5
$\begingroup$

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?

  • 0
    I thought the join and meet spitted out a single value. That is, they are defined on a partially ordered set, and one gives the supremum of the sets, while the other the infimum...So they are entirely different things from the union (which gives a set)!2011-09-29
  • 0
    @Swlabr - my terminology might be wrong, I have edited my question to show the lattice's they were drawing ( a lattice of sets - if that makes sence)2011-09-29
  • 0
    I think the exterior product is something completely different from the join in lattice & poset theory. So I don't understand this remark2011-09-29
  • 0
    @anon, they do go on to define the meet in terms of the duals (as edited) (i.e. it obeys the de Morgan rule)2011-09-29
  • 0
    @ThomasRot I get the impression that the set answer is the more typical, and maybe the geometric algebra book was showing how to bring lattice alegra into geometric algebra - something that was lost on me because I read the GA book before knowing anything about lattices.) Would that be your view?2011-09-29
  • 2
    The symbol $\wedge$ is the "meet". It stands for the greatest lower bound of two elements in a poset. The symbol $\vee$ is the "join". It stands for the least upper bound of two elements in a poset. I have downvoted because the text does not match the symbol in the question, or in the Hasse diagram. See http://en.wikipedia.org/wiki/Lattice_%28order%29#Lattices_as_posets2011-09-29
  • 0
    @CarlMummert, The book I'm using identifies the geometric outer product $\wedge$ with the join - it raises the grade of the space from vectors to areas, from areas to volumes etc. This notation may be unstandard - and may in part be behind my confusion - but it is definately what is used in the geometric algebra community.2011-09-29
  • 0
    Yes, that would make it more difficult. In your Hasse diagram, arranged as it is, the join of $\{a\}$ and $\{b\}$ is $\{a\} \cap \{b\} = \emptyset$. This has nothing to do with an outer product, it is simply the greatest lower bound in the poset.2011-09-29
  • 0
    @Tom: would you mind posting a reference to the book? I would like to see how they phase things2011-09-29
  • 0
    @CarlMummert Have a look at Universal Geometric Algebra by Hestenes, available here: http://geocalc.clas.asu.edu/html/GeoAlg.html, in particular section 2.2011-09-29
  • 0
    @CarlMummert Ba! The book I was using was `Geometric Algebra for physicists`. Just noticed that in the article I linked they were a little more careful of their definition of the join, stating that it matches the exterior product only when the exterior product is not equal to zero! I guess that solves my problem!2011-09-29

4 Answers 4