Notation and the name choice for meet and join (in order theory)
I have two simple questions:
From where do the names meet and join come from? I don't see any intuition between those names in context of order theory.
From where does the notation come? I have to admit, I always had an impression that notation we use is somewhat backwards... just take the look at the picture at the wikipedia!
I used this notation a lot of times and I have to admit that often I have to stop for a minute and convince myself that I'm indeed using it right. I'm hoping that learning a bit about origin and context of name choice and notation will make it easier.
I believe the terms meet and join come from lattice theory. A lattice, after all, can be defined as a partially ordered set in which any two elements have a meet and a join. In practice, a lattice typically arises as a collection of "closed" sets (with respect to some kind of algebraic closure) ordered by set inclusion; typical examples would be the lattice of all subgroups of a group, or the lattice of all subspaces of a vector space.
Consider the lattice of subspaces of a vector space. The meet of two subspaces is their set-theoretic intersection; e.g., for two $2$-dimensional subspaces of $\mathbb R^3$, their meet is the line where the two planes meet. The join of two subspaces is what we get when the two subspaces join together to make a bigger subspace; in general it's not just the set-theoretic union, but the linear span of the union.
You also wanted to know where the symbols $\vee$ and $\wedge$ come from. I don't know but I'd guess they are derived from the symbols $\cup$ and $\cap$ for union and intersection, the lattice operations in the lattice of all subsets of a set. As for the symbols $\cup$ and $\cap$, my wild guess is that they are stylized versions of the letters u (for union) and n (for intersection). And if that's not the true history, it's good enough for a mnemonic, isn't it?
For me, there are more pairs of similar operations: intersection and union for sets, and and or in logic. meet and join are abstractions of these pairs and so have abstract names. In context of algebraic lattices these are the primitive operations, in equivalent context of order theory these are names for inf and sup of two elements. Also note the similarity in symbols for operations $∧, ∨, ∩, ∪$ and in connection symbols for order $<, >, ⊂, ⊃$.