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L(Q)-preservation theorems
Published online by Cambridge University Press: 12 March 2014
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Much effort has been spent to prove that the reduced product operation preserves, and sometimes even strengthens the L-equivalence of structures, where L is some infinitary language. A similar result is suggested by the following well-known fact:
Assume D is a nonprincipal ultrafilter on ω and, for n Є ω, Cn is a set. If the ultra-product ΠωCn/D is infinite, it has a cardinality ≥ 
Hence, by Łos' theorem, (i) if 
and φ(x) is a first-order formula, then 
iff 
where Q is the unary quantifier “there are many.”
We shall prove some generalizations of (i). In particular, we show
(ii) if D is a nonprincipal ultrafilter over I = ω, and 
then 
where L(Q) is the language obtained from the first-order language by adding the quantifier Q.
(ii) remains true, if D is an ω-regular or an atomless filter over a set I.
Lipner [7] proved that if is regular, then the L(Q)-equivalence is preserved under direct products. We show that the assumption “ is regular” is necessary.
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- Copyright © Association for Symbolic Logic 1975
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