Distal and non-distal Pairs

The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.

[1]  Ayhan Günaydin,et al.  Dependent pairs , 2011, J. Symb. Log..

[2]  Chris Miller,et al.  Expansions of o-minimal structures by dense independent sets , 2016, Ann. Pure Appl. Log..

[3]  Philipp Hieronymi.  Defining the set of integers in expansions of the real field by a closed discrete set , 2009, 0906.4972.

[4]  Ayhan Gunaydin,et al.  The real field with the rational points of an elliptic curve , 2009, 0906.0528.

[5]  A. Chernikov,et al.  Externally definable sets and dependent pairs , 2010, Israel Journal of Mathematics.

[6]  Pierre Simon,et al.  Distal and non-distal NIP theories , 2011, Ann. Pure Appl. Log..

[7]  Pantelis E. Eleftheriou,et al.  Structure theorems in tame expansions of o-minimal structures by a dense set , 2015 .

[8]  M. Baaz,et al.  Tameness in expansions of the real field , 2016 .

[9]  Lou van den Dries.  Dense pairs of o-minimal structures , 1998 .

[10]  S. Shelah.  Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory , 1971 .

[11]  Lou van den Dries,et al.  T-Convexity and Tame Extensions , 1995, J. Symb. Log..

[12]  B. Zilber,et al.  The model theory of the field of reals with a subgroup of the unit circle , 2008 .

[13]  Lou van den Dries,et al.  The field of reals with a predicate for the powers of two , 1985 .

[14]  Lou van den Dries,et al.  The Fields of Real and Complex Numbers with a Small Multiplicative Group , 2006 .

[15]  Alf Onshuus,et al.  The independence property in generalized dense pairs of structures , 2011, J. Symb. Log..

[16]  Gareth Boxall.  NIP for some pair-like theories , 2011, Arch. Math. Log..

[17]  Alfred Dolich,et al.  Dp-Minimality: Basic Facts and Examples , 2011, Notre Dame J. Formal Log..