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?



Finite-join antichains in lattices
I will add a few comments to Bjørn Kjos-Hanssen's answer.



> 
> 1. 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).



> 
> 2. 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.



> 
> 3. 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).


