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On the elementary theory of restricted elementary functions. (English) Zbl 0698.03023
As a contribution to definability theory in the spirit of Tarski’s classical work on $$({\mathbb{R}},<,0,1,+,\cdot)$$ we extend here part of his results to the structure ${\mathbb{R}}^{RE}=({\mathbb{R}},<,0,1,+,\cdot,\exp |_{[0,1]},\sin |_{[0,\pi]}).$ Here $$\exp |_{[0,1]}$$ and $$\sin |_{[0,\pi]}$$ are the restrictions of the exponential and sine function to the closed intervals indicated; formally we identify these restricted functions with their graphs and regard these as binary relations on $${\mathbb{R}}$$. The superscript “RE” stands for “restricted elementary” since, given any elementary function, one can in general only define certain restrictions of it in $${\mathbb{R}}^{RE}.$$
Let $$({\mathbb{R}}^{RE}$$, constants) be the expansion of $${\mathbb{R}}^{RE}$$ obtained by adding a name for each real number to the language. We can now formulate our main result as follows. Theorem. $$({\mathbb{R}}^{RE}$$, constants) is strongly model-complete.
This means that every formula $$\phi (X_ 1,...,X_ m)$$ in the natural language of $$({\mathbb{R}}^{RE}$$, constants) is equivalent to an existential formula $$\exists Y_ 1...Y_ n$$ $$\psi (X_ 1,...,X_ m,Y_ 1,...,Y_ n)$$ with the extra property that for each $$x\in {\mathbb{R}}^ m$$ such that $$\phi$$ (x) is true in $${\mathbb{R}}^{RE}$$ there is exactly one $$y\in {\mathbb{R}}^ n$$ such that $$\psi$$ (x,y) is true in $${\mathbb{R}}^{RE}$$. (Here $$\psi$$ is quantifier free.) In other words, strong model- completeness is the next best thing after quantifier elimination, and is of course stronger than model-completeness.

##### MSC:
 03C40 Interpolation, preservation, definability
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##### References:
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