Harremoës Peter
Copenhagen Business College, Nørre Voldgade 34, 1358 Copenhagen K, Denmark.
Entropy (Basel). 2018 Oct 12;20(10):784. doi: 10.3390/e20100784.
We study entropy inequalities for variables that are related by functional dependencies. Although the powerset on four variables is the smallest Boolean lattice with non-Shannon inequalities, there exist lattices with many more variables where the Shannon inequalities are sufficient. We search for conditions that exclude the existence of non-Shannon inequalities. The existence of non-Shannon inequalities is related to the question of whether a lattice is isomorphic to a lattice of subgroups of a group. In order to formulate and prove the results, one has to bridge lattice theory, group theory, the theory of functional dependences and the theory of conditional independence. It is demonstrated that the Shannon inequalities are sufficient for planar modular lattices. The proof applies a gluing technique that uses that if the Shannon inequalities are sufficient for the pieces, then they are also sufficient for the whole lattice. It is conjectured that the Shannon inequalities are sufficient if and only if the lattice does not contain a special lattice as a sub-semilattice.
我们研究由函数依赖关系相关的变量的熵不等式。尽管四个变量上的幂集是具有非香农不等式的最小布尔格，但存在变量更多的格，其中香农不等式就足够了。我们寻找排除非香农不等式存在的条件。非香农不等式的存在与一个格是否同构于一个群的子群格的问题相关。为了阐述和证明这些结果，必须在格理论、群理论、函数依赖理论和条件独立理论之间架起桥梁。结果表明，香农不等式对于平面模格是足够的。证明应用了一种胶合技术，该技术利用如果香农不等式对于各个部分是足够的，那么它们对于整个格也是足够的。据推测，当且仅当格不包含一个特殊格作为子半格时，香农不等式才是足够的。
