Boolean Dimension and Local Dimension

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form “is x < y?”. This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a very new one. We determine precisely which values of dimension/boolean dimension/local dimension imply that the two other parameters are bounded.