Generalized Directable Automata

In [18] the last three authors introduced the notion of a generalized directable automaton as a generalization of many already known types of automata. The algorithms for testing whether a finite automaton belongs to some of important subclasses of the class of generalized directable automata are studied by the authors in [20]. In this paper structural properties of finite and infinite generalized directable automata are considered, tests for membership of a finite automaton in the pseudovarieties of generalized directable and locally directable automata are given, and the least generalized directing and locally directing congruences on a finite automaton are described.