Boolean-type Retractable State-finite Automata Without Outputs

An automaton A is called a retractable automaton if, for every subau-tomaton B of A, there is at least one homomorphism of A onto B which leaves the elements of B fixed (such homomorphism is called a retract ho-momorphism of A onto B). We say that a retractable automaton A=(A,X,δ) is Boolean-type if there exists a family {λ B | B is a subautomaton of A} of retract homomorphisms λ B of A such that, for arbitrary subautomata B 1 and B 2 of A, the condition B 1 ⊆ B 2 implies Kerλ B 2 ⊆ Kerλ B 1. In this paper we describe the Boolean-type retractable state-finite automata without outputs.