What do you call a lattice whose meet operation preserves disjointness of subsets?

Question

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?

Answer 1

Many lattices do have this property (for example completely distributive, or 5-element nondistributive). Here is a small counterexample.

Let $L$ be the 9-element lattice $$ L=\{s_1,t_1,s^*,t^*,s_1\wedge t_1,s_1\wedge t^*, s^*\wedge t_1,0,1\} $$ where all elements are incomparable except that $0$, $1$, $\wedge$ have the usual meaning. Then $S=\{s_1,s^*\}$ and $T=\{t_1,t^*\}$ are both mutually-disjoint, but $S\wedge T$ is not, since $$ 0< s_1\wedge t_1=(s_1\wedge t_1)\wedge \left[ (s_1\wedge t^*)\vee (s^*\wedge t_1)\right]. $$

Answer 2

Here's an answer to a related question which involves much more standard terminology.

Say $A \subset L$ is an antichain iff $\forall x, y \in A(x \neq y \rightarrow x \wedge y = 0)$. If $A_1$ and $A_2$ are two antichains, then $A_1 \wedge A_2$ is yet another antichain which we say "refines" both $A_1$ and $A_2$.

Proof: If $x_i, y_i \in A_i$ with $x_i \wedge y_i = 0$, then clearly $(x_1 \wedge x_2) \wedge (y_1 \wedge y_2) = 0. \\ \\ \square $

The notion of antichain makes sense even if the joins $\bigvee (S - \{ x \})$ aren't defined. If $L$ satisfies $\forall x \in L\ \forall S \subset L\ (x \wedge \bigvee S = \bigvee (\{ x \} \wedge S)$ then the notions of "antichain" and "mutually disjoint" coincide.