A principal filter is the mathematical label used to describe a filter over a partially ordered set (c.f. also lattice) that gets generated by a single element. To create such a filter we start with a single element and then build a set that contains all the elements of the lattice that are greater than this element. It can easily be shown that such a set satisfies all the properties required of a filter.
As ideals are dual to filters so principal ideals are dual to principal filters.
Related mathematical concepts include filters, ideals, rings, topology, universal algebra, model theory and lattices. | 
