Finite-join antichains in lattices 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?
 A: 
  
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*Does property (A) have a name in the literature? Is it a studied notion?
  

Don't know, but we could call it the property of being a strong antichain (since by taking $m=n=1$, it implies being an antichain).


  
*When does (A) coincide with being an antichain? ... 
  

I don't have a general answer...

... Only in distributive lattices?

No. It coincides in any lattice all of whose antichains have size $\le 2$, and these can be non-distributive. Example: the non-modular (hence non-distributive) 5-element lattice $N=\{0,a,b,c,1\}$ where $0<a<1$, $0<b<c<1$, and $a$ is incomparable with $b$ and $c$.


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

No, consider the 5-element modular non-disitributive lattice $M=\{0,a_1,a_2,a_3,1\}$ with $0<a_i<1$ for each $i$, and the $a_i$ incomparable.
Then $\{a_1,a_2\}$ is a maximal strong antichain, but not a maximal antichain since $\{a_1,a_2,a_3\}$ is an antichain and $a_1\wedge (a_2\vee a_3)=a_1\wedge 1 = a_1\ne 0$.
A: I will add a few comments to Bjørn Kjos-Hanssen's answer.

  
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*Does property (A) have a name in the literature? Is it a studied notion?
  

The property is a kind of independence property. A set-theorist's antichain $S$ of a lattice $L$ is called weakly independent if
whenever $a_1,\ldots,a_{n+1}\in S$ are distinct, then $(a_1\vee\cdots\vee a_n)\wedge a_{n+1} = 0$. It is called strongly independent if whenever $I$ and $J$ are finite sets of indices, then $(\bigvee_{i\in I} a_i)\wedge 
(\bigvee_{j\in I} a_j)=(\bigvee_{k\in I\cap J} a_k).$ Property (A) lies between these two concepts. In a modular lattice all three notions agree (weak independence, Property (A), and strong independence).


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

It coincides in any pseudocomplemented lattice. These need not be distributive. For example, the congruence lattice of a semilattice is pseudocomplemented and almost never distributive. The order-dual of the lattice of convex subsets of a finite chain is pseudocomplemented, but not distributive when the chain has more than 2 elements.


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

I would guess that being maximal with respect to (A) rarely means being a maximal antichain for lattices that are not pseudocomplemented. For a simple example, if $S$ is an antichain in the lattice $L$ of subspaces of a vector space $V$, and $S$ satisfies Property (A), then the size of $S$ cannot exceed the dimension of $V$. But $L$ can have antichains  larger than the dimension of $V$. For example, if $V$ has finite dimension greater than one and $T\subseteq L$ is the set of 1-dimensional subspaces, then $T$ is larger than any antichain in $L$ that has Property (A).
