On Hamiltonian closure on class of algebras with one operator

V. L. Usol'tsev

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 284-302

Résumé

In this article we study Hamiltonian simple algebras and lattices of Hamiltonian closed subalgebras in class of algebras with one operator. Obtained for algebras with arbitrary basic signature results are used for the description of Hamiltonian simple algebras and lattices of Hamiltonian closed subalgebras from class of unars with Mal'tsev operation that by V. K. Kartashov were defined. Unar with Mal'tsev operation is an algebra with one Mal'tsev operation $p(x,y,z)$ and one unary operation acting as endomorphism with respect to operation $p(x,y,z)$. Universal algebra $A$ is called Hamiltonian if every subuniverse of $A$ is a block of some congruence of the algebra $A$. A. G. Pinus defined a Hamiltonian closure on an arbitrary universal algebra. Precisely, the Hamiltonian closure $\overline{B}$ of a subalgebra $B$ of a universal algebra $A$ is the smallest subalgebra of algebra $A$ containing $B$ that coincides with some block of some congruence on algebra $A$. Subalgebra $B$ of universal algebra $A$ is called Hamiltonian closed if $\overline{B} = B$. Set of all Hamiltonian closed subalgebras of algebra $A$ with added empty set is lattice with respect to inclusion. A universal algebra $A$ is called a Hamiltonian simple algebra if $\overline{B} = A$ for each non-empty and non-one-element subalgebra $B$ of $A$. We found necessary conditions of Hamiltonian simplicity for arbitrary algebras with one operator and idempotent basic operations of positive arity. For these algebras families of their subalgebras forming chains with respect to inclusion in their lattices of Hamiltonian closed subalgebras are constructed. We also found necessary conditions of Hamiltonian simplicity for arbitrary algebras with one operator and with connected unary reduct. It is showed these conditions are not sufficient. For arbitrary algebras with one operator and idempotent basic operations necessary conditions of their lattice of Hamiltonian closed subalgebras is chain are obtained. We found necessary and sufficient conditions of Hamiltonian simplicity for unars with Mal'tsev operation that by V. K. Kartashov were defined. The structure of lattices of Hamiltonian closed subalgebras for algebras from this class is described. For these lattices necessary and sufficient conditions of their distributivity and modularity are obtained. We also found necessary and sufficient conditions when a lattice of Hamiltonian closed subalgebras of algebras from given class is a chain. The structure of atoms and coatoms of such lattices is described. Bibliography: 22 titles.

Détail

Keywords: Hamiltonian closure of a subalgebra, Hamiltonian simple algebra, lattices of Hamiltonian closed subalgebras, algebra with operators, Mal'tsev operation.