Dimension of Restricted Classes of Interval Orders

TL;DR

This paper extends Rabinovitch's 1978 result by showing that certain restricted classes of interval orders, including those with mixed open and closed unit intervals and those with closed intervals of length 0 or 1, have order dimension at most 3.

Contribution

It generalizes the known dimension bound to new classes of interval orders with specific interval representations.

Findings

Interval orders with mixed open and closed unit intervals have dimension at most 3.

Interval orders with closed intervals of length 0 or 1 have dimension at most 3.

The dimension bound applies to broader classes of interval orders than previously known.

Abstract

Rabinovitch showed in 1978 that the interval orders having a representation consisting of only closed unit intervals have order dimension at most 3. This article shows that the same dimension bound applies to two other classes of posets: those having a representation consisting of unit intervals (but with a mixture of open and closed intervals allowed) and those having a representation consisting of closed intervals with lengths in $\{0,1\}$.