A Generalized Cantor Theorem

Abstract

A well known theorem of Cantor asserts that the cardinal of the power-set of a given set always exceeds the cardinal of the original set. An analogous result for sets having additional structure is the well known theorem that the set of initial segments of a well ordered set always has order type greater than the original set. These two theorems suggest that there should be a similar result for general partially ordered sets. In formulating such a theorem an extension to partially ordered sets of the notion of an initial segment of a well ordered set is required. Of the several possibilities for this choice, the most natural one is the concept of an order ideal. If P is a partially ordered set with order relation ≦, then a subset I of P is an order ideal if a≦b∈P implies a∈P. The set ℐ(P) of all order ideals of P is easily seen to be a complete partially ordered set when ordered by set inclusion, since the union and intersection of any set of order ideals is again an order ideal. Note that the empty set is specifically included among the order ideals.