Directed Partial Ordering

Last updated Nov 1, 2022

Let $A$ be a Partial Ordering. We call $A$ a Directed Partial Ordering if $\forall \alpha, \beta \in A$ $\exists \gamma \in A$ s.t. $\alpha \leq \gamma$ and $\beta \leq \gamma$.

1. We are effectively trying to capture the idea that we can always find a “later” element in the Partial Ordering after any two elements.
2. All Total Orderings are Directed Partial Ordering. Indeed for Total Ordering $(T, \leq)$, for $\alpha, \beta \in T$, we must either have $\alpha \leq \beta$ or $\beta \leq \alpha$. Then $\max\limits(\alpha, \beta) \in A$ and $\alpha \leq \max\limits(\alpha, \beta)$ and $\beta \leq \max\limits(\alpha, \beta)$.