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\iteman{io-port 02247423}
\itemau{Rothmaler, Philipp}
\itemti{Elementary epimorphisms.}
\itemso{J. Symb. Log. 70, No. 2, 473-487 (2005).}
\itemab
Inspired by the dual pair of concepts, pure monomorphism and pure epimorphism, in the context of the model theory of modules, the author defines an elementary epimorphism to be a surjective homomorphism $f:A\rightarrow B$ between ${\cal L}$-structures, for some language ${\cal L}$, such that for every formula $\phi \in {\cal L}$ (or in some specified subset of ${\cal L}$) and for every $\overline{b} \in\phi (B)$ there is $\overline{a}\in \phi (A)$ such that $f(\overline{a})=\overline{b}$. He also defines a strict elementary epimorphism to be an elementary epimorphism $f$ where witnesses to existential quantifiers can be pulled back along $f$. After investigating satisfaction of formulas in inverse limits, a ``dual'' of the elementary chain theorem is proved (in a general form which specialises to both an absolute and a pure version). The situation is not, however, as simple as for elementary monomorphisms, and a number of examples are presented showing the necessity of various restrictions in the main theorem and its corollaries. Finally the results are applied to suitable inverse limits of flat and absolutely pure modules.
\itemrv{Mike Prest (Manchester)}
\itemcc{}
\itemut{epimorphism; elementary map; pure epimorphism; positive primitive formula; inverse limit}
\itemli{doi:10.2178/jsl/1120224724}
\end