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.