Vaught's Conjecture for Almost Chainable Theories

DOI10.1017/JSL.2021.60arXiv1905.05531WikidataQ113858285 ScholiaQ113858285MaRDI QIDQ6318698

Miloš S. Kurilić

Publication date: 14 May 2019

Abstract: A structure $mathbbY$ of a relational language $L$ is called almost chainable iff there are a finite set $FsubsetY$ and a linear order $<$ on the set $YsetminusF$ such that for each partial automorphism $varphi$ (i.e., local automorphism, in Fra"{i}ss'{e}'s terminology) of the linear order $langleYsetminusF,<angle$ the mapping ${mathrmid}_{F}cupvarphi$ is a partial automorphism of $mathbbY$. By a theorem of Fra"{i}ss'{e}, if $|L|<omega$, then $mathbbY$ is almost chainable iff the profile of $mathbbY$ is bounded; namely, iff there is a positive integer $m$ such that $mathbbY$ has $leqm$ non-isomorphic substructures of size $n$, for each positive integer $n$. A complete first order $L$-theory $mathcalT$ having infinite models is called almost chainable iff all models of $mathcalT$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of $mathcalT$. In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

Mathematics Subject Classification ID

Total orders (06A05) Models with special properties (saturated, rigid, etc.) (03C50) Model theory of denumerable and separable structures (03C15) Categoricity and completeness of theories (03C35)

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