arXiv subjects

John Goodrick

Publications and source records attributed to John Goodrick.

21 records · Page 2Linked to original sources

arXiv2009

Dp-minimality: basic facts and examples

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. The rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

math.LO↗

arXiv2007

When does elementary bi-embeddability imply isomorphism?

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if a countable theory T has the Schroder-Bernstein property then it is classifiable (it is superstable and has NDOP and NOTOP) and satisfies a slightly stronger condition than nonmultidimensionality, namely: there cannot be a model M of T, a type p over M, and an automorphism f of M such that for every two distinct natural numbers i and j, f^i(p) is orthogonal to f^j(p). We also make some conjectures about how the class of theories with the Schroder-Bernstein property can be characterized.

math.LO↗

arXiv2007

The Schroder-Bernstein property for theories of abelian groups

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the Schroder-Bernstein property; 2. Th(G, +) is omega-stable; 3. G is the direct sum of a divisible group and a torsion group of bounded exponent; 4. Th(G, +) is superstable, and if (H, +) is a saturated elementary extension of (G,+), every map in Aut(H/H^0) is unipotent.

math.LO↗