The 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 Ne\v{s}et\v{r}il and Pudl\'{a}k 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 permutations of the elements of P. We prove that posets with
cover graphs of bounded tree-width have bounded boolean dimension. This stays
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?Comment: one more reference added; paper revised along the suggestion of three
reviewer