On Hamiltonian ternary algebras with operators

V. L. Usol'tsev

Čebyševskij sbornik, Tome 15 (2014) no. 3, pp. 100-113.

Résumé

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 $\Omega_1 \cup \Omega_2$, where $\Omega_1$ is an arbitrary signature containing near-unanimity function and $\Omega_2$ is a set of operators. It is proved that $V$ not contains nontrivial Abelian algebras.

Détail

Keywords: Hamiltonian algebra, Abelian algebra, algebra with operators, ternary operation, near-unanimity function.

Bibliographie

[1] Csákány B., “Abelian properties of primitive classes of universal algebras”, Acta. Sci. Math., 25 (1964), 202–208 | MR | Zbl

[2] Shoda K., “Zur theorie der algebraischen erweiterungen”, Osaka Math. Journal, 4 (1952), 133–143 | MR | Zbl

[3] McKenzie R., “Congruence extencion, Hamiltonian and Abelian properties in locally finite varieties”, Algebra Universalis, 28 (1991), 589–603 | DOI | MR | Zbl

[4] Valeriot M., “Finite simple Abelian algebras are strictly simple”, Proc. of the Amer. Math. Soc., 108 (1990), 49–57 | DOI | MR | Zbl

[5] Kiss E., Valeriot M., “Abelian algebras and the Hamiltonian property”, J. Pure Appl. Algebra, 87:1 (1993), 37–49 | DOI | MR | Zbl

[6] Klukovits L., “Hamiltonian varieties of universal algebras”, Acta. Sci. Math., 37 (1975), 11–15 | MR | Zbl

[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, 4:1 (1974), 99–105 | DOI | MR | Zbl

[9] Hagemann J., Herrmann C., “Arithmetically locally equational classes and representation of partial functions”, Universal algebra (Estergom, Hungary), Colloq. Math. Soc. Janos Bolyai, 29, 1982, 345–360 | MR | Zbl

[10] Khobbi D., Makkenzi R., Stroenie konechnykh algebr, Mir, M., 1993, 287 pp.

[11] Kiss E., Valeriot M., “Strongly Abelian varieties and the Hamiltonian property”, The Canadian J. of Mathematics, 43 (1991), 331–346 | DOI | MR | Zbl

[12] Stepanova A. A., Trikashnaya N. V., “Abelevy i gamiltonovy gruppoidy”, Fundam. i priklad. matematika, 15:7 (2009), 165–177 | Zbl

[13] Kurosh A. G., Obschaya algebra. Lektsii 1969–1970 uchebnogo goda, Nauka, M., 1974, 160 pp.

[14] Kilp M., Knauer U., Mikhalev A. V., Monoids, Acts and Categories with Applications to Wreath Products and Graphs, Walter de Gruyter, Berlin, 2000, 529 pp. | MR | Zbl

[15] Skornyakov L. A., “Unars”, Universal Algebra (Esztergom, 1977), Colloq. Math. Soc. J. Bolyai, 29, 1982, 735–743 | MR | Zbl

[16] Wenzel G. H., “Subdirect irreducibility and equational compactness in unary algebras $\langle A; f \rangle$”, Arch. Math. (Basel), 21 (1970), 256–264 | DOI | MR | Zbl

[17] Pixley A. F., “Distributivity and permutability of congruence relations in equational classes of algebras”, Proc. Amer. Math. Soc., 14:1 (1963), 105–109 | DOI | MR | Zbl

[18] Maróti M., McKenzie R., “Existence theorems for weakly symmetric operations”, Algebra Universalis, 59:3–4 (2008), 463–489 | DOI | MR | Zbl

[19] Marković P., McKenzie R., “Few subpowers, congruence distributivity and near-unanimity terms”, Algebra Universalis, 58 (2008), 119–128 | DOI | MR | Zbl

[20] Usoltsev V. L., “O podpryamo nerazlozhimykh unarakh s maltsevskoi operatsiei”, Izv. Volgogradskogo gos. ped. un-ta. Ser. “Estestv. i fiz.-mat. nauki”, 2005, no. 4(13), 17–24 | MR

[21] Usoltsev V. L., “Prostye i psevdoprostye algebry s operatorami”, Fundam. i priklad. matematika, 14:7 (2008), 189–207 | MR | Zbl

[22] Kartashov V. K., “Ob unarakh s maltsevskoi operatsiei”, Universalnaya algebra i ee prilozheniya, Tez. dokl. mezhd. seminara, posv. pamyati prof. L. A. Skornyakova (Volgograd, 1999), 31–32

[23] Usoltsev V. L., “Svobodnye algebry mnogoobraziya unarov s maltsevskoi operatsiei $p$, zadannogo tozhdestvom $p(x,y,x)=y$”, Chebyshevskii sb., 12:2(38) (2011), 127–134 | MR | Zbl

[24] Usoltsev V. L., “O polinomialno polnykh i abelevykh unarakh s maltsevskoi operatsiei”, Uch. zap. Orlovskogo gos. un-ta, 6(50):2 (2012), 229–236 | Zbl

[25] Hagemann J., Herrmann C., “A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity”, Arch. d. Math., 32:3 (1979), 234–245 | DOI | MR | Zbl