Kelly--Trotter猜想与偏序集乘积的维数
The Kelly--Trotter conjecture and dimension of poset products
- School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究有限偏序集乘积的序维数，否证了Kelly--Trotter猜想，构造了具有$(3,d)$-覆盖性质的偏序集$Q_d$，并给出$(r,s)$-覆盖的等价刻画。
AI中文摘要：
我们研究有限偏序集笛卡尔积的序维数。Kelly和Trotter在1982年猜想，对于所有有限偏序集$P$和$Q$，有$\dim(P\times Q)\ge\dim P+\dim Q-2$。对于$m\ge3$，令$R_m$表示$m$个顶点上的完全图的关联偏序集。我们证明存在一个常数$C$，使得对于所有足够大的$m$，有$\dim(R_m\times R_m)\le\left(1+\frac{2}{\log_2 6}\right)\dim R_m+C$。由于$1+2/\log_2 6<2$，这否证了Kelly--Trotter猜想，并表明$2\dim R_m-2-\dim(R_m\times R_m)$可以随$\dim R_m$线性增长。对于每个整数$d\ge8$，我们构造一个关联偏序集$Q_d$，使得$\dim Q_d=d$且$Q_d$具有$(3,d)$-覆盖性质。因此，对于每个满足$\dim P\ge3$的偏序集$P$，有$\dim(P\times Q_d)\le\dim P+d-3$。从而，对于每个整数$d\ge8$，偏序集$Q_d$与所有维数至少为3的偏序集一起违反Kelly--Trotter猜想。最后，我们证明一个偏序集$Q$具有$(r,s)$-覆盖当且仅当对于每个至少有两个元素的有限链$C$，有$\dim(Q\times C^r)\le s$。
英文摘要：
We study the order dimension of Cartesian products of finite posets. Kelly and Trotter conjectured in 1982 that $\dim(P\times Q)\ge\dim P+\dim Q-2$ for all finite posets $P$ and $Q$. For $m\ge3$, let $R_m$ denote the incidence poset of the complete graph on $m$ vertices. We prove that there is a constant $C$ such that, for all sufficiently large $m$, $\dim(R_m\times R_m)\le\left(1+\frac{2}{\log_2 6}\right)\dim R_m+C$. Since $1+2/\log_2 6<2$, this disproves the Kelly--Trotter conjecture and shows that $2\dim R_m-2-\dim(R_m\times R_m)$ can grow linearly with $\dim R_m$. For every integer $d\ge8$, we construct an incidence poset $Q_d$ such that $\dim Q_d=d$ and $Q_d$ has the $(3,d)$-covering property. Consequently, $\dim(P\times Q_d)\le\dim P+d-3$ for every poset $P$ with $\dim P\ge3$. Thus, for every integer $d\ge8$, the poset $Q_d$ violates the Kelly--Trotter conjecture with every poset of dimension at least $3$. Finally, we prove that a poset $Q$ has an $(r,s)$-covering if and only if $\dim(Q\times C^r)\le s$ for every finite chain $C$ with at least two elements.
