Preview

Chebyshevskii Sbornik

Advanced search

ON HAMILTONIAN TERNARY ALGEBRAS WITH OPERATORS

https://doi.org/10.22405/2226-8383-2014-15-3-100-113

Abstract

In this work is given the description of Hamiltonian algebras in some subclasses of class of algebras with operators having one ternary basic operation and one operator. Universal algebra A is a Hamiltonian algebra if every subuniverse of A is the block of some congruence of the algebra A. Algebra with operators is an universal algebra with additional system of the unary operations acting as endomorphisms with respect to basic operations. These operations are called permutable with basic operations. An algebra with operators is ternary if it has exactly one basic operation and this operation is ternary. It is obtained the sufficient condition of Hamiltonity for arbitrary universal algebras with operators. It is described Hamiltonian algebras in classes of ternary algebras with one operator and with basic operation that is either Pixley operation, or minority function, or majority function of special view. Let V be a variety of algebras with operators and V has signature Ω1 ∪Ω2, where Ω1 is an arbitrary signature containing near-unanimity function and Ω2 is a set of operators. It is proved that V not contains nontrivial Abelian algebras.

About the Author

V. L. Usol’tsev
Volgograd State Social and Pedagogical University
Russian Federation


References

1. Cs´ak´any B. Abelian properties of primitive classes of universal algebras // Acta. Sci. Math. 1964. Vol. 25. P. 202–208.

2. Shoda K. Zur theorie der algebraischen erweiterungen // Osaka Math. Journal. 1952. Vol. 4. P. 133–143.

3. McKenzie R. Congruence extencion, Hamiltonian and Abelian properties in locally finite varieties // Algebra Universalis. 1991. N 28. P. 589–603.

4. Valeriot M. Finite simple Abelian algebras are strictly simple // Proc. of the Amer. Math. Soc. 1990. N 108. P. 49–57.

5. Kiss E., Valeriot M. Abelian algebras and the Hamiltonian property // J. Pure Appl. Algebra. 1993. Vol. 87. N 1. P. 37–49.

6. Klukovits L. Hamiltonian varieties of universal algebras // Acta. Sci. Math. 1975. Vol. 37. P. 11–15.

7. Freese R., McKenzie R. Commutator theory for congruence modular varieties. London, 1987.

8. Werner H. Congruences on products of algebras and functionally complete algebras // Algebra Universalis. 1974. Vol. 4. N 1. P.99–105.

9. Hagemann, J., Herrmann C. Arithmetically locally equational classes and representation of partial functions // Universal algebra, Estergom (Hungary). Vol. 29. Colloq. Math. Soc. Janos Bolyai. 1982. P. 345-360.

10. Хобби Д., Маккензи Р. Строение конечных алгебр. М.: Мир, 1993. 287 с.

11. Kiss E., Valeriot M. Strongly Abelian varieties and the Hamiltonian property // The Canadian J. of Mathematics. 1991. V. 43. P. 331–346.

12. Степанова А.А., Трикашная Н.В. Абелевы и гамильтоновы группоиды // Фундам. и приклад. математика. 2009. Т. 15, вып. 7. C. 165–177.

13. Курош А.Г. Общая алгебра. Лекции 1969-1970 учебного года. М.: Наука, 1974. 160 с.

14. Kilp M., Knauer U., Mikhalev A.V. Monoids, Acts and Categories with Applications to Wreath Products and Graphs. Berlin: Walter de Gruyter, 2000. 529 p.

15. Skornyakov L.A. Unars // Colloq. Math. Soc. J. Bolyai. 1982. Vol. 29. Universal Algebra (Esztergom 1977). P. 735–743.

16. Wenzel G.H. Subdirect irreducibility and equational compactness in unary algebras ⟨A; f⟩ // Arch. Math. (Basel) 21. 1970. P. 256–264.

17. Pixley A.F. Distributivity and permutability of congruence relations in equational classes of algebras // Proc. Amer. Math. Soc. 1963. Vol. 14. N 1. P.105– 109.

18. Mar´oti M., McKenzie R. Existence theorems for weakly symmetric operations // Algebra Universalis. 2008. Vol. 59. N 3-4. P.463–489.

19. Markovi´c P., McKenzie R. Few subpowers, congruence distributivity and nearunanimity terms // Algebra Universalis, 2008. Vol. 58, P. 119–128.

20. Усольцев В.Л. О подпрямо неразложимых унарах с мальцевской операцией // Изв. Волгоградского гос. пед. ун-та. Сер. "Естеств. и физ.-мат. науки". 2005. N4(13). С. 17–24.

21. Усольцев В. Л. Простые и псевдопростые алгебры с операторами // Фундам. и приклад. математика. 2008. Т. 14, вып. 7. С. 189–207.

22. Карташов В.К. Об унарах с мальцевской операцией // Универсальная алгебра и ее приложения: тез. докл. межд. семинара, посв. памяти проф. Л. А. Скорнякова. Волгоград, 1999. С. 31–32.

23. Усольцев В.Л. Свободные алгебры многообразия унаров с мальцевской операцией p, заданного тождеством p(x, y, x) = y // Чебышевский сб. 2011. Т. 12. Вып. 2(38). С. 127–134.

24. Усольцев В.Л. О полиномиально полных и абелевых унарах с мальцевской операцией // Уч. зап. Орловского гос. ун-та. 2012. Т. 6(50). Ч. 2. С. 229–236.

25. Hagemann J., Herrmann C. A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity // Arch. d. Math. 1979. Vol. 32. N 3. P.234–245.


Review

For citations:


Usol’tsev V.L. ON HAMILTONIAN TERNARY ALGEBRAS WITH OPERATORS. Chebyshevskii Sbornik. 2014;15(3):100-113. (In Russ.) https://doi.org/10.22405/2226-8383-2014-15-3-100-113

Views: 463


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 2226-8383 (Print)