Geodesic
Congruence preserving operations on the ring $\mathbb {Z}_{p^3}$
Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 519-535
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/MB.2022.0155-21
Classification : 03B50, 08A40
Keywords: congruence; clone; polynomial 
@article{10_21136_MB_2022_0155_21,
     author = {Gavala, Cyril and Plo\v{s}\v{c}ica, Miroslav and Varga, Ivana},
     title = {Congruence preserving operations on the ring $\mathbb {Z}_{p^3}$},
     journal = {Mathematica Bohemica},
     pages = {519--535},
     year = {2023},
     volume = {148},
     number = {4},
     doi = {10.21136/MB.2022.0155-21},
     mrnumber = {4673835},
     zbl = {07790601},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2022.0155-21/}
}
TY  - JOUR
AU  - Gavala, Cyril
AU  - Ploščica, Miroslav
AU  - Varga, Ivana
TI  - Congruence preserving operations on the ring $\mathbb {Z}_{p^3}$
JO  - Mathematica Bohemica
PY  - 2023
SP  - 519
EP  - 535
VL  - 148
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2022.0155-21/
DO  - 10.21136/MB.2022.0155-21
LA  - en
ID  - 10_21136_MB_2022_0155_21
ER  - 
%0 Journal Article
%A Gavala, Cyril
%A Ploščica, Miroslav
%A Varga, Ivana
%T Congruence preserving operations on the ring $\mathbb {Z}_{p^3}$
%J Mathematica Bohemica
%D 2023
%P 519-535
%V 148
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2022.0155-21/
%R 10.21136/MB.2022.0155-21
%G en
%F 10_21136_MB_2022_0155_21
Gavala, Cyril; Ploščica, Miroslav; Varga, Ivana. Congruence preserving operations on the ring $\mathbb {Z}_{p^3}$. Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 519-535. doi: 10.21136/MB.2022.0155-21

[1] Aichinger, E., Mayr, P.: Polynomial clones on groups of order $pq$. Acta Math. Hung. 114 (2007), 267-285. | DOI | MR | JFM

[2] Aichinger, E., Mudrinski, N.: Some applications of higher commutators in Mal'cev algebras. Algebra Univers. 63 (2010), 367-403. | DOI | MR | JFM

[3] Bulatov, A. A.: Polynomial reducts of modules I. Rough classification. Mult.-Valued Log. 3 (1998), 135-154. | MR | JFM

[4] Bulatov, A. A.: Polynomial reducts of modules II. Algebras of primitive and nilpotent functions. Mult.-Valued Log. 3 (1998), 173-193. | MR | JFM

[5] Bulatov, A. A.: On the number of finite Mal'tsev algebras. Contributions to General Algebra 13 Johannes Heyn, Klagenfurt (2001), 41-54. | MR | JFM

[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. | DOI | MR | JFM

[7] Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties. London Mathematical Society Lecture Note Series 125. Cambridge University Press, Cambridge (1987). | MR | JFM

[8] Gavala, C.: Compatible Operations on Rings of Integers Modulo $n$: Master Thesis. Šafárik University Košice, Košice (2016), Slovak.

[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. | DOI | MR | JFM

[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. | DOI | MR | JFM

[11] Idziak, P. M.: Clones containing Mal'tsev operations. Int. J. Algebra Comput. 9 (1999), 213-226. | DOI | MR | JFM

[12] Mayr, P.: Polynomial clones on squarefree groups. Int. J. Algebra Comput. 18 (2008), 759-777. | DOI | MR | JFM

[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. | DOI | MR | JFM

[14] Opršal, J.: A relational description of higher commutators in Mal'cev varieties. Algebra Univers. 76 (2016), 367-383. | DOI | MR | JFM

[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. | MR | JFM

[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. | DOI | MR | JFM

[17] Salomaa, A.: On infinitely generated sets of operations in finite algebras. Ann. Univ. Turku., Ser. A I 74 (1964), 13 pages. | MR | JFM

[18] Shaw, J.: Commutator relations and the clones of finite groups. Algebra Univers. 72 (2014), 29-52. | DOI | MR | JFM

[19] Szendrei, Á.: Idempotent reducts of abelian groups. Acta Sci. Math. 38 (1976), 171-182. | MR | JFM

[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. | MR | JFM

Cité par Sources :