Set Theory – Clarify Definitions of Relation and 0-Ary Relation

Question

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.

Answer 1

Your approach is adopted in Bruno Poizat, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic (2000 - ed or 1985), page 32-33:

Note: In certain cases, it can be convenient to introduce $0$-ary relations. As a function from $X$ to $Y$ associates to every element of $X$ one and only one element of $Y$, there is always one function from $\emptyset$ to $E$, whether or not $E$ is empty. The graph of this function is $\emptyset$; in other words, there is one $0$-tuple of elements of $E$. Consequently, there are always two nullary relations with universe $E$, namely $\{ \emptyset \}$ and $\emptyset$; the first can be called the true, the second the false. In contrast, if $m > 0$, there are no $m$-tuples of elements of $\emptyset$, and therefore only one $m$-ary relation with universe $\emptyset$, which is $\emptyset$. (We have often said that $\emptyset$ is always a local isomorphism between two $m$-ary relations for $m > 0$.)

A nullary relation symbol is therefore interpreted in a structure either by the true or by the false. These nullary relations are not very interesting in and of themselves, but occur naturally as a calculation tool: If a formula in $n$ free variables represents an $n$-ary relation (the set of $n$-tuples that satisfy it), a statement represents a $0$-ary relation, either the true or the false.