Mathematica Bohemica, Vol. 148, No. 4, pp. 519-535, 2023
Congruence preserving operations on the ring $\mathbb{Z}_{p^3}$
Cyril Gavala, Miroslav Ploščica, Ivana Varga
Received October 11, 2021. Published online October 11, 2022.
Abstract: 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.
References: [1] E. Aichinger, P. Mayr: Polynomial clones on groups of order $pq$. Acta Math. Hung. 114 (2007), 267-285. DOI 10.1007/s10474-006-0530-x | MR 2296547 | Zbl 1121.08004
[2] E. Aichinger, N. Mudrinski: Some applications of higher commutators in Mal'cev algebras. Algebra Univers. 63 (2010), 367-403. DOI 10.1007/s00012-010-0084-1 | MR 2734303 | Zbl 1206.08003
[3] A. A. Bulatov: Polynomial reducts of modules I. Rough classification. Mult.-Valued Log. 3 (1998), 135-154. Zbl 0909.08003
[4] A. A. Bulatov: Polynomial reducts of modules II. Algebras of primitive and nilpotent functions. Mult.-Valued Log. 3 (1998), 173-193. Zbl 0923.08002
[5] A. A. Bulatov: On the number of finite Mal'tsev algebras. Contributions to General Algebra 13. Johannes Heyn, Klagenfurt (2001), 41-54. MR 1854568 | Zbl 0986.08003
[6] A. A. Bulatov: 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 10.1080/10236620215291 | MR 1957653 | Zbl 1022.08001
[7] R. Freese, R. McKenzie: Commutator Theory for Congruence Modular Varieties. London Mathematical Society Lecture Note Series 125. Cambridge University Press, Cambridge (1987). MR 0909290 | Zbl 0636.08001
[8] C. Gavala: Compatible Operations on Rings of Integers Modulo $n$: Master Thesis. Šafárik University Košice, Košice (2016). (In Slovak.)
[9] G. P. Gavrilov: 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 10.1515/dma.1996.6.4.405 | MR 1422350 | Zbl 0863.03010
[10] G. P. Gavrilov: 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 10.1515/dma.1997.7.3.231 | MR 1468067 | Zbl 0965.03029
[11] P. M. Idziak: Clones containing Mal'tsev operations. Int. J. Algebra Comput. 9 (1999), 213-226. DOI 10.1142/S021819679900014X | MR 1703074 | Zbl 1023.08003
[12] P. Mayr: Polynomial clones on squarefree groups. Int. J. Algebra Comput. 18 (2008), 759-777. DOI 10.1142/S0218196708004597 | MR 2428154 | Zbl 1147.08003
[13] D. G. Meshchaninov: Superstructures of the class of polynomials in $P_k$. Math. Notes 44 (1988), 850-854; translation from Mat. Zametki 44 (1988), 673-681. DOI 10.1007/BF01158427 | MR 0980588 | Zbl 0669.03012
[14] J. Opršal: A relational description of higher commutators in Mal'cev varieties. Algebra Univers. 76 (2016), 367-383. DOI 10.1007/s00012-016-0391-2 | MR 3556818 | Zbl 1357.08002
[15] M. Ploščica, I. Varga: Clones of compatible operations on rings $\Bbb{Z}_{p^k}$. J. Mult.-Val. Log. Soft Comput. 36 (2021), 391-404. Zbl 07536110
[16] A. B. Remizov: 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 10.1515/dma.1991.1.1.9 | MR 1072635 | Zbl 0726.03014
[17] A. Salomaa: On infinitely generated sets of operations in finite algebras. Ann. Univ. Turku., Ser. A I 74 (1964), 13 pages. MR 0169781 | Zbl 0123.00503
[18] J. Shaw: Commutator relations and the clones of finite groups. Algebra Univers. 72 (2014), 29-52. DOI 10.1007/s00012-014-0287-y | MR 3229950 | Zbl 1309.08003
[19] Á. Szendrei: Idempotent reducts of abelian groups. Acta Sci. Math. 38 (1976), 171-182. MR 0422118 | Zbl 0307.20032
[20] Á. Szendrei: 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 0648640 | Zbl 0487.08002
