Geodesic
Test elements and the Retract Theorem for monounary algebras
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 975-986 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The term “Retract Theorem” has been applied in literature in connection with group theory. In the present paper we prove that the Retract Theorem is valid (i) for each finite structure, and (ii) for each monounary algebra. On the other hand, we show that this theorem fails to be valid, in general, for algebras of the form $\mathcal{A}=(A,F)$, where each $f\in F$ is unary and $\operatorname{card}F >1$.
The term “Retract Theorem” has been applied in literature in connection with group theory. In the present paper we prove that the Retract Theorem is valid (i) for each finite structure, and (ii) for each monounary algebra. On the other hand, we show that this theorem fails to be valid, in general, for algebras of the form $\mathcal{A}=(A,F)$, where each $f\in F$ is unary and $\operatorname{card}F >1$.
Classification : 08A60
Keywords: monounary algebra; retract; test element 
@article{CMJ_2007_57_3_a14,
     author = {Jakub{\'\i}kov\'a-Studenovsk\'a, Danica and P\'ocs, Jozef},
     title = {Test elements and the {Retract} {Theorem} for monounary algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {975--986},
     year = {2007},
     volume = {57},
     number = {3},
     mrnumber = {2356934},
     zbl = {1174.08302},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a14/}
}
TY  - JOUR
AU  - Jakubíková-Studenovská, Danica
AU  - Pócs, Jozef
TI  - Test elements and the Retract Theorem for monounary algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2007
SP  - 975
EP  - 986
VL  - 57
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a14/
LA  - en
ID  - CMJ_2007_57_3_a14
ER  - 
%0 Journal Article
%A Jakubíková-Studenovská, Danica
%A Pócs, Jozef
%T Test elements and the Retract Theorem for monounary algebras
%J Czechoslovak Mathematical Journal
%D 2007
%P 975-986
%V 57
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a14/
%G en
%F CMJ_2007_57_3_a14
Jakubíková-Studenovská, Danica; Pócs, Jozef. Test elements and the Retract Theorem for monounary algebras. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 975-986. http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a14/

[1] D. Jakubíková-Studenovská: Retract irreducibility of connected monounary algebras I. Czech. Math. J. 46 (1996), 291–308. | MR

[2] B. Jónsson: Topics in universal algebra. Springer-Verlag, Berlin, Heidelberg, New York, 1972. | MR

[3] E. Nelson: Homomorphism of mono-unary algebras. Pacif. J. Math. 99, 2 (1982), 427–429. | DOI

[4] J. C. O’Neill and E. C. Turner: Test elements and the Retract Theorem in Hyperbolic groups. New York J. of Math. 6 (2000), 107–117. | MR

[5] J. Nielsen: Die Automorphismen der allgemeiner unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 78 (1918), 385–397.

[6] M. Novotný: Über Abbildungen von Mengen. Pacif. J. Math. 13 (1963), 1359–1369. | MR

[7] M. Novotný: Mono-unary algebras in the work of Czechoslovak mathematicians. Arch. Math. (Brno) 26 (1990), 155–164. | MR

[8] E. C. Turner: Test words for automorphisms of free groups. Bull. London Math. Soc. 28 (1996), 255–263. | DOI | MR | Zbl

[9] D. A. Voce: Test words and stable image of an endomorphism. PhD Thesis, Univ. at Albany, 1995.