discriminator function

Let $A$ be a non-empty set. The ternary discriminator on $A$ is the ternary operation $t$ on $A$ such that

In other words, $t$ is a function that determines whether or not a pair of elements in $A$ are the same, hence the name discriminator.

It is easy to see that, by setting two of the three variables the same, $t$ becomes a constant function: $t(a,b,a)=a$, $t(a,a,b)=b$, and $t(a,b,b)=a$.

More generally, the quaternary discriminator or the switching function on $A$ is the quaternary operation $q$ on $A$ such that

However, this generalization is really an equivalent concept in the sense that one can derive one type of discriminator from another: given $q$ above, set $t(a,b,c)=q(a,b,c,a)$. Conversely, given $t$ above, set $q(a,b,c,d)=t(t(a,b,c),t(a,b,d),d)$.

Remark. The following ternary functions $t_1,t_2:A^3\to A$ could also serve as discriminator functions:

But they are really no different from the ternary discriminator $t$:

$$t_1(a,b,c)=t(b,a,c)\mathit{\hspace{1em}}\text{ and }\mathit{\hspace{1em}}t(a,b,c)=t_1(b,a,c),$$

$$t_2(a,b,c)=t(c,t(a,b,c),a)\mathit{\hspace{1em}}\text{ and }\mathit{\hspace{1em}}t(a,b,c)=t_2(a,t_2(a,b,c),c).$$