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?