Boolean-type Retractable State-finite Automata Without Outputs

An automaton $\mathbf{A}$ is called a retractable automaton if, for every subautomaton $\mathbf{B}$ of $\mathbf{A}$, there is at least one homomorphism of $\mathbf{A}$ onto $\mathbf{B}$ which leaves the elements of $B$ fixed (such homomorphism is called a retract homomorphism of $\mathbf{A}$ onto $\mathbf{B}$). We say that a retractable automaton $\mathbf{A}$=(A,X,$\delta$) is Boolean-type if there exists a family $\{{\lambda }_B\mid \text{ B is a subautomaton of A }\}$ of retract homomorphisms ${\lambda }_B$ of $\mathbf{A}$ such that, for arbitrary subautomata ${\mathbf{B}}_1$ and ${\mathbf{B}}_2$ of $\mathbf{A}$, 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.