On Rees closure in some classes of algebras with an operator
https://doi.org/10.22405/2226-8383-2021-22-2-271-287
Abstract
In this paper we introduce the concept of Rees closure for subalgebras of universal algebras.
We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a
Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains
𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ on the lattice of subalgebras of universal algebra. Consequently, in the general case, a lattice
of Rees subalgebras is not a sublattice of lattice of subalgebras.
A non-one-element universal algebra 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 is trivial. We characterize Rees simple algebras in terms of Rees closure.
Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. We described Rees simple algebras in some subclasses of the class of algebras with one operator and a ternary basic operation. For algebras from these classes, the structure of lattice of Rees subalgebras is described. Necessary and sufficient conditions for the lattice of Rees subalgebras of algebras from these classes to be a chain are obtained.
About the Author
Vadim Leonidovich Usol’tsev
Volgograd State Social-Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
References
1. Pinus, A. G. 2014, "Hamiltonian closure on universal algebras" , Sibirskiy Matematicheskiy Zhurnal, vol. 55, no. 3(325), pp. 610–616 (Russian); translated in Siberian Mathematical Journal, vol. 55, no. 3(325), pp. 498–502. DOI: 10.1134/S0037446614030112
2. Pinus, A. G. 2014, "The classical Galois closure for universal algebras" , Izvestiya vysshikh uchebnykh zavedeniy. Matematika, no. 2, pp. 47–53 (Russian); translated in Russian Mathematics, vol. 58, no. 2, pp. 39–44. DOI: 10.3103/S1066369X14020066
3. Pinus, A. G. 2015, "On some of logical closures on universal algebras" , Sibirskie Elektronnye Matematicheskie Izvestiya (Siberian Electronic Mathematical Reports), vol. 12, pp. 698–703
4. (Russian). DOI: 10.17377/semi.2015.12.055
5. P¨oschel, R. 2004, "Galois connections for operations and relations" , pp. 231–258. In: Denecke, K., Ern´e, M. & Wismath, S. L. (Eds). 2004, "Galois connections and applications" , Kluwer
6. academic publishers, Dordrecht; Boston, 501 p.
7. Cs´ak´any, B. 1964, "Abelian properties of primitive classes of universal algebras" , Acta Scientiarum Mathematicarum, vol. 25, pp. 202–208. (Russian)
8. Shoda, K. 1952, "Zur theorie der algebraischen erweiterungen" , Osaka Mathematical Journal, vol. 4, pp. 133–143.
9. Tichy, R. F. 1981, "The Rees congruences in universal algebras" , Publications de l’Institut Mathematique (Beograd), vol. 29, pp. 229–239.
10. Chajda, I., Eigenthaler, G. & Langer, H. 2003, "Congruence classes in universal algebra" , Heldermann Verlag, Vienna, 192 pp.
11. Chajda, I. & Duda, J. 1985, "Rees algebras and their varieties" , Publicationes Mathematicae (Debrecen), vol. 32, pp. 17–22.
12. ˇSeˇselja, B. & Tepavˇcevi´c, A. 1995, "On a characterization of Rees varieties" , Tatra Mountains Mathematical Publications, vol. 5, pp. 61–69.
