In which posets is the set of compact elements downwards closed? In a poset $(D, \sqsubseteq)$, a compact element is an element $d \in D$ such that for every directed set $A$ which happens to have a supremum $\bigsqcup A \in D$ with $d \sqsubseteq \bigsqcup A$, it is $d \sqsubseteq a$ for some $a \in A$ (e.g., Definition I-4.1 of [Gierz et al 2003]).

My (elementary) question. Suppose $d$ is compact and $c \sqsubseteq d$. What are typical extra impositions on the poset, that ensure that $c$ must also be compact?
The only natural condition I can think of is Berry's "I condition" ([1], Sect. 12.3):

(I) each compact element dominates finitely many elements.

The above condition is the defining property of Berry's dI-domains, which are distributive (that's what the "d" stands for) algebraic domains satisfying condition I.  This is a widely known and well studied class of domains, which served as the basis for what is known as "stable" semantics (an attempt to capture the sequential behavior of the $\lambda$-calculus).  Girard's coherence spaces [2] give a particularly simple and useful example of dI-domains.  A coherence space may be seen as the set of cliques of a reflexive graph, ordered by inclusion.  The compact elements are the finite cliques, which obviously satisfy your requirement.

I don't know if generalizations of the I condition have been studied.  A counterexample to your property is given by the ordinal $\omega+2$ ($\omega+1$ is compact but $\omega$ is not), so certainly generalizing "finite" to "well-founded+no infinite antichains" does not work.

[1]  Roberto Amadio and Pierre-Louis Curien.  Domains and Lambda-Calculi.  Cambridge Tracts in Theoretical Computer Science, 1998.

[2] Jean-Yves Girard, Yves Lafont and Paul Taylor.  Proofs and Types.   Cambridge Tracts in Theoretical Computer Science, 1989.