Retractable state-finite automata without outputs

A homomorphism of an automaton A without outputs onto a subau-tomaton B of A is called a retract homomorphism if it leaves the elements of B fixed. An automaton A is called a retractable automaton if, for every subautomaton B of A, there is a retract homomorphism of A onto B. In [1] and [3], special retractable automata are examined. The purpose of this paper is to give a construction for state-finite retractable automata without outputs. In this paper, by an automaton we mean an automaton without outputs, that is, a system A = (A, X, δ) consisting of a non-empty state set A, a non-empty input set X and a transition function δ : A × X → A. If A has only one element then the automaton A will be called trivial. The function δ is extended to A × X * (X * denotes the free monoid over X) as follows. If a is an arbitrary state of A then δ(a, e) = a for the empty word e, and δ(a, qx) = δ(δ(a, q), x) for every q ∈ X * , x ∈ X. If B is a non-empty subset of the state-set of an automaton A = (A, X, δ) such that δ(b, x) ∈ B for every b ∈ B and x ∈ X, then B = (B, X, δ B) is an automaton, where δ B denotes the restriction of δ to B × X. This automaton is called a subautomaton (more precisely, an A-subautomaton) of A. A subautoma-ton B of an automaton A is called a proper subautomaton of A if B is a proper subset of A. A subautomaton B of an automaton A is said to be a minimal subautomaton of A if B has no proper subautomaton. If a subautomaton B of an automaton A has only one state then B is minimal; the state of B is called a trap of A. If an automaton A = (A, X, δ) contains only one trap denoted by a 0 then A is called a one-trap automaton (or an OT-automaton). This fact will be denoted by (A, X, δ; a 0). If an automaton A has a subautomaton which is contained in every subautomaton of A then it is called the kernel of A. The kernel of A is denoted by KerA. Let A = (A, X, δ) be …