  Given the family $P$ of all nonempty subsets of a set $U$ of alternatives, a
choice over $U$ is a function $c \colon \Omega \to P$ such that $\Omega
\subseteq P$ and $c(B) \subseteq B$ for all menus $B \in \Omega$. A choice is
total if $\Omega = P$, and partial otherwise. In economics, an agent is
considered rational whenever her choice behavior satisfies suitable axioms of
consistency, which are properties quantified over menus. Here we address the
following lifting problem: Given a partial choice satisfying one or more axioms
of consistency, is it possible to extend it to a total choice satisfying the
same axioms? After characterizing the lifting of some choice properties that
are well-known in the economics literature, we study the decidability of the
connected satisfiability problem for unquantified formulae of an elementary
fragment of set theory, which involves a choice function symbol, the Boolean
set operators, the singleton, the equality and inclusion predicates, and the
propositional connectives. In two cases we prove that the satisfiability
problem is NP-complete, whereas in the remaining cases we obtain
NP-completeness under the additional assumption that the number of choice terms
is constant.
