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# Model Completeness of an Algebra of Languages

David Haussler
Proceedings of the American Mathematical Society
Vol. 83, No. 2 (Oct., 1981), pp. 371-374
DOI: 10.2307/2043531
Stable URL: http://www.jstor.org/stable/2043531
Page Count: 4
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## Abstract

An algebra $\langle \mathscr{L}, f, g \rangle$ of languages over a finite alphabet $\Sigma = \{a_1, \ldots, a_n\}$ is defined with operations $f(L_1,\ldots, L_n) = a_1L_1 \cup \cdots \cup a_nL_n \cup \{\lambda\}$ and $g(L_1,\ldots, L_n) = a_1L_1 \cup \cdots \cup a_nL_n$ 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 $\langle \mathscr{L}, f, g \rangle$ is given and it is shown that the subalgebra $\langle \mathscr{R}, f, g \rangle$ where $\mathscr{R}$ is the set of regular languages is a prime model for the theory of $\langle \mathscr{L}, f, g \rangle$. We show also that the theory of $\langle \mathscr{L}, f, g \rangle$ is decidable.

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