# Article

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Keywords:
monounary algebra; variety; normal variety; choice algebra
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 $L$.
References:
[1] I. Chajda: Normally presented varieties. Algebra Universalis 34 (1995), 327–335. MR 1350845 | Zbl 0842.08007
[2] E. Graczyńska: On normal and regular identities. Algebra Universalis 27 (1990), 387–397. MR 1058483
[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
[4] D. Jakubíková-Studenovská: Endomorphisms and connected components of partial monounary algebras. Czechoslovak Math. J. 35 (1985), 467–490. MR 0803041
[5] D. Jakubíková-Studenovská: On completions of partial monounary algebras. Czechoslovak Math. J. 38 (1988), 256–268. MR 0946294
[6] O. Kopeček and M. Novotný: On some invariants of unary algebras. Czechoslovak Math. J. 24 (1974), 219–246. MR 0347703
[7] I. I. Mel’nik: Nilpotent shifts of varieties. Math. Notes (New York) 14 (1973), 692–696. MR 0366782
[8] M. Novotný: Über Abbildungen von Mengen. Pacific J. Math. 13 (1963), 1359–1369. MR 0157143

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