Intersecting elements of posets without least element

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

Definition 1. Let $latex \mathfrak{A}$ is a poset with least element $latex 0$. I will call elements $latex a$, $latex b$ in $latex \mathfrak{A}$ intersecting when exists c such that $latex c\ne 0$ and $latex c\subseteq a$ and $latex c\subseteq b$.

Today I’ve got the following idea:

We may drop the requirement that $latex \mathfrak{A}$ contains least element and change the requirement $latex c\ne 0$ to simple “$latex c$ is not the least element of $latex \mathfrak{A}$”.

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.