# discriminator function

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

 $t(a,b,c):=\left\{\begin{array}[]{ll}a&\textrm{if }a\neq b,\\ c&\textrm{otherwise.}\end{array}\right.$

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

 $q(a,b,c,d):=\left\{\begin{array}[]{ll}d&\textrm{if }a\neq b,\\ c&\textrm{otherwise.}\end{array}\right.$

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:

 $t_{1}(a,b,c):=\left\{\begin{array}[]{ll}b&\textrm{if }a\neq b,\\ c&\textrm{otherwise.}\end{array}\right.\hskip 28.452756ptt_{2}(a,b,c):=\left\{% \begin{array}[]{ll}c&\textrm{if }a\neq b,\\ a&\textrm{otherwise.}\end{array}\right.$

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

 $t_{1}(a,b,c)=t(b,a,c)\quad\mbox{ and }\quad t(a,b,c)=t_{1}(b,a,c),$
 $t_{2}(a,b,c)=t(c,t(a,b,c),a)\quad\mbox{ and }\quad t(a,b,c)=t_{2}(a,t_{2}(a,b,% c),c).$

## References

• 1 G. Grätzer: , 2nd Edition, Springer, New York (1978).
• 2 S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).
Title discriminator function DiscriminatorFunction 2013-03-22 18:20:58 2013-03-22 18:20:58 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 08A40 switching function ternary discriminator quaternary discriminator