TY - JOUR
T1 - Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels
AU - Myasnikov, A. G.
AU - Romanovskii, N. S.
N1 - Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2018/3/1
Y1 - 2018/3/1
N2 - A group G is said to be rigid if it contains a normal series G = G1G2GmGm+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 m 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 m 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 = G1G2GmGm+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 m 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 m admits quantifier elimination down to a Boolean combination of ∀∃-formulas.
KW - divisible rigid group
KW - model
KW - saturation
KW - stability
KW - theory
KW - ∀∃-formula
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U2 - 10.1007/s10469-018-9476-7
DO - 10.1007/s10469-018-9476-7
M3 - Article
AN - SCOPUS:85047136866
SN - 0002-5232
VL - 57
SP - 29
EP - 38
JO - Algebra and Logic
JF - Algebra and Logic
IS - 1
ER -