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**Weakly maximal decidable structures.**
*(English)*
Zbl 1149.03015

A structure \(M = (S, P_1,\dots,P_n)\) is a set \(S\) on which some predicates \(P_1,\dots,P_n\) are defined. First-order (FO) and monadic second-order (MSO) theories of \(M\) are considered. With the help of the Feferman-Vaught composition theorem on the reducibility of the FO theory of a generalized product of some structures to the theory of the factor structures and to the MSO theory of the index structures, Shelah’s composition theorem, and Büchi’s results on the connection beween MSO of the structure \((N,<)\) and finite automata, the authors construct a structure \(M\) whose MSO theory is decidable and such that the FO theory of every expansion of \(M\) by a non-definable constant is undecidable.

Reviewer: Alex Nabebin (Moskva)

### MSC:

03B25 | Decidability of theories and sets of sentences |

03C57 | Computable structure theory, computable model theory |

03D05 | Automata and formal grammars in connection with logical questions |

### Keywords:

logical structure; first-order logic; monadic second-order theories; maximality of theory; definability in logic; decidability; finite automaton; recognizability by automaton; relationship between definability and recognizability; rich words
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\textit{A. Bès} and \textit{P. Cégielski}, Theor. Inform. Appl. 42, No. 1, 137--145 (2008; Zbl 1149.03015)

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