  We continue the investigation of tabular algebras with trace (a certain class
of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished
bases) by determining the extent to which the tabular structure may be
recovered from a knowledge of the structure constants. This problem is
equivalent to understanding a certain category (the category of table data
associated to a tabular algebra) which we introduce. The main result is that
this category is equivalent to another category (the category of based posets
associated to a tabular algebra) whose structure we describe explicitly.
