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Semi-bounded relations in ordered modules. (English) Zbl 1095.03024

This paper continues the development of the model theory of ordered modules, building on and extending previous work of the author and of others on ordered abelian groups [O. Belegradek, in: L. BĂ©lair et al. (eds.), Model theory and applications. Based on the Euro-Conference in model theory and applications, Ravello, Italy, May 27–June 1, 2002. Rome: Aracne. Quaderni di Matematica 11, 15–39 (2002; Zbl 1081.03036); O. Belegradek, V. Verbovskiy, and F. O. Wagner, Ann. Pure Appl. Logic 121, 113–143 (2003; Zbl 1025.03029)] and ordered vector spaces [J. Loveys and Y. Peterzil, Isr. J. Math. 81, 1–30 (1993; Zbl 0797.03034)]. The author presents a list of interesting examples: the theory being developed is clearly non-trivial.
A relation on a linearly ordered set is bounded if it is bounded from above and from below in all coordinates. A relation on a linearly ordered structure \(M\) is called semi-bounded if it is first-order definable in the expansion of \(M\) by all bounded relations on \(M\). A function on \(M\) is semi-bounded if its graph is semi-bounded. A relation on \(M\) is eventually definable if it is the union of a bounded relation and a first-order definable relation.
This paper studies the structure of semi-bounded relations in an ordered module \(M\) over an ordered commutative ring \(R\). Necessarily \(R\) is a domain and \(M/rM\) is finite for every non-zero \(r\in R\). A \(k\)-strip is a subset of \(M^{k}\) defined by a relation of the form \(| \bar{r}\cdot\bar{x}| \leq m\), where \(\bar{r}\) is a \(k\)-tuple in \(R\) and \(m\in M\) is positive. There are many varied and interesting results exploring the types of quantifier elimination that occur in such structures, of which the following are highlighted in the author’s abstract: (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 endomorphism of the additive group of \(M\) which is semi-bounded in \(M\) is just multiplication by some element of the field of fractions of \(R\).
The examples are developed to show that various assumptions are essential, or that obvious improvements won’t work. For instance, it follows that the finiteness requirement mentioned above is unavoidable: if every semi-bounded relation on \(M\) is eventually definable, then \(M/rM\) is finite for every non-zero \(r\in R\).
To tackle any particular “application” model-theoretically we need first of all a good handle on the structure of the definable sets; the various results leading up to the main results of the paper provide us with several useful tools.

MSC:

03C64 Model theory of ordered structures; o-minimality
03C10 Quantifier elimination, model completeness, and related topics
03C60 Model-theoretic algebra
16W80 Topological and ordered rings and modules
06F25 Ordered rings, algebras, modules
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References:

[1] Rocky Mountain Journal of Mathematics 19 pp 871– (1980)
[2] A structure theorem for semibounded sets in the reals 57 pp 779– (1992) · Zbl 0768.03019
[3] Semi-bounded relations in ordered abelian groups (2002) · Zbl 1081.03036
[4] The model theory of groups pp 138– (1989)
[5] DOI: 10.1016/S0168-0072(02)00084-2 · Zbl 1025.03029
[6] DOI: 10.1007/BF02761295 · Zbl 0797.03034
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