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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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