Semi-bounded relations in ordered modules

Journal of Symbolic Logic 69 (2):499-517 (2004)
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Abstract

A relation on a linearly ordered structure is called semi-bounded if it is definable in an expansion of the structure by bounded relations. We study ultimate behavior of semi-bounded relations in an ordered module M over an ordered commutative ring R such that M/r M is finite for all nonzero r∈ R. We consider M as a structure in the language of ordered R-modules augmented by relation symbols for the submodules rM, and prove several quantifier elimination results for semi-bounded relations and functions in M. We show that these quantifier elimination results essentially characterize the ordered modules M with finite indices of the submodules rM. It is proven that any semi-bounded k-ary relation on M is equal, outside a finite union of k-strips, to a k-ary relation quantifier-free definable in M, any semi-bounded function from Mk to M is equal, outside a finite union of k-strips, to a piecewise linear function, and any semi-bounded in M endomorphism of the additive group of M is of the form x ↦ σ x, for some σ from the field of fractions of R.

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Coset-minimal groups.Oleg Belegradek, Viktor Verbovskiy & Frank O. Wagner - 2003 - Annals of Pure and Applied Logic 121 (2-3):113-143.

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