# Article

 Title: A note on normal varieties of monounary algebras  (English) Author: Chajda, Ivan Author: Länger, Helmut Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 Volume: 52 Issue: 2 Year: 2002 Pages: 369-373 Summary lang: English . Category: math . Summary: 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 $L$. Keyword: monounary algebra Keyword: variety Keyword: normal variety Keyword: choice algebra MSC: 08A60 MSC: 08B15 idZBL: Zbl 1011.08006 idMR: MR1905444 . Date available: 2009-09-24T10:51:56Z Last updated: 2012-05-31 Stable URL: http://hdl.handle.net/10338.dmlcz/127725 . Reference: [1] I. Chajda: Normally presented varieties.Algebra Universalis 34 (1995), 327–335. Zbl 0842.08007, MR 1350845 Reference: [2] E. Graczyńska: On normal and regular identities.Algebra Universalis 27 (1990), 387–397. MR 1058483 Reference: [3] E. Jacobs and R. Schwabauer: The lattice of equational classes of algebras with one unary operation.Am. Math. Monthly 71 (1964), 151–155. MR 0162740 Reference: [4] D. Jakubíková-Studenovská: Endomorphisms and connected components of partial monounary algebras.Czechoslovak Math. J. 35 (1985), 467–490. MR 0803041 Reference: [5] D. Jakubíková-Studenovská: On completions of partial monounary algebras.Czechoslovak Math. J. 38 (1988), 256–268. MR 0946294 Reference: [6] O. Kopeček and M. Novotný: On some invariants of unary algebras.Czechoslovak Math. J. 24 (1974), 219–246. MR 0347703 Reference: [7] I. I. Mel’nik: Nilpotent shifts of varieties.Math. Notes (New York) 14 (1973), 692–696. MR 0366782 Reference: [8] M. Novotný: Über Abbildungen von Mengen.Pacific J. Math. 13 (1963), 1359–1369. MR 0157143 .

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