I am interested in complete lattices $L$ (with least element $0$ and greatest element 1) satisfying either of the following properties.
Property 1 (stronger): Given any totally ordered subset $I \subseteq L$ and any $a \in L$,
$$a \wedge \left(\bigvee_{i\in I} i \right )= \bigvee_{i\in I} (a \wedge i).$$
Property 2 (weaker): Given any totally ordered subset $I \subseteq L$ and any $a \in L$ such that $a \wedge i = 0$ for all $i \in I$,
$$a \wedge \left(\bigvee_{i\in I} i \right )= \bigvee_{i\in I} (a \wedge i) = 0.$$
Questions 1: Are there names for these properties? They seem related to continuity.
Question 2: Does the answer to question 1 change if $L$ has one or both of the following properties: (a) $L$ is modular, (b) $\bigvee J=1$ for some join-irredundant family of atoms $J$?
Note 2: This question is similar to the one asked here.
Question 1
Property 1: At least one author (see third paragraph of first page in On characterizations of atomistic lattices) asserts that this property is upper-continuity. The most common definition of upper-continuity I've seen appears, e.g. in Continuous lattices and domains as the condition that $x \wedge \bigvee D = \bigvee x \wedge D$ for every directed set $D$. I could image these definitions being equivalent if, for example, it were true that every directed set in a complete lattices contains a totally ordered subset (chain) with equal supremum. Not convinced that this is the case, however.
Property 2: If $L$ is a semi-ortholattice under the relation $x \perp y \iff x \wedge y = 0$, and if one can use chain and directed set interchangeably in this context, as with upper-continuity, then $L$ has this property iff it is ortho-continuous (see Maeda and Maeda, Theory of Symmetric Lattices, p9).
Question 2
Property 1: The notion of upper-continuity applies to all complete lattices.
Property 2: Every modular lattice is a semi-ortholattice under the relation $x \perp y \iff x \wedge y = 0$.