| 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 (print) |
| ISSN:
|
1572-9141 (online) |
| 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 $L$. (English) |
| 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:
|
2020-07-03 |
| 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, 10.1007/BF01182089 |
| Reference:
|
[2] E. Graczyńska: On normal and regular identities.Algebra Universalis 27 (1990), 387–397. MR 1058483, 10.1007/BF01190718 |
| 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, 10.2307/2311743 |
| 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, 10.2140/pjm.1963.13.1359 |
| . |
