Strong compact elements in lattice

Question

I wonder whether there is a name for the following stronger notion of compactness in lattices, namely that if for a respective element $a$ we have that $a \le \bigvee\limits_{i\in I} b_i$ then there exists $j\in I$ such that $a \leq b_j$. Clearly, if the order is total then this is precisely the concept of compact element. But what about in the general case? Thanks.

Answer 1

That will be a completely join-prime element.
A join-prime element $x$ is one that satisfies $$x \leq a \vee b \quad\Rightarrow x \leq a \;\text{ or }\; x \leq b.$$ This can be generalized to finitely many elements: $$x \leq \bigvee_{i=1}^n a_i \quad\Rightarrow\quad x \leq a_i, \;\text{ for some}\; i \in \{1,\ldots,n\}.$$ If for any set $I$, $$x \leq \bigvee_{i \in I} a_i \quad\Rightarrow\quad x \leq a_i, \;\text{ for some}\; i \in I,$$ then $x$ is said to be completely join-prime.