Geodesic
A note on normal varieties of monounary algebras
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 2, pp. 369-373
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $(A,f)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form ${\mathrm Mod}(f^n(x)=x)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in ${\mathrm HSC}({\mathrm Mod}(f^{mn}(x)=x))$ for every $m>1$ where ${\mathrm C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of $L$ is isomorphic to $L$.
A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $(A,f)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form ${\mathrm Mod}(f^n(x)=x)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in ${\mathrm HSC}({\mathrm Mod}(f^{mn}(x)=x))$ for every $m>1$ where ${\mathrm C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of $L$ is isomorphic to $L$.
Classification : 08A60, 08B15
Keywords: monounary algebra; variety; normal variety; choice algebra 
@article{CMJ_2002_52_2_a12,
     author = {Chajda, Ivan and L\"anger, Helmut},
     title = {A note on normal varieties of monounary algebras},
     journal = {Czechoslovak Mathematical Journal},
     pages = {369--373},
     year = {2002},
     volume = {52},
     number = {2},
     mrnumber = {1905444},
     zbl = {1011.08006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2002_52_2_a12/}
}
TY  - JOUR
AU  - Chajda, Ivan
AU  - Länger, Helmut
TI  - A note on normal varieties of monounary algebras
JO  - Czechoslovak Mathematical Journal
PY  - 2002
SP  - 369
EP  - 373
VL  - 52
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/CMJ_2002_52_2_a12/
LA  - en
ID  - CMJ_2002_52_2_a12
ER  - 
%0 Journal Article
%A Chajda, Ivan
%A Länger, Helmut
%T A note on normal varieties of monounary algebras
%J Czechoslovak Mathematical Journal
%D 2002
%P 369-373
%V 52
%N 2
%U http://geodesic.mathdoc.fr/item/CMJ_2002_52_2_a12/
%G en
%F CMJ_2002_52_2_a12
Chajda, Ivan; Länger, Helmut. A note on normal varieties of monounary algebras. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 2, pp. 369-373. http://geodesic.mathdoc.fr/item/CMJ_2002_52_2_a12/

[1] I. Chajda: Normally presented varieties. Algebra Universalis 34 (1995), 327–335. | DOI | MR | Zbl

[2] E. Graczyńska: On normal and regular identities. Algebra Universalis 27 (1990), 387–397. | DOI | MR

[3] E. Jacobs and R. Schwabauer: The lattice of equational classes of algebras with one unary operation. Am. Math. Monthly 71 (1964), 151–155. | DOI | MR

[4] D. Jakubíková-Studenovská: Endomorphisms and connected components of partial monounary algebras. Czechoslovak Math. J. 35 (1985), 467–490. | MR

[5] D. Jakubíková-Studenovská: On completions of partial monounary algebras. Czechoslovak Math. J. 38 (1988), 256–268. | MR

[6] O. Kopeček and M. Novotný: On some invariants of unary algebras. Czechoslovak Math. J. 24 (1974), 219–246. | MR

[7] I. I. Mel’nik: Nilpotent shifts of varieties. Math. Notes (New York) 14 (1973), 692–696. | MR

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