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  • An $n$-ary relation is a set of $n$-tuples; inside which a tuple either exists or it doesn't.
  • A nullary relation is a $0$-ary relation, in other words it is a set that either contains the only $0$-tuple or it doesn't.
  • A homogeneous relation is a relation in which all the values in its tuples are elements of a specific set. Edit: In other words, it is a subset of the set of all $(a_0, a_1, ..., a_{n-1})$ in which $a_i \in A$.

But the tuple of a nullary relation has no values at all, so is the nullary relation homogeneous, heterogeneous, or neither?

I'm developing a formal system in a programming language (Java) that for each relation, you can ask if it is homogeneous or not, which returns a value of two-valued logic (Boolean type, true or false), where true means it is homogeneous and false means it is heterogeneous. What should this be said when asked that question about a nullary relation?

And for those saying that a homogeneous relation is always binary, no. Mathematics is all about generalization; it'd be a shame to introduce a property like homogeneity for relations and study it for only binary ones. It would be like studying closure for only binary operations.

SMMH
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Your definition of homogeneous relation seems very vague to me. I would define it as an element of the power set of $X^n.$ In this sense, $X^0$ is a set consisting of a single element: The unique function with domain empty-set. The power set contains both of your nullary relations, and so yes, to me, a nullary relation is homogeneous in the most obvious way of defining it. But if you're going to define a notion of homogeneous relation for more than two variables, you can pick whatever convention you want.

  • But $X^0$ is ${()}$ and so its power set is ${Ø, {()}}$. This erases any trace of our origianl set $X$, so how can we be sure a nullary relation originated from $X^0$ and not $Y^0$ where $X \neq Y$? – SMMH Jun 21 '21 at 05:34
  • @SMMH Why does it matter? The nullary relation is a homogeneous relation on any set. But like I said, it depends on your definition. If you want to define homogeneous relation differently, then feel free to exclude it. It is a matter of opinion, not mathematics. –  Jun 21 '21 at 05:47