On congruence lattices of algebras with an operator and the symmetric main operation

V. L. Usoltsev

Čebyševskij sbornik, Tome 25 (2024) no. 1, pp. 103-115

Résumé

In this paper we study properties of congruence lattices 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 a chain algebra if $A$ has a linearly ordered congruence lattice. 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 part which can contain arbitrary operations, and the additional part consisting of operators. The operators are unary operations that act as endomorphisms with respect to the main operations, i.e., one are permutable with the main operations. An unar is an algebra with one unary operation. If $f$ is the unary operation from the signature $\Omega$ then the unar $\langle A, f\rangle$ is called an unary reduct of algebra $\langle A, \Omega\rangle$. A description of algebras with one operator and the main symmetric operation that have a linear ordered congruence lattice is obtained. It shown that algebra of given class is a chain algebra if and only if one is subdirectly irreducible. For algebras of given class we obtained necessary and sufficient conditions for the coincidence of their congruence lattices and congruence lattices of unary reducts these algebras.

Détail

Keywords: congruence lattice, algebra with operators, unary reduct of algebra, chain algebra, subdirectly irreducible algebra.