The aim of the present paper is to describe all connected monounary algebras for which there exists a representation by means of connected monounary algebras which are retract irreducible in the class ${\mathcal U}_c$ (or in ${\mathcal U} $).
For a subalgebra ${\mathcal B}$ of a partial monounary algebra ${\mathcal A}$ we define the quotient partial monounary algebra ${\mathcal A}/{\mathcal B}$. Let ${\mathcal B}$, ${\mathcal C}$ be partial monounary algebras. In this paper we give a construction of all partial monounary algebras ${\mathcal A}$ such that ${\mathcal B}$ is a subalgebra of ${\mathcal A}$ and ${\mathcal C}\cong {\mathcal A}/{\mathcal B}$.
The term “Retract Theorem” has been applied in literature in connection with group theory. In the present paper we prove that the Retract Theorem is valid (i) for each finite structure, and (ii) for each monounary algebra. On the other hand, we show that this theorem fails to be valid, in general, for algebras of the form $\mathcal{A}=(A,F)$, where each $f\in F$ is unary and $\operatorname{card}F >1$.