  It is known from Grzegorczyk's paper \cite{grze-1951} that the lattice of
real semi-algebraic closed subsets of ${\mathbb R}^n$ is undecidable for every
integer $n\geq 2$. More generally, if $X$ is any definable set over a real or
algebraically closed field $K$, then the lattice $L(X)$ of all definable
subsets of $X$ closed in $X$ is undecidable whenever $\dim X\geq 2$.
Nevertheless, we investigate in this paper the model theory of the class ${\rm
SC\_{def}}(K,d)$ of all such lattices $L(X)$ with $\dim X\leq d$ and $K$ as
above or a henselian valued field of characteristic zero. <p> We show that the
universal theory of ${\rm SC\_{def}}(K,d)$, in a natural expansion by
definition of the lattice language, is the same for every such field $K$. We
give a finite axiomatization of it and prove that it is locally finite and
admits a model-completion, which turns to be decidable as well as all its
completions. We expect $L({\mathbb Q}\_p^d)$ to be a model of (a little variant
of) this model-completion. This leads us to a new conjecture in $p$-adic
semi-algebraic geometry which, combined with the results of this paper, would
give decidability (via a natural recursive axiomatization) and elimination of
quantifiers for the complete theory of $L({\mathbb R}\_p^d)$, uniformly in $p$.
