Any shortcut to remember least upper bound $\vee$ and greatest lower bound $\wedge$ in Lattice concept

Question

I know this is silly but I am every time forgetting lub (least upper bound) in lattice as $\vee$ and glb (greatest lower bound) as $\wedge$. Is there any shortcut or mnemonic for remembering which one is join and which one is meet? Is there any historical reason for choosing such symbols?

Answer 1

If you think about logical operations as binary operations on $\{0,1\}$ under the identification $\text{True}=1$ and $\text{False}=0$, then $\land$ coincides with $\min$ and $\lor$ coincides with $\max$.

Answer 2

I too have struggled with this notation when I first learned about lattice theory. I am glad to see that I am not alone in this confusion. Something making the notation particularly confusing is there is a "v" shape in the Hasse diagram of a lattice formed by $x \wedge y$, $x$ and $y$ (if $x$ and $y$ are incomparable yadda yadda...) and a wedge shape in the Hasse diagram of a lattice formed by $x \vee y$, $x$ and $y$. But something I have not thought about until now is that if we draw Hasse diagrams upside-down, this is no longer an issue.

Like Angina mentioned, $\cup$ and $\cap$ mimic the shape of $\vee$ and $\wedge$. My hypothesis is that, since Boolean algebras were studied before general lattices, that $\cup$ and $\cap$ were the only notation. Then, somewhere along the way, the symbols were drawn a bit differently to distinguish between sets and elements of an abstract lattice.

In my opinion, the symbol $\wedge$ is used way to frequently in mathematics. E.g. the smash product in topology, exterior algebra of a vector space (in particular, differential forms), lattice meet, logical AND etc. (there are probably more, maybe a use in analysis?). Unfortunately, while you and I may not like the meet/join notation, we are stuck with it as the notation has stood the test of time.