Origin of min/max notation Here I am referring to the notation $x \wedge y = \min \{ x,y \}$ and $x \vee y = \max \{ x,y \}$. These seem to reference the corresponding usages in logic, where $\wedge$ means "and" and $\vee$ means "or". That is, they reference the usages in logic in the sense that both systems form Boolean lattices.
But this interpretation of the notation works exactly when we are dealing with $>$ and $\geq$ inequalities. That is, $x \wedge y > z$ can be read as "the minimum of $x$ and $y$ is greater than $z$" or as "$x$ and $y$ are greater than $z$". The analogous correspondence works with $\vee$. 
My question is: since analysts much more frequently deal with $<$ and $\leq$, why do we not use $\wedge$ for supremm and $\vee$ for infimum? Did this happen in order theory outside analysis and then transfer over? Historical references would be fantastic.
(Also, I blindly guessed how to tag this. Feel free to edit my tags.)

It is not essential to write it the way you have written it. For example, we could have written
$$z\leq x\wedge y$$
and this reads as "$z$ is less than or equal to $x$ and $y$". So if you interpret the way you have written it as backwards, then write it the other way and it is not backwards.
