TY - JOUR

T1 - Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels

AU - Myasnikov, A. G.

AU - Romanovskii, N. S.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - A group G is said to be rigid if it contains a normal series G = G1 > G2 > . . . > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]- modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory Im of divisible m-rigid groups is complete. Here, it is proved that this theory is ω-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated m-rigid groups. Also, it is proved that the theory Im admits quantifier elimination down to a Boolean combination of ∀∃-formulas.

AB - A group G is said to be rigid if it contains a normal series G = G1 > G2 > . . . > Gm > Gm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]- modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory Im of divisible m-rigid groups is complete. Here, it is proved that this theory is ω-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated m-rigid groups. Also, it is proved that the theory Im admits quantifier elimination down to a Boolean combination of ∀∃-formulas.

KW - divisible rigid group

KW - model

KW - saturation

KW - stability

KW - theory

KW - ∀∃-formula

KW - for all there exists-formula

UR - http://www.scopus.com/inward/record.url?scp=85047136866&partnerID=8YFLogxK

U2 - 10.1007/s10469-018-9476-7

DO - 10.1007/s10469-018-9476-7

M3 - Article

AN - SCOPUS:85047136866

VL - 57

SP - 29

EP - 38

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 1

ER -