TY - JOUR

T1 - Enriching a predicate and tame expansions of the integers

AU - Conant, Gabriel

AU - D'Elbee, Christian

AU - Halevi, Yatir

AU - Jimenez, Leo

AU - Rideau-Kikuchi, Silvain

N1 - Publisher Copyright:
© 2023 World Scientific Publishing Co. Pte Ltd. All rights reserved.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - Given a structure M and a stably embedded -definable set Q, we prove tameness preservation results when enriching the induced structure on Q by some further structure Q. In particular, we show that if T = Th(M) and Th(Q) are stable (respectively, superstable, -stable), then so is the theory T[Q] of the enrichment ofMby Q. Assuming simplicity of T, elimination of hyperimaginaries and a further condition on Q related to the behavior of algebraic closure, we also show that simplicity and NSOP1 pass from Th(Q) to T[Q]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of (Z,+). More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of (Z, +) by some unary predicate A N.

AB - Given a structure M and a stably embedded -definable set Q, we prove tameness preservation results when enriching the induced structure on Q by some further structure Q. In particular, we show that if T = Th(M) and Th(Q) are stable (respectively, superstable, -stable), then so is the theory T[Q] of the enrichment ofMby Q. Assuming simplicity of T, elimination of hyperimaginaries and a further condition on Q related to the behavior of algebraic closure, we also show that simplicity and NSOP1 pass from Th(Q) to T[Q]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of (Z,+). More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of (Z, +) by some unary predicate A N.

KW - Stably embedded sets

KW - preservation of dividing lines

KW - tame expansions of the integers

UR - http://www.scopus.com/inward/record.url?scp=85178101380&partnerID=8YFLogxK

U2 - 10.1142/S0219061324500119

DO - 10.1142/S0219061324500119

M3 - Article

AN - SCOPUS:85178101380

SN - 0219-0613

JO - Journal of Mathematical Logic

JF - Journal of Mathematical Logic

M1 - 2450011

ER -