Just a heads up, there are two types of semigroups out there, and I think that automata theory uses the weird (i.e., non-algebraic) kind, so don't be too disappointed if this doesn't hook in with other standard maths. I have only seen this second usage of semigroups in automata theory, and only heard it used by an automata theorist. A quick Google search didn't turn up this other usage, so it seems to be very domain-specific lingo.On the other hand, you can look into algebraic automata theory (a subject I know nothing about) and play with monoids (the kleene closure of an alphabet with the concatenation operation is the free monoids over that alphabet).
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