Abstract
A monotone structure (\(\mathfrak{A} ;\);μ) consists of a structure\(\mathfrak{A}\) and a monotone systemμ over the domain of\(\mathfrak{A}\).L(Q n) is\(L_{\omega \omega }\), enlarged by a newn-ary quantifierQ n.\(Q^n \overline x \varphi \left( {\overline x } \right)\) says in (\(\mathfrak{A}\);μ) that there isU ∈μ such thatϕ[ā] is valid in (\(\mathfrak{A}\);μ) for allā ∈U n. If
is a class of monotone structures,
means thatϕ is valid in all expansions of monotone structures in
. We show for the class\(\mathfrak{u}\) of all ultrafilters that interpolation with respect to
holds forL(Q n) exactly in casen=1. Then we prove for a large class of
(e.g. the class of topological groups) thatL(Q n) satisfies interpolation with respect to
for alln ≧ 1. Counterexamples indicate that the class of
is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI n instead ofQ n, thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.
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Ebbinghaus, H.D., Ziegler, M. Interpolation in Logiken monotoner systeme. Arch math Logik 22, 1–17 (1980). https://doi.org/10.1007/BF02318022
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DOI: https://doi.org/10.1007/BF02318022