Model completeness of an algebra of languages
David H. Haussler · Proceedings of the American Mathematical Society · 1981
An algebra $\left \langle {\mathcal {L},f,g} \right \rangle$ of languages over a finite alphabet $\Sigma = \{ {a_1}, \ldots ,{a_n}\}$ is defined with operations $f({L_1}, \ldots ,{L_n}) = {a_1}{L_1} \cup \cdots \cup {a_n}{L_n} \cup \{ \lambda \}$ and $g({L_1}, \ldots ,{L_n}) = {a_1}{L_1} \cup \cdots \cup {a_n}{L_n}$ and its first order theory is shown to be model complete. A characterization of the regular languages as unique solutions of sets of equations in $\left \langle {\mathcal {L},f,g} \right \rangle$ is given and it is shown that the subalgebra $\left \langle {\mathcal {R},f,g} \right \rangle$ where $\mathcal {R}$ is the set of regular languages is a prime model for the theory of $\left \langle {\mathcal {L},f,g} \right \rangle$. We show also that the theory of $\left \langle {\mathcal {L},f,g} \right \rangle$ is decidable.