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



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



"Interval" terminology for (partially) ordered sets
These sets are usually called convex sets. See for example [this paper](http://dml.cz/bitstream/handle/10338.dmlcz/118617/CommentatMathUnivCarolRetro_34-1993-3_23.pdf).


