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We consider several ways of decomposing models into parts of bounded size forming a congruence over a base, and show that admitting any such decomposition is equivalent to mutual algebraicity at the level of theories. We also show that a theory T T is mutually algebraic if and only if there is a uniform bound on the number of coordinate-wise non-algebraic types over every model, regardless of its cardinality.
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Uniformly Bounded Arrays and Mutually Algebraic Structures
We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.
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- Award ID(s):
- 1855789
- PAR ID:
- 10162382
- Date Published:
- Journal Name:
- Notre Dame journal of formal logic
- Volume:
- 61
- Issue:
- 2
- ISSN:
- 0029-4527
- Page Range / eLocation ID:
- 265–282
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/doi:10.1215/00294527-2020-0004
