Confluent partial orders

Question

Let $(P, \le)$ be a poset such that $$ \forall a, b, c \in P: b \ge a \le c \implies \exists d \in P: b \le d \ge c. $$ I am looking for literature where such confluent partial orders are studied.

Answer 1

In a context slightly more general than yours, this is called the right Ore condition. If you treat your poset as a category where there is a unique morphism from $p$ to $q$ if and only if $p \geq q$ and no other morphisms whatsoever, then your situation becomes a special case of the one described in the linked page.

Ore conditions come up all the time when you are localizing categories, because they make the localized morphisms much easier to understand --- one only needs two-step zigzags instead of arbitrarily long finite ones; all this is described in the first chapter of Gabriel-Zisman's textbook Calculus of fractions and homotopy theory.