How do you call this feature and property of relation?

Question

If an operator can be defined for $k$ operands, $k \in \mathbb N$, how do you call this feature of the operator? For example, "+" is such an operator.

Similarly, for a relation $R$ on a set $X$, $R$ can be defined as k-ary relation for all $k \in \mathbb N$ in some consistent way. What do you call this feature of $R$? For example, mutual independence can be defined for any $k$ number of random variables.

Furthermore, if $R$ has a property that if $R$ holds for some $k_1$ elements of $X$, then $R$ holds for any $k_2$ of the $k_1$ elements, for any $k_2 \leq k_1$, how do you call this property of $R$?

Thanks!

Answer 1

In the case of an operator, you're looking for associativity.

• Any binary operator "$*$" that satisfies the associativity law $a*(b*c)=(a*b)*c$, can be treated as a $k$-ary operator for all $k\in \mathbb{N}$ and interpreted as applying $k-1$ times the corresponding binary relation, keeping the given order of the terms. Matrix multiplication is such an operation.
• If you also assume commutativity ($a*b=b*a$) then, the new operator can be interpreted as applying $k-1$ times the corresponding binary relation, but this time order of terms doesn't matter. Addition is such an operation.

As for a relation, maybe a partial order fits for you? For example, if "$\leq$" is a partial order, then you can define a new relation "$\sqsubseteq$" as follows.

$$\leq_k=\{(a_1,\ldots,a_k):a_1\leq\cdots\leq a_k\}$$ and $$\sqsubseteq=\bigcup_{k\in\mathbb{N}}\leq_k=\bigcup_{k\in\mathbb{N}}\{(a_1,\ldots,a_k):a_1\leq\cdots\leq a_k\}$$

Answer 2

Neither of these is a property of a single object, but rather a relationship between different objects.

The first thing is a construction: if you have any associative binary operation $*\colon X^2\to X$ on a set $X$, it naturally extends to a $k$-ary operation $*^{(k)}\colon X^k\to X$ for any natural $k\geq 1$. But this extension is formally a different object, it's a sequence of maps from $X^k\to X$, or, putting it differently, a map $\bigcup_k X^k\to X$. In any case, this is not $*$, though it's certainly related to it in the obvious way.

The second thing is even more complicated: the relation of independence of two random variables does not tell us much about independence of $n$ random variables for $n>2$, and again, the latter relation lives on a different object, so they don't even induce each other, and they're certainly not the same thing, even if we call it the same.

Similarly to the operation context, if you have an arbitrary binary relation $R$ on a set $X$ you can define an $n$_ary relation on this set by saying that a sequence is in this relation if each pair is in $R$, or if each pair of successive elements of the sequence is in $R$. But none of these is a property of $R$, but rather a relationship between this and some other relation.