I know that the set of functions $X\to Y$ can be denoted $Y^X$ (using cardianl exponentiation), and I've seen cardinal factorials used for the set of bijections from a set to itself. Personally, I've also used $\text{Bi}(X,Y)$, $\text{In}(X,Y)$, and $\text{Sur}(X,Y)$ to denote the set of bijections, injections, and surjections, respectively, and I've generally done the same with other types of functions (ex. $\text{Iso}(X,Y)$ for isomorphism, etc).
Well today, I was working on a problem involving "the set of all well-orders on a given set," "the set of binary operations on a well-orderd set," and "the set of homomorphisms between magmas over the same." While I was doing this, I couldn't help but think "it would be awfully handy if there were a standard way to abbreviate all these sets!"
So, how about it? Is there a standard for denoting specific sets of relations/functions on a set (ex. total-orders, equivalence relations, group operations)? Perhaps an index of commonly used notations?
Note: I am referring to functions/relations over a given set, rather than in general. I've seen notations for "the class/category of < structure >" (ex. Ord - category of preordered sets, $\mathfrak{G}$ - class of groups), but I'm not aware of these being used to refer to structures over a specific set (e.g. $\textbf{Ord}(X)$ for the set of preorders on $X$). I bring this up because, in most cases, "the set of relations/functions on $X$ such that $Y$" can be identified with the set of structures over $X$ that satisfy $Y$ (for example, the set of rings whose underlying set is $X$).