Intersecting elements of posets without least element

From the preprint of my article “Filters on Posets and Generalizations” (with little rewording):

Definition 1. Let is a poset with least element . I will call elements , in intersecting when exists c such that and and .

Today I’ve got the following idea:

We may drop the requirement that contains least element and change the requirement to simple “ is not the least element of “.

This allows to generalize some of notions in my article.

I am going to rewrite both this article and my draft Pointfree funcoids to allow posets without least element.