Set representations of orders and a structural equivalent of saturation

Set representations are useful in the theory of knowledge spaces. A set representation of an order is an isomorphic mapping of its base set into the power set of some set ordered by set inclusion. Such a representation is basic if the union of the representing sets of the predecessors of an element contains strictly less elements than the representing set of this element, and it is parsimonious if the difference is exactly one element. This paper investigates the properties of the minimal number of elements which must be used in a parsimonious representation. This value is studied for several order operations. Moreover, orders which allow essentially only one parsimonious set representation are structurally characterized. These orders are called saturated. Finally, the way to apply these results to knowledge spaces are outlined.