# Fixed points of endomorphisms of certain free products

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2012)

- Volume: 46, Issue: 1, page 165-179
- ISSN: 0988-3754

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topSilva, Pedro V.. "Fixed points of endomorphisms of certain free products." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 46.1 (2012): 165-179. <http://eudml.org/doc/273026>.

@article{Silva2012,

abstract = {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.},

author = {Silva, Pedro V.},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {endomorphisms; fixed points; free products; fixed points of endomorphisms; fixed point submonoids; SC monoids; confluent rewriting systems},

language = {eng},

number = {1},

pages = {165-179},

publisher = {EDP-Sciences},

title = {Fixed points of endomorphisms of certain free products},

url = {http://eudml.org/doc/273026},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Silva, Pedro V.

TI - Fixed points of endomorphisms of certain free products

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2012

PB - EDP-Sciences

VL - 46

IS - 1

SP - 165

EP - 179

AB - 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.

LA - eng

KW - endomorphisms; fixed points; free products; fixed points of endomorphisms; fixed point submonoids; SC monoids; confluent rewriting systems

UR - http://eudml.org/doc/273026

ER -

## References

top- [1] M. Benois, 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. Zbl0643.68119MR903667
- [2] J. Berstel, Transductions and Context-free Languages. Teubner, Stuttgart (1979). Zbl0424.68040MR549481
- [3] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups. Ann. Math.135 (1992) 1–51. Zbl0757.57004MR1147956
- [4] O. Bogopolski, A. Martino, O. Maslakova and E. Ventura, The conjugacy problem is solvable in free-by-cyclic groups. Bull. Lond. Math. Soc.38 (2006) 787–794. Zbl1116.20027MR2268363
- [5] R.V. Book and F. Otto, String-Rewriting Systems. Springer-Verlag, New York (1993). Zbl0832.68061MR1215932
- [6] J. Cassaigne and P.V. Silva, Infinite words and confluent rewriting systems: endomorphism extensions. Int. J. Algebra Comput.19 (2009) 443–490. Zbl1213.68477MR2536187
- [7] J. Cassaigne and P.V. Silva, Infinite periodic points of endomorphisms over special confluent rewriting systems. Ann. Inst. Fourier59 (2009) 769–810. Zbl1166.68021MR2521435
- [8] D.J. Collins and E.C. Turner, Efficient representatives for automorphisms of free products. Mich. Math. J.41 (1994) 443–464. Zbl0820.20035MR1297701
- [9] D. Cooper, Automorphisms of free groups have finitely generated fixed point sets. J. Algebra111 (1987) 453–456. Zbl0628.20029MR916179
- [10] S.M. Gersten, Fixed points of automorphisms of free groups. Adv. Math.64 (1987) 51–85. Zbl0616.20014MR879856
- [11] R.Z. Goldstein and E.C. Turner, Monomorphisms of finitely generated free groups have finitely generated equalizers. Invent. Math.82 (1985) 283–289. Zbl0582.20023MR809716
- [12] R.Z. Goldstein and E.C. Turner, Fixed subgroups of homomorphisms of free groups. Bull. Lond. Math. Soc.18 (1986) 468–470. Zbl0576.20016MR847985
- [13] D. Hamm and J. Shallit, Characterization of finite and one-sided infinite fixed points of morphisms on free monoids. Technical Report CS-99-17 (1999).
- [14] T. Head, Fixed languages and the adult languages of 0L schemes. Int. J. Comput. Math.10 (1981) 103–107. Zbl0472.68034MR645626
- [15] S. Lyapin, Semigroups. Fizmatgiz. Moscow (1960). English translation by Am. Math. Soc. (1974). Zbl0100.02301
- [16] O.S. Maslakova, 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. Zbl1031.20014MR2017513
- [17] M. Petrich and P.V. Silva, On directly infinite rings. Acta Math. Hung.85 (1999) 153–165. Zbl0991.16024MR1713097
- [18] J. Sakarovitch, Éléments de Théorie des Automates. Vuibert, Paris (2003). Zbl1178.68002
- [19] P.V. Silva, Rational subsets of partially reversible monoids. Theoret. Comput. Sci.409 (2008) 537–548. Zbl1155.68047MR2473928
- [20] P.V. Silva, Fixed points of endomorphisms over special confluent rewriting systems. Monatsh. Math.161 (2010) 417–447. Zbl1236.20063MR2734969
- [21] M. Sykiotis, Fixed subgroups of endomorphisms of free products. J. Algebra315 (2007) 274–278. Zbl1130.20032MR2344345
- [22] E. Ventura, Fixed subgroups of free groups: a survey. Contemp. Math.296 (2002) 231–255. Zbl1025.20012MR1922276

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