# “Interval” terminology for (partially) ordered sets

Let $(X,\preceq)$ be a poset.

Is there a standard, generally recognised term for a set $A \subset X$ satisfying $$\forall x,y,z \in X, \ (x \in A \ \textrm{ and } \ z \in A \ \textrm{ and } \ x \preceq y \preceq z) \ \Rightarrow \ y \in A \ ?$$

Of course, the natural term to use would be "interval". And yet, I've read that the term "interval" refers specifically to a set taking one of the following forms:

$\{x \in X : a \preceq x \preceq b\}$

$\{x \in X \setminus \{a\} : a \preceq x \preceq b\}$

$\{x \in X \setminus \{b\} : a \preceq x \preceq b\}$

$\{x \in X \setminus \{a,b\} : a \preceq x \preceq b\}$

where $a,b \in X$.

[So for example, under this terminology, we have the statement (from Wikipedia), "A poset is locally finite if every interval is finite." Indeed, if $(X,\preceq)=(\mathbb{N},\leq)$, then $\mathbb{N}$ is not an interval, even though it clearly satisfies the condition that I am interested in.]