| Title:
|
Congruence preserving operations on the ring $\mathbb {Z}_{p^3}$ (English) |
| Author:
|
Gavala, Cyril |
| Author:
|
Ploščica, Miroslav |
| Author:
|
Varga, Ivana |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
148 |
| Issue:
|
4 |
| Year:
|
2023 |
| Pages:
|
519-535 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the interval $I(p^3)$ in the lattice of clones on the ring $\mathbb {Z}_{p^3}$ between the clone of polynomial operations and the clone of congruence preserving operations. All clones in this interval are known and described by means of generators. In this paper, we characterize each of these clones by the property of preserving a small set of relations. These relations turn out to be in a close connection to commutators. (English) |
| Keyword:
|
congruence |
| Keyword:
|
clone |
| Keyword:
|
polynomial |
| MSC:
|
03B50 |
| MSC:
|
08A40 |
| idZBL:
|
Zbl 07790601 |
| idMR:
|
MR4673835 |
| DOI:
|
10.21136/MB.2022.0155-21 |
| . |
| Date available:
|
2023-11-23T12:37:22Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151972 |
| . |
| Reference:
|
[1] Aichinger, E., Mayr, P.: Polynomial clones on groups of order $pq$.Acta Math. Hung. 114 (2007), 267-285. Zbl 1121.08004, MR 2296547, 10.1007/s10474-006-0530-x |
| Reference:
|
[2] Aichinger, E., Mudrinski, N.: Some applications of higher commutators in Mal'cev algebras.Algebra Univers. 63 (2010), 367-403. Zbl 1206.08003, MR 2734303, 10.1007/s00012-010-0084-1 |
| Reference:
|
[3] Bulatov, A. A.: Polynomial reducts of modules I. Rough classification.Mult.-Valued Log. 3 (1998), 135-154. Zbl 0909.08003, MR 1447082 |
| Reference:
|
[4] Bulatov, A. A.: Polynomial reducts of modules II. Algebras of primitive and nilpotent functions.Mult.-Valued Log. 3 (1998), 173-193. Zbl 0923.08002, MR 1447082 |
| Reference:
|
[5] Bulatov, A. A.: On the number of finite Mal'tsev algebras.Contributions to General Algebra 13 Johannes Heyn, Klagenfurt (2001), 41-54. Zbl 0986.08003, MR 1854568 |
| Reference:
|
[6] Bulatov, A. A.: Polynomial clones containing the Mal'tsev operation of the groups $\Bbb{Z}_{p^2}$ and $\Bbb{Z}_p \times \Bbb{Z}_p$.Mult.-Valued Log. 8 (2002), 193-221. Zbl 1022.08001, MR 1957653, 10.1080/10236620215291 |
| Reference:
|
[7] Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties.London Mathematical Society Lecture Note Series 125. Cambridge University Press, Cambridge (1987). Zbl 0636.08001, MR 0909290 |
| Reference:
|
[8] Gavala, C.: Compatible Operations on Rings of Integers Modulo $n$: Master Thesis.Šafárik University Košice, Košice (2016), Slovak. |
| Reference:
|
[9] Gavrilov, G. P.: On the superstructure of the class of polynomials in multivalued logics.Discrete Math. Appl. 6 (1996), 405-412 translation from Diskretn. Mat. 8 1996 90-97. Zbl 0863.03010, MR 1422350, 10.1515/dma.1996.6.4.405 |
| Reference:
|
[10] Gavrilov, G. P.: On the closed classes of multivalued logic containing the polynomial class.Discrete Math. Appl. 7 (1997), 231-242 translation from Diskretn. Mat. 9 1997 12-23. Zbl 0965.03029, MR 1468067, 10.1515/dma.1997.7.3.231 |
| Reference:
|
[11] Idziak, P. M.: Clones containing Mal'tsev operations.Int. J. Algebra Comput. 9 (1999), 213-226. Zbl 1023.08003, MR 1703074, 10.1142/S021819679900014X |
| Reference:
|
[12] Mayr, P.: Polynomial clones on squarefree groups.Int. J. Algebra Comput. 18 (2008), 759-777. Zbl 1147.08003, MR 2428154, 10.1142/S0218196708004597 |
| Reference:
|
[13] Meshchaninov, D. G.: Superstructures of the class of polynomials in $P_k$.Math. Notes 44 (1988), 850-854 translation from Mat. Zametki 44 1988 673-681. Zbl 0669.03012, MR 0980588, 10.1007/BF01158427 |
| Reference:
|
[14] Opršal, J.: A relational description of higher commutators in Mal'cev varieties.Algebra Univers. 76 (2016), 367-383. Zbl 1357.08002, MR 3556818, 10.1007/s00012-016-0391-2 |
| Reference:
|
[15] Ploščica, M., Varga, I.: Clones of compatible operations on rings $\Bbb{Z}_{p^k}$.J. Mult.-Val. Log. Soft Comput. 36 (2021), 391-404. Zbl 07536110, MR 4578809 |
| Reference:
|
[16] Remizov, A. B.: Superstructure of the closed class of polynomials modulo $k$.Discrete Math. Appl. 1 (1991), 9-22 translation from Diskretn. Mat. 1 1989 3-15. Zbl 0726.03014, MR 1072635, 10.1515/dma.1991.1.1.9 |
| Reference:
|
[17] Salomaa, A.: On infinitely generated sets of operations in finite algebras.Ann. Univ. Turku., Ser. A I 74 (1964), 13 pages. Zbl 0123.00503, MR 0169781 |
| Reference:
|
[18] Shaw, J.: Commutator relations and the clones of finite groups.Algebra Univers. 72 (2014), 29-52. Zbl 1309.08003, MR 3229950, 10.1007/s00012-014-0287-y |
| Reference:
|
[19] Szendrei, Á.: Idempotent reducts of abelian groups.Acta Sci. Math. 38 (1976), 171-182. Zbl 0307.20032, MR 0422118 |
| Reference:
|
[20] Szendrei, Á.: Clones of linear operations on finite sets.Finite Algebra and Multiple-Valued Logic Colloquia Mathematica Societatis János Bolyai 28. North-Holland, Amsterdam (1981), 693-738. Zbl 0487.08002, MR 0648640 |
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