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Semi-bounded relations in ordered modules

Published online by Cambridge University Press:  12 March 2014

Oleg Belegradek
Oleg Belegradek*
Affiliation:
Department of Mathematics, Istanbul Bilgi University, 80370, Dolapdere-Istanbul, Turkey, E-mail: olegb@bilgi.edu.tr
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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/rM is finite for all nonzero r ϵ 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 (1) 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, (2) any semibounded function from Mk to M is equal, outside a finite union of k-strips, to a piecewise linear function, and (3) any semi-bounded in M endomorphism of the additive group of M is of the form x ↦ σx, for some σ from the field of fractions of R.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 499 - 517
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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