  A new information theoretic condition is presented for reconstructing a
discrete random variable $X$ based on the knowledge of a set of discrete
functions of $X$. The reconstruction condition is derived from Shannon's 1953
lattice theory with two entropic metrics of Shannon and Rajski. Because such a
theoretical material is relatively unknown and appears quite dispersed in
different references, we first provide a synthetic description (with complete
proofs) of its concepts, such as total, common and complementary informations.
Definitions and properties of the two entropic metrics are also fully detailed
and shown compatible with the lattice structure. A new geometric interpretation
of such a lattice structure is then investigated that leads to a necessary (and
sometimes sufficient) condition for reconstructing the discrete random variable
$X$ given a set $\{ X_1,\ldots,X_{n} \}$ of elements in the lattice generated
by $X$. Finally, this condition is illustrated in five specific examples of
perfect reconstruction problems: reconstruction of a symmetric random variable
from the knowledge of its sign and absolute value, reconstruction of a word
from a set of linear combinations, reconstruction of an integer from its prime
signature (fundamental theorem of arithmetic) and from its remainders modulo a
set of coprime integers (Chinese remainder theorem), and reconstruction of the
sorting permutation of a list from a minimal set of pairwise comparisons.
