Boolean Dimension and Tree-Width

Dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if P has dimension d, then to know whether x ≤ y in P it is enough to check whether x ≤ y in each of the d linear extensions of a witnessing realizer. Focusing on the encoding aspect, Nešetřil and Pudlák defined a more expressive version of dimension. A poset P has Boolean dimension at most d if it is possible to decide whether x ≤ y in P by looking at the relative position of x and y in only d linear orders on the elements of P (not necessarilly linear extensions). We prove that posets with cover graphs of bounded tree-width have bounded Boolean dimension. This stands in contrast with the fact that there are posets with cover graphs of tree-width three and arbitrarily large dimension. This result might be a step towards a resolution of the long-standing open problem: Do planar posets have bounded Boolean dimension?