Abstract
Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of \(\text {Th}(A)\) up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite (in particular, if there are no unary predicates), then the theory is \(\aleph _0\)-categorical or Borel complete; this generalizes a theorem due to Schirmann (Theories des ordres totaux et relations dequivalence. Master’s thesis, Universite de Paris VII, 1997).
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The author was partially supported by NSF Research Grant DMS-1308546. The author is greatly indebted to the anonymous Reviewer #1 for numerous improvements for readability and notation.
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Rast, R. The complexity of isomorphism for complete theories of linear orders with unary predicates. Arch. Math. Logic 56, 289–307 (2017). https://doi.org/10.1007/s00153-017-0525-z
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DOI: https://doi.org/10.1007/s00153-017-0525-z