On \(\kappa\)-complete reduced products
type: Item
instance of: scholarly article
title: On \(\kappa\)-complete reduced products
zbMATH Open document ID: 0755.03015
DOI: 10.1007/BF01269947
author: Tapani Hyttinen
published in: Archive for Mathematical Logic
publication date: 1992-09-26T00:00:00Z
review text: Keisler's theorem states that (under GCH) elementarily equivalent structures have isomorphic ultrapowers. The author generalizes (the proof of) this result as follows: (1) For any two structures \(\mathfrak A\), \(\mathfrak B\), let \(F_ \alpha({\mathfrak A},{\mathfrak B})\) be the corresponding Ehrenfeucht-Fraissé game of length \(\alpha\). If --- under suitable cardinal assumptions --- for each \(\alpha<\kappa\) player \(E\) has a winning strategy for \(F_ \alpha({\mathfrak A},{\mathfrak B})\) and \(F\) is a \(\kappa\)-semigood filter over \(\lambda\), \(F'\) a \(\kappa\)-descendingly incomplete filter over \(\xi\), \(\xi\leq\lambda\), then the ultrapowers (of \(\mathfrak A\) and \(\mathfrak B\) resp.) modulo any filter \(D\) over \(\xi\times\lambda\) are isomorphic, provided \(F'\times F\subseteq D\). (2) Filters \(F\) as required in (1) exist. (3) The filter \(D\) in (1) can be \(\kappa\)-complete. (4) There are applications to a generalized notion of universality (w.r.t. infinitary languages).
reviewed by: K.Gloede
Mathematics Subject Classification ID: 03C20
Mathematics Subject Classification ID: 03C75
zbMATH DE Number: 55778
zbMATH Keywords: filters
zbMATH Keywords: elementarily equivalent structures
zbMATH Keywords: Ehrenfeucht-Fraissé game
zbMATH Keywords: ultrapowers
zbMATH Keywords: universality
zbMATH Keywords: infinitary languages
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01269947
OpenAlex ID: W2026593029
description: scientific article; zbMATH DE number 55778