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The order dimension of divisibility
Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2020-12-24 , DOI: 10.1016/j.jcta.2020.105391
David Lewis , Victor Souza

The Dushnik-Miller dimension of a partially-ordered set P is the smallest d such that one can embed P into a product of d linear orders. We prove that the dimension of the divisibility order on the interval {1,,n}, is equal to (logn)2(loglogn)Θ(1) as n goes to infinity.

We prove similar bounds for the 2-dimension of divisibility in {1,,n}, where the 2-dimension of a poset P is the smallest d such that P is isomorphic to a suborder of the subset lattice of [d]. We also prove an upper bound for the 2-dimension of posets of bounded degree and show that the 2-dimension of the divisibility poset on the set (αn,n] is Θα(logn) for α(0,1). At the end we pose several problems.



中文翻译:

整除的顺序维

部分有序集合P的Dushnik-Miller维是最小的d,因此可以将P嵌入到d个线性阶积中。我们证明了区间上的除数阶的维{1个ñ},等于 日志ñ2日志日志ñ-Θ1个n达到无穷大时。

我们证明了2维除数的相似界 {1个ñ},其中位姿P的2维维是最小的d,使得P同构为P的子集晶格的子阶[d]。我们还证明了有界度的二维态的一个上限,并证明了该集上可除性二维集的二维αññ]Θα日志ñ 对于 α01个。最后,我们提出了几个问题。

更新日期:2020-12-24
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