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?

2. When does (A) coincide with being an antichain? Only in distributive lattices?
3. Does maximal with respect to (A) imply being a maximal antichain?