To join the seminar, connect to the Zoom meeting a few minutes prior to the beginning at (or manually by entering the meeting ID 812 2079 3393 in the Zoom app).Kleene iteration, or Kleene star, is one of the most intriguing algebraic operations appearing in theoretical computer science. Usually, the Kleene star is interpreted as the limit of n-th powers of an element. This is how it behaves in algebras of formal languages and in relational algebras. In the talk, we shall show some examples of structures with non-standard Kleene iteration, and apply them to proving some results for corresponding logical theories.