This article proposes a new information theoretic necessary condition 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 the Shannon's Lattice of Information (LoI) \cite{Shannon53} and two entropic metrics proposed respectively by Shannon and Rajski. This theoretical material being relatively unknown and/or dispersed in different references, we provide a complete and synthetic description of the LoI concepts like the total, common and complementary informations with complete proofs. The two entropic metrics definitions and properties are also fully detailled and showed compatible with the LoI structure. A new geometric interpretation of the Lattice structure is then investigated that leads to a new necessary condition for reconstructing the discrete random variable $X$ given a set $\{ X_0$,...,$X_{n-1} \}$ of elements of the lattice generated by $X$. Finally, this condition is derived in five specific examples of reconstruction of $X$ from a set of deterministic functions of $X$: the reconstruction of a symmetric random variable from the knowledge of its sign and of its absolute value, the reconstruction of a binary word from a set of binary linear combinations, the reconstruction of an integer from its prime signature (Fundamental theorem of arithmetics) and from its reminders modulo a set of coprime integers (Chinese reminder theorem), and the reconstruction of the sorting permutation of a list from a set of 2-by-2 comparisons. In each case, the necessary condition is shown compatible with the corresponding well-known results.

翻译：本文提出了一种新的信息论必要条件，用于基于离散随机变量$X$的一组离散函数知识来重构该变量。重构条件源自Shannon信息格（LoI）\cite{Shannon53}以及分别由Shannon和Rajski提出的两种熵度量。由于这些理论材料相对鲜为人知或分散于不同文献，我们提供了LoI概念的完整且综合的描述（如总信息、共有信息与互补信息），并附上完整证明。两种熵度量的定义与性质也得到详细阐述，并证明了它们与LoI结构的兼容性。随后，我们探索了格结构的一种新的几何解释，从而得出了一个必要条件，用于在给定由$X$生成的格元素集合$\{ X_0$,...,$X_{n-1} \}$的情况下重构离散随机变量$X$。最后，该条件被推导至五个具体示例中，这些示例涉及从一组$X$的确定性函数重构$X$：从符号与绝对值知识重构对称随机变量；从一组二元线性组合重构二进制字；从质数签名（算术基本定理）及一组互质整数的余数（中国剩余定理）重构整数；以及从一组两两比较重构列表的排序排列。在每个案例中，该必要条件被证明与相应已知结果兼容。