The dimension of a knowledge space

The dimension of a partial order P on a set A is a well-known and thoroughly studied concept. It is defined as the smallest number of linear orders on A whose intersection equals P. It can also be characterized as the smallest family F of mappings f from A to the reals such that a P b if and only if f(a) is less or equal to f(b) for all the mappings in F. The interest for this characterization stems from the fact that it qualifies the problem of determining the dimension of a partial order as a measurement theoretical question (Roberts, 1985). The states in a knowledge structure and, in particular, in a knowledge space are partially ordered by set inclusion. Therefore the dimension of a knowledge structure is a well-defined concept, and it corresponds to the dimension of the restriction of the set inclusion relation to the knowledge structure itself. We show that, under a rather general condition, named “join escape”, the dimension of a finite knowledge space equals the length d of a maximum antichain in its basis. If, for a given knowledge space, join escape does not hold, then d only provides an upper bound to the dimension of the knowledge space. It is further found that, under the stated condition, the dimension of a knowledge space coincides with the least number of maximal chains whose element-wise union reconstructs the entire knowledge space.