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.

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