What is known about the class of languages recognized by finite automata having the same initial and accepting state?  This is a proper subset of the regular languages (since every such language contains the empty string), but how weak is it?  Is there a simple algebraic characterization?Ditto for languages recognized by non-deterministic automata having the same set of initial and accepting states.
(N)DFA with same initial/accepting state(s)
fl.formal languages;automata theory;algebra;regular language
This question is solved for deterministic automata and for unambiguous automata in the book [1][1] J. Berstel, D. Perrin, C, Reutenauer, Codes and automata, Vol. 129 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2009.In the case of deterministic automata, the characterization is given in Proposition 3.2.5. Recall that a submonoid $M$ of $A^*$ is right unitary if, for all $u, v \in M$, $u, uv \in M$ implies $v \in M$.  Proposition. Let $L$ be a regular subset of $A^*$. The following conditions are equivalent:$L$ is a right unitary submonoid,$L = P^*$ for some prefix code $P$,The minimal automaton of $L$ has a unique final state, namely the initial state.There exists a deterministic automaton recognizing $L$ having the initial state as unique final state.For unambiguous automata, the characterization follows from Theorem 4.2.2 and can be stated as follows:Proposition. Let $L$ be a regular subset of $A^*$. The following conditions are equivalent:$L$ is a free submonoid of $A^*$,$L = C^*$ for some code $C$,There exists an unambiguous automaton recognizing $L$ having the initial state as unique final state.Finally, for nondeterministic automata, the characterization is simply that $L$ is a submonoid of $A^*$.