June 2004 Semi-bounded relations in ordered modules
Oleg Belegradek
J. Symbolic Logic 69(2): 499-517 (June 2004). DOI: 10.2178/jsl/1082418540


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 (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 semi-bounded 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 ↦ σ x, for some σ from the field of fractions of R.


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Oleg Belegradek. "Semi-bounded relations in ordered modules." J. Symbolic Logic 69 (2) 499 - 517, June 2004. https://doi.org/10.2178/jsl/1082418540


Published: June 2004
First available in Project Euclid: 19 April 2004

zbMATH: 1095.03024
MathSciNet: MR2058186
Digital Object Identifier: 10.2178/jsl/1082418540

Primary: 03C64
Secondary: 03C10 , 03C60 , 06F25

Rights: Copyright © 2004 Association for Symbolic Logic


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Vol.69 • No. 2 • June 2004
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