (2003) 422-472. English translation in Algebra Logic 42 (2003) 237-265. | MR 2017513 | Zbl 1031.20014
[17] and , On directly infinite rings. Acta Math. Hung. 85 (1999) 153-165. | MR 1713097 | Zbl 0991.16024
[18] , Éléments de Théorie des Automates. Vuibert, Paris (2003). | Zbl 1178.68002
[19] , Rational subsets of partially reversible monoids. Theoret. Comput. Sci. 409 (2008) 537-548. | MR 2473928 | Zbl 1155.68047
[20] , Fixed points of endomorphisms over special confluent rewriting systems. Monatsh. Math. 161 (2010) 417-447. | MR 2734969 | Zbl 1236.20063
[21] , Fixed subgroups of endomorphisms of free products. J. Algebra 315 (2007) 274-278. | MR 2344345 | Zbl 1130.20032
[22] , Fixed subgroups of free groups: a survey. Contemp. Math. 296 (2002) 231-255. | MR 1922276 | Zbl 1025.20012