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There are classical books of Eilenberg with this kind of subject, and for finite automata any book from compiler theory, like Aho, Ullman…and of course in Knuth’s Art…
I have added a reference to automaton and linked to various other entries.
I elaborated a bit in the “Idea” section, and also removed what I think was a redundancy in the “Definition” section: the definition of “automaton” given here was actually already exactly the notion of “deterministic, sequential, Moore automaton” from Joy of Cats, rather than a more general notion. I haven’t read Joy of Cats (other than to look this up), so I’m not sure why they focused on this particular notion (and built a category out of it).
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