13. Chajda, I. 1997, "Rees ideal algebras" , Mathematica Bohemica, vol. 122, no. 2, pp. 125–130.
14. Gumm, P. H. & Ursini, A. 1984, "Ideals in universal algebras" , Algebra Universalis, vol. 19, pp. 45–54.
15. Sz´asz, G. 1968, "Rees factor lattices" , Publicationes Mathematicae (Debrecen), vol. 15, pp. 259– 266.
16. Usol’tsev, V. L. 2016, "Rees algebras and Rees congruence algebras of one class of algebras with operator and basic near-unanimity operation" , Chebyshevskii Sbornik, vol. 17, issue 4(60), pp. 157–166. (Russian)
17. Usol’tsev, V. L. 2018, "Rees congruence algebras in classes of unars and algebras with operators" , Mezhdunarodnaya algebraicheskaya conferentsia posvyashchennaya 110-letiyu so
18. dnya rozhdeniya professora A. G. Kurosha. Tezisy dokladov (International Algebraic Conference dedicated to the 110th anniversary of the birth of Professor A. G. Kurosh. Abstracts), Moscow State University Publishing House, Moscow, pp. 199–201. (Russian)
19. Kartashov, V. K. 1999, "On unars with Mal’tsev operation" , Universal’naya algebra i ee prilozheniya: Tezisy soobshcheniy uchastnikov mezhdunarodnogo seminara, posvyashchennogo
20. pamyati prof. Mosk. gos. un-ta L.A. Skornyakova (Universal algebra and application: theses of International workshop dedicated memory of prof. L.A. Skornyakov), Volgograd, pp. 31–32. (Russian)
21. Usol’tsev, V. L. 2008, "Unars with ternary Mal’tsev operation" , Uspekhi matematicheskikh nauk, vol. 63, no. 5, pp. 201–202 (Russian); translated in Russian mathematical surveys, 2008,
22. vol. 63, no. 5, pp. 986-988. DOI: 10.1070/RM2008v063n05ABEH004572
23. Usoltsev, V. L. 2010, "Simple and pseudosimple algebras with operators" , Journal of Mathematical Sciences, 2010, vol. 164, no. 2, pp. 281-293. DOI: 10.1007/S1095800997306
24. Usol’tsev, V. L. 2014, "On Hamiltonian ternary algebras with operators" , Chebyshevskii Sbornik, vol. 15, issue 3(51), pp. 100–113. (Russian)
25. Usol’tsev, V. L. 2015, "On Hamiltonian closure on class of algebras with one operator" , Chebyshevskii Sbornik, vol. 16, issue 4(56), pp. 284–302. (Russian)
26. Usol’tsev, V. L. 2012, "On polynomially complete and Abelian unars with Mal’tsev operation" , Uchenye Zapiski Orlovskogo Gosudarstvennogo Universiteta, vol. 6(50), part 2, pp. 229–236.
27. (Russian)
28. Usol’tsev, V. L. 2005, "On subdirect irreducible unars with Mal’tsev operation" , Izvestiya VGPU. Seriya "Estestvennye i fiziko-matematicheskie nauki", no. 4(13), pp. 17-24. (Russian)
29. Lata, A. N. 2017, "On congruence coherent Rees algebras and algebras with an operator" , Chebyshevskii Sbornik, vol. 18, issue 2(62), pp. 154–172. (Russian)
30. Usol’tsev, V. L. 2011, "Free algebras in the variety of unars with Mal’tsev operation that defined by identity 𝑝(𝑥, 𝑦, 𝑥) = 𝑦" , Chebyshevskii Sbornik, vol. 12, issue 2(38), pp. 127–134. (Russian)
31. Szendrei, A. 1986, "Clones in universal algebra" , Les presses de l’Universit´e de Montr´eal, Montr´eal, 166 pp.
32. Baker, K. A. & Pixley, A. 1975, "Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems" , Mathematische Zeitschrift, vol. 143, pp. 165–174. DOI: 10.1007/
33. BF01187059
34. Markovi´c, P. & McKenzie, R. 2008, "Few subpowers, congruence distributivity and nearunanimity terms" , Algebra Universalis, vol. 58, pp. 119–128. DOI: 10.1007/s00012-008-2049-1
35. Usol’tsev, V. L. 2013, "On strictly simple ternary algebras with operators" , Chebyshevskii Sbornik, vol. 14, issue 4(48), pp. 196–204. (Russian)
36. Usol’tsev, V. L. 2016, "On congruence lattices of algebras with one operator and basic nearunanimity operation" , Nauchno-tekhnicheskiy vestnik Povolzhya, issue 2, pp. 28–30. (Russian)
Review
For citations:
Usol’tsev V.L. On Rees closure in some classes of algebras with an operator. Chebyshevskii Sbornik. 2021;22(2):271-287. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-2-271-287
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