Rees congruence algebras in classes of unars and algebras with operators

V. L. Usoltsev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 1, pp. 219-235

Résumé

In the paper, some kinds of algebras associated with the concept of a Rees congruence is considered. A congruence $\theta$ of a universal algebra $A$ is called a Rees congruence if there exists a subalgebra $B$ of the algebra $A$ such that $\theta$ is the set-theoretical union of $B^2$ and the identity relation on $A$. An algebra $A$ is called a Rees congruence algebra if any congruence of the algebra $A$ is a Rees congruence. A non-one-element algebra is called a Rees simple algebra if all its Rees congruences are trivial. An algebra with one unary operation is called a unar. An algebra with operators is a universal algebra with an additional system of operators, i.e., of unary operations acting as endomorphisms relative to operations from the basic signature. The description of Rees congruence algebras and of Rees simple algebras in the class of unars is obtained. A necessary condition to be a Rees congruence algebra for algebras $\langle A, \Omega \cup \{f\} \rangle$ with operator $f$ and arbitrary basic signature $\Omega$ is found. It is shown that this necessary condition is not a sufficient condition in the general case. The description of Rees congruence algebras in some subclasses of the class of algebras with one operator and with a ternary basic operation is obtained.