From mathworld.wolfram.com:
A relation is any subset of a Cartesian product
But if so, then the null set is all of: 0-ary, 1-ary, 2-ary etc. Wouldn’t it be better to define it as:
A relation is any non-empty subset of a Cartesian product
Second question, from this answer I understand that in set theory a 0-ary function is commonly taken to mean ${ emptyset }$, also called a constant. If a 0-ary relation is a non-empty subset of some set $D^{0}$ ($D$ is the domain), isn’t it right to say a 0-ary relation is also a constant? I haven’t seen model theory texts include 0-ary relations, which makes sense as they’re including 0-ary functions instead. To be technically the same as 0-ary functions (as they’re defined in the linked answer), wouldn’t we have to define relation as:
An n-ary relation is any non-empty subset of $D^{{0,1,dots,n-1}}$
What do you think?
EDIT: I should add that you can still define the empty set to be a relation of arity 0. From the comments below it seems it’s a matter of convenience more than technical correctness as to how relations are defined.