Belegradek, O2024-07-182024-07-1820040022-48121943-5886https://doi.org/10.2178/jsl/1082418540https://hdl.handle.net/11411/8121A 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 is an element of 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 M-k 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 --> sigmax, for some sigma from the field of fractions of R.eninfo:eu-repo/semantics/closedAccessArticle2-s2.0-294269000810.2178/jsl/10824185405172Q149969Q3WOS:000221710700012