# Why is the symbol for the exterior product a meet rather than a join?

It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (quite reasonably) the "meet product", but write that as $A\vee B$.

I'm aware the answer might be "it just turned out that way", but was this choice of symbol motivated by a natural way to think of $A\wedge B$ as a meet in some lattice or other?

If not, which historical figure should I blame for this situation? I understand Grassmann wrote $A\wedge B$ as $[A, B]$, which would let him off the hook.

EDIT: Intuitively, if I have two vectors $a$ and $b$, $a\wedge b$ is the smallest object that contains both $a$ and $b$. If you order elements of the exterior algebra by inclusion, which seems very natural if you're doing geometry, you get a lattice where the join operation is the exterior product.

This mostly seems to come to the fore in Clifford-algebra-related approaches to geometry. Examples of people mentioning this relationship and complaining about / altering the standard notation:

While digging up these references I also turned up a claim that Peano wrote the exterior product using $\vee$, whereas Cartan used $\wedge$, but it's not clear to me whether Cartan originated (or popularised) this notation or whether he had a particular reason for choosing it [or indeed whether this claim is correct -- it was an offhand remark with no citation].

• Symbols have more than one meaning in mathematics. What makes you think $\vee$ must always have some relation to joins and $\wedge$ must always have some relation to meets? Notations don't have to be "logical", what's logical about the symbol "+"? And why do you think it's necessary to "blame" someone for this? – Najib Idrissi Feb 22 '16 at 9:30
• Also it's not clear to me in what way the exterior product is at all like a join. Can you clarify? – Najib Idrissi Feb 22 '16 at 9:34
• I'll edit something into the question about why it "looks like" a join (at least to me). I'm fine if the answer is that there's no rationale for this notation -- but if that's the case here I'd like to confirm that. I'm working up some educational material that includes both lattices and exterior products and would like to be able to tell my students something more than "I don't know why you're seeing the same symbol used with opposite meanings". – helveticat Feb 22 '16 at 9:55
• Well it's still not clear to me why you say "opposite meanings". The symbols $\vee$ and $\wedge$ are also used for wedge sum and smash product, and it never occurred to me to think that this had anything to do with lattices -- it's just notation, sometimes the same notation is used for different things in different areas of math... Anyway, are you aware of History of Science and Mathematics? – Najib Idrissi Feb 22 '16 at 9:59
• I've added a bit of detail on this. Thanks for the pointer to HSM, I wasn't aware of that; if I don't get a clear answer here I'll take the question there. Thanks for encouraging me to beef up the question BTW. – helveticat Feb 22 '16 at 10:11