Dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if P has dimension d, then to know whether x ≤ y in P it is enough to check whether x ≤ y in each of the d linear extensions of a witnessing realizer. Focusing on the encoding aspect, Nešetřil and Pudlák defined a more expressive version of dimension. A poset P has Boolean dimension at most d if it is possible to decide whether x ≤ y in P by looking at the relative position of x and y in only d linear orders on the elements of P (not necessarilly linear extensions). We prove that posets with cover graphs of bounded tree-width have bounded Boolean dimension. This stands in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. This result might be a step towards a resolution of the long-standing open problem: Do planar posets have bounded Boolean dimension?

中文翻译：

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布尔维数和树宽

维度是衡量偏序集复杂度的关键指标。小维度允许简洁编码。实际上，如果 P 的维数为 d，那么要知道 P 中的 x ≤ y 是否足以检查见证实现者的 d 条线性扩展中的每一个中的 x ≤ y 是否足够。专注于编码方面，Nešetřil 和 Pudlák 定义了一个更具表现力的维度版本。如果可以通过在 P 的元素上仅以 d 个线性顺序查看 x 和 y 的相对位置（不一定是线性扩展）来确定 P 中的 x ≤ y，则偏序 P 最多具有 d 布尔维数。我们证明了具有有界树宽覆盖图的偏序集具有有界布尔维数。这与存在具有树宽 3 和任意大维度的覆盖图的偏序集形成对比。