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- W2899504139 abstract "We work with a pre-$lambda$-frame, which is an abstract elementary class (AEC) endowed with a collection of basic types and a non-forking relation satisfying certain natural properties with respect to models of cardinality $lambda$. We investigate the density of uniqueness triples in a given pre-$lambda$-frame $mathfrak s$, that is, under what circumstances every basic triple admits a non-forking extension that is a uniqueness triple. Prior results in this direction required strong hypotheses on $mathfrak s$. Our main result is an improvement, in that we assume far fewer hypotheses on $mathfrak s$. In particular, we do not require $mathfrak s$ to satisfy the extension, uniqueness, stability, or symmetry properties, or any form of local character, though we do impose the amalgamation and stability properties in $lambda^+$, and we do assume $diamondsuit(lambda^+)$. As a corollary, by applying our main result to the trivial $lambda$-frame, it follows that in any AEC $mathbf K$ satisfying modest hypotheses on $mathbf K_lambda$ and $mathbf K_{lambda^+}$, the set of $*$-domination triples in $mathbf K_lambda$ is dense among the non-algebraic triples. We also apply our main result to the non-splitting relation, obtaining the density of uniqueness triples from very few hypotheses." @default.
- W2899504139 created "2018-11-09" @default.
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- W2899504139 date "2018-04-29" @default.
- W2899504139 modified "2023-09-27" @default.
- W2899504139 title "Density of uniqueness triples from the diamond axiom" @default.
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