To make my question more precise and compact (and probably more intuitive), let me define the following:

A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x \rbrace)$ is defined and $x \wedge \bigvee(S - \{x\}) = \varnothing$.

If for every two mutually disjoint subsets $S_1$ and $S_2$ of $L$, $S_1 \wedge S_2 = \lbrace s_1\wedge s_2 \mid s_1 \in S_1, s_2 \in S_2\rbrace$ is also mutually disjoint, we say that $\wedge$ of $L$ preserves disjointness.

Now my question is: What do you call a lattice with this property? Is this property equivalent to a well-known property?

To make my question more precise and compact (and probably more intuitive), let me define the following:

A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x \rbrace)$ is defined and $x \wedge \bigvee(S - \lbrace x \rbrace) = \varnothing$.

If for every two mutually disjoint subsets $S_1$ and $S_2$ of $L$, $S_1 \wedge S_2 = \lbrace s_1\wedge s_2 \mid s_1 \in S_1, s_2 \in S_2\rbrace$ is also mutually disjoint, we say that $\wedge$ of $L$ preserves disjointness.

Now my question is: What do you call a lattice with this property? Is this property equivalent to a well-known property?

To make my question more precise and compact (and probably more intuitive), let me define the following:

A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $x \wedge \bigvee(S - \{x\}) = \varnothing$.

If for every two mutually disjoint subsets $S_1$ and $S_2$ of $L$, $S_1 \wedge S_2 = \lbrace s_1\wedge s_2 \mid s_1 \in S_1, s_2 \in S_2\rbrace$ is also mutually disjoint, we say that $\wedge$ of $L$ preserves disjointness.

Now my question is: What do you call a lattice with this property? Is this property equivalent to a well-known property?

To make my question more precise and compact (and probably more intuitive), let me define the following:

A subset $S$ of a lattice is mutually disjoint if for each $x \in S$, $\bigvee(S - \lbrace x \rbrace)$ is defined and $x \wedge \bigvee(S - \{x\}) = \varnothing$.

If for every two mutually disjoint subsets $S_1$ and $S_2$ of $L$, $S_1 \wedge S_2 = \lbrace s_1\wedge s_2 \mid s_1 \in S_1, s_2 \in S_2\rbrace$ is also mutually disjoint, we say that $\wedge$ of $L$ preserves disjointness.

Now my question is: What do you call a lattice with this property? Is this property equivalent to a well-known property?