Simple and pseudosimple algebras with operators

In this work, algebras with operators, i.e., algebras with an additional system of unary operations acting as endomorphisms with respect to the basic operations, are considered. We find necessary conditions for the simplicity and pseudosimplicity of an arbitrary universal algebra with a fixed operator and explore how several properties of congruences of this algebra depend on the structure of the unary reduct of a given algebra. For algebras with one ternary operation, defined in the standard way and satisfying Mal'cev's identities, and one operator (i.e., unars with Mal'cev's operation), necessary and sufficient conditions for their simplicity and pseudosimplicity are obtained.