Inequalities in dimension theory for posets

Abstract:

The dimension of a poset $(X,P)$, denoted $\dim (X,P)$, is the minimum number of linear extensions of $P$ whose intersection is $P$. It follows from Dilworth’s decomposition theorem that $\dim (X,P) \leq \operatorname {width} (X,P)$. Hiraguchi showed that $\dim (X,P) \leq |X|/2$. In this paper, $A$ denotes an antichain of $(X,P)$ and $E$ the set of maximal elements. We then prove that $\dim (X,P) \leq |X - A|;\dim (X,P) \leq 1 + \operatorname {width} (X - E);$ and $\dim (X,P) \leq 1 + 2\operatorname {width} (X - A)$. We also construct examples to show that these inequalities are sharp.