Let $(L,\vee,\wedge,0,1)$ be a lattice with unique least and greatest elements $0$ and $1$, respectively. I'll say that an *antichain* $A$ in $L$ is a subset of $L\setminus\{0\}$ such that for every $a,b\in A$ distinct, $a\wedge b=0$. (This is a set-theorist's antichain.)

I want to consider the following "finite-join" variation on an antichain: a subset $A$ of $L\setminus\{0\}$ has property (A) if for all $a_1,\ldots,a_n,b_1,\ldots,b_m\in A$ distinct, $(a_1\vee\cdots \vee a_n)\wedge(b_1\vee\cdots\vee b_n)=0$.

Note that in a distributive lattice, (A) is equivalent to being an antichain.

To see that they're not equivalent, consider the lattice of subspaces of a vector space of dimension 6 with basis vectors $e_1,e_2,e_3,e_4,e_5,e_6$. Let $a$ be the subspace generated by $(e_1,e_4)$, $b_1$ the subspace generated by $(e_2+e_3,e_5+e_6)$, and $b_2$ the subspace generated by $(e_1+e_2+e_3,e_4+e_5+e_6)$. Then, $a\wedge b_1=a\wedge b_2=\{0\}$, but $a\wedge(b_1\vee b_2)=a$.

My questions: 1. Does property (A) have a name in the literature? Is it a studied notion?

When does (A) coincide with being an antichain? Only in distributive lattices?

Does maximal with respect to (A) imply being a maximal antichain?