The subdirect irreducibility and the atoms of congruence lattices of algebras with one operator and the symmetric main operation

V. L. Usoltsev

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 257-270

Résumé

In that paper we study atoms of congruence lattices and subdirectly irreducibility of algebras with one operator and the main symmetric operation. A ternary operation $d(x,y,z)$ satisfying identities $d(x, y, y) = d(y, y, x) = d(y, x, y) = x$ is called a minority operation. The symmetric operation is a minority operation defined by specific way. An algebra $A$ is called subdirectly irreducible if $A$ has the smallest nonzero congruence. An algebra with operators is an universal algebra whose signature consists of two nonempty non-intersectional parts: the main one which can contain arbitrary operations, and the additional one consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one that permutable with main operations. A lattice $L$ with zero is called atomic if any element of $L$ contains some atom. A lattice $L$ with zero is called atomistic if any nonzero element of $L$ is a join of some atom set. It shown that congruence lattices of algebras with one operator and main symmetric operation are atomic. The structure of atoms in the congruence lattices of algebras in given class is described. The full describe of subdirectly irreducible algebras and of algebras with an atomistic congruence lattice in given class is obtained.

Détail

Keywords: subdirectly irreducible algebra, congruence lattice, atom of congruence lattice, atomic lattice, algebra with operators.