Ultrafilter extensions do not preserve elementary equivalence
نویسندگان
چکیده
منابع مشابه
Ultrafilter Extensions for Coalgebras
This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is wellknown that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence. Stone-coalgebras (coalgebras over the category of Stone spaces), on the ot...
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Let TX be the full transformation semigroups on the set X. For an equivalence E on X, let TE(X) = {α ∈ TX : ∀(x, y) ∈ E ⇔ (xα, yα) ∈ E}It is known that TE(X) is a subsemigroup of TX. In this paper, we discussthe Green's *-relations, certain *-ideal and certain Rees quotient semigroup for TE(X).
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Proof: By assumption, φinf is preserved under ultrafilter extensions. The second conjunct of θ, i.e., ∀xy.(x = y → Rxy) is also preserved under ultrafilter extension, since it is modally definable using global modality. Finally, consider the third conjunct of θ. From the fact that M |= ∀xy.(x = y → Rxy), we can derive that the denotation of R in ueM includes all pairs of ultrafilters (u, v) suc...
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u1 = {Z | there are finitely many sets X1, . . . Xk such that Z = X1 ∩ · · · ∩Xk}. That is, u1 is the set of finite intersections of sets from u0. Note that u0 ⊆ u1, since u0 has the finite intersection property, we have ∅ 6∈ u1, and by definition u1 is closed under finite intersections. Now, define u2 as follows: u′ = {Y | there is a Z ∈ u1 such that Z ⊆ Y } ∗UMD, Philosophy. Webpage: ai.stanf...
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ژورنال
عنوان ژورنال: Mathematical Logic Quarterly
سال: 2019
ISSN: 0942-5616,1521-3870
DOI: 10.1002/malq.201900045