1
$\begingroup$

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.]

$\endgroup$
2
$\begingroup$

These sets are usually called convex sets. See for example this paper.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.