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Connectedness of the theory of non-surjective injections. (English) Zbl 0840.03043
Summary: It is shown that the first-order theory \(T\) of a non-surjective injection (of the universe into itself) is connected. By a criterion for connectedness, due to J. Mycielski, one reduces this result to the following theorem: If \({\mathfrak A}\), \({\mathfrak B}\) are any structures and \(\varepsilon\) is a definable equivalence relation on their disjoint product \({\mathfrak A} \dot \times {\mathfrak B}\) such that \(({\mathfrak A} \dot \times {\mathfrak B})/ \varepsilon\) is a model of \(T\), then either \({\mathfrak A}\) has the property that for some finite structure \({\mathfrak S}\) and a definable equivalence relation \(\simeq\) on \({\mathfrak S} \dot \times {\mathfrak A}\), \(({\mathfrak S} \dot \times {\mathfrak A})/ \simeq\) is a model of \(T\), or the analogous property holds for \({\mathfrak B}\).
MSC:
03F25 Relative consistency and interpretations
03C07 Basic properties of first-order languages and structures
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