Scott Density __hot__ ❲Extended❳

A poset $P$ is said to be Scott dense if it has a dense subset $D$ such that for every directed subset $S \subseteq D$, the supremum of $S$ in $P$ is also in $D$. Here, a directed subset is a subset $S$ such that for every two elements $x, y \in S$, there exists an element $z \in S$ with $x \leq z$ and $y \leq z$. The requirement that the supremum of a directed subset $S \subseteq D$ lies in $D$ ensures that $D$ is not only dense but also "closed" under certain operations.

Scott density has far-reaching implications in various areas of mathematics and computer science: scott density

Lower Scott density usually indicates higher powder "fluffiness," more irregular shape, or higher surface friction. A poset $P$ is said to be Scott