Many ternary properties are interesting, and many of them are learned in school before properties of binary relations. I define any ternary relation as a relation that is expressed as a three-place predicate. For example, consider the three place predicate, '_____added to_____yields_____'. This particular 3-adic relation has a number of interesting properties, including:
the commutative property: $\forall x \forall y \forall z : R(x,y,z) \implies R(y,x,z)$
the associative property: $\forall w \forall x \forall y \forall z : R(s(w,x),y,z) \implies R(w,s(x,y),z)$, where the function $s$ takes the sum of its inputs.
Note that these two properties are also properties of the relation '_____multiplied by_____yields_____'.
These two ternary relations differ, however, on this property (the additive identity property):
$\forall x : R(x,0,x)$
For all quantities, $x$ added to $0$ yields $x$, but it is not the case that for all quantities $x$, $x$ multiplied by $0$ yields $x$.
Notice that the 3-adic relation '_____and_____sit to either side of_____on the sofa' also shares the commutative property.