# Intersecting elements of posets without least element

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

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

Today I’ve got the following idea:

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

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.