TITLE: difference between join and union
QUESTION [5 upvotes]: Is there a difference between the join operator, $\wedge$, and the union of a set?
In particular, what is the join of $a \wedge b $ and $b \wedge c$? Is it $a\wedge b \wedge c$ or is it $0$?
I seem to have read both answers (in physics textbooks where they have skimmed over the details of how they define their operators).  
The answer $0$ comes from a geometric algebra book studying projective geometry, where they identify the geometric exterior product (= Grassmann's exterior product) with the join operator.  Since the exterior product is anti-commutative and associative it follows that for vectors $a$, $b$, $c$,
$a\wedge a=0 \implies (a\wedge b)\wedge(b\wedge c) = 0$.  
They went on to define the meet in terms of the exterior product
$(a\vee b)^* = a^*\wedge b^*$
where the star denotes the dual of the (in this case) vectors $a$ and $b$.
(see for example Universal Geometric Algebra by David Hestenes)
The set union answer comes from a discussion of lattices and probabilities (different book) where they identify join and meet with set union and intersection.  So for example, (since my terminology might be wrong), they drew a lattice such as follows,
    {a,b}
   /    \
 {a}    {b}
   \    /
     {} 

So in this case the join is $a \cup b$.
Does join/meet have a strict definition that is distinct from union/intersection - or can you define it however you like given the circumstances?  If its the latter, which is the more usual definition?

REPLY [2 votes]: After being prompted by Carl I tried to find the original references I found the answer.  My question, it turns out, was explicitly dealt with in
Projective Geometry with Clifford Algebra, by Hestenes and Ziegler, where it says:

We define the join $J$ of blades $A$ and $B$ (to within a scalar factor) as
  a common dividend of lowest step. The support of $J$ is then the usual
  ‘lattice join’ of the supports of $A$ and $B$. We define $J$ explicitly by 
$J =A\wedge B$ (3.4) 
when $A\wedge B$ does not vanish.... When $Step A + Step B \ge n$ and the join
  of $A$ and $B$ can be identified with the unit pseudoscalar $I$.

This entirely answers my question - for I was worried about the case when $A\wedge B = 0$, where the join is identified as the pseudoscalar (as a special case).