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.

翻译：本文给出了基于一组离散函数知识重构离散随机变量$X$的一个新信息论条件。该重构条件源自Shannon 1953年提出的格论，并结合了Shannon和Rajski的两种熵度量。由于此类理论材料相对鲜为人知且分散于不同文献，我们首先对其概念（如总信息、共有信息和互补信息）进行综合描述（附完整证明）。两种熵度量的定义和性质也被详尽阐述，并证明其与格结构兼容。随后，我们探讨了此类格结构的一种新几何解释，这导出了基于$X$生成的格中元素集$\{ X_1,\ldots,X_{n} \}$重构离散随机变量$X$的一个必要条件（有时也是充分条件）。最后，通过五个完美重构问题的具体实例说明该条件：根据符号和绝对值重构对称随机变量、根据一组线性组合重构单词、根据素因子签名重构整数（算术基本定理）、根据一组互素整数模余数重构整数（中国剩余定理），以及根据最小成对比较集重构列表的排序置换。