  An automaton $\bf A$ is called a retractable automaton if, for every
subautomaton $\bf B$ of $\bf A$, there is at least one homomorphism of $\bf A$
onto $\bf B$ which leaves the elements of $B$ fixed (such homomorphism is
called a retract homomorphism of $\bf A$ onto $\bf B$). We say that a
retractable automaton ${\bf A}$=(A,X,$\delta$) is Boolean-type if there exists
a family $\{\lambda_B \mid \textrm{ B is a subautomaton of A } \}$ of retract
homomorphisms $\lambda _B$ of $\bf A$ such that, for arbitrary subautomata
${\bf B}_1$ and ${\bf B}_2$ of $\bf A$, the condition $B_1\subseteq B_2$
implies $Ker\lambda _{B_2}\subseteq Ker\lambda _{B_1}$. In this paper we
describe the Boolean-type retractable state-finite automata without outputs.
