How is this thing called in order theory?

Question

Let $(P, \le)$ be poset and $x \in P$.

Let $X$ be union of all chains containing $x$. How is $X$ called in order theory?

For example if I pick $x$ to be the element marked with red arrow, what is the name for the set of elements colored in blue?

Answer 1

It's the cone generated by $x$. It is all the elements which are comparable with $x$, since $y$ is comparable with $x$ if and only if $\{x,y\}$ is a chain.

Often we talk about the upper or lower cones, which would be only "upwards" or "downwards". Namely, $y$ is in the upper cone if and only if $\{x,y\}$ is a chain and $x$ is the minimum element of it